Maths for engineers Books

Book SynopsisDiscover the benefits of applying algorithms to solve scientific, engineering, and practical problems Providing a combination of theory, algorithms, and simulations, Handbook of Applied Algorithms presents an all-encompassing treatment of applying algorithms and discrete mathematics to practical problems in hot application areas, such as computational biology, computational chemistry, wireless networks, and computer vision. In eighteen self-contained chapters, this timely book explores: * Localized algorithms that can be used in topology control for wireless ad-hoc or sensor networks * Bioinformatics algorithms for analyzing data * Clustering algorithms and identification of association rules in data mining * Applications of combinatorial algorithms and graph theory in chemistry and molecular biology * Optimizing the frequency planning of a GSM network using evolutioTable of ContentsPreface. Abstracts. Contributors. 1. Generating All and Random Instances of A combinatorial Object (Ivan Stojmenovic) 2. Backtracking and Isomorph-Free Generation of Polyhexes (Lucia Moura and Ivan Stojmenovic) 3. Graph Theoretic Models in Chemistry and Molecular Biology (Debra Knisley and Jeff Knisley) 4. Algorithmic Methods for the Analysis of Gene Expression Data (Hongbo Xie, Uros Midic, Slobodan Vucetic, and Zoran Obradovic) 5. Algorithms of Reaction-Diffusion Computing (Andrew Adamatzky) 6. Data Mining Algorithms I: Clustering (Dan A. Simovici) 7. Data Mining Algorithms II: Frequent Item Sets (Dan A. Simovici) 8. Algorithms for Data Streams (Camil Demetrescu and Irene Finocchi) 9. Applying Evolutionary Algorithms to Solve the Automatic Frequency Planning Problem (Francisco Luna, Enrique Alba, Antonio J. Nero, Patrick Nauru, and Salvador Pedraza) 10. Algorithmic Game Theory and Application s(Marios Mavronicolas, Vicky Papdopoulou, and Paul Spirakis) 11. Algorithms for Real-Time Object Detection in Images (Milos Stojmenovic) 12. 2D Shape Measures for Computer Vision (Paul L. Rosin and Jovisa Zunic) 13. Cryptographic Algorithms (Binal Roy and Amiya Nayak) 14. Secure Communication in Distributed Sensor Networks (DSN) (Subhamoy Maitra and Bimal Roy) 15. Localized Topology Control Algorithms for Ad Hoc and Sensor Networks (Hannes Frey and David Simplot-Ryl) 16. A Novel Admission Control for Multimedia LEO Satellite Networks (Syed R. Rizvi, Stephan Olariu, and Mona E. Rizvi) 17. Resilient Recursive Routing in Communication Networks (Costas C. Constantinou, Alexander S. Stepanenko, Theodoros N. Arvanitis, Kevin J. Baughan, and Bin Liu) 18. Routing Algorithms on WDM Optical Networks (Qian-Ping Gu) Index.

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Book SynopsisThe objective of this book is to motivate an appreciation of contemporary statistical techniques within the context of engineering. The author presents an optimum blend between statistical thinking and statistical methodology through emphasis of a broad sweep of tools rather than endless streams of seemingly unrelated methods and formulae.Trade Review"Overall this is an excellent book, which defines a broader mandate than many of its competing texts. By providing, clear, understandable discussion of the basics of statistics through to more advanced methods commonly used by engineers, this book is an essential reference for practitioners, and an ideal text for a two semester course introducing engineers to the power and utility of statistics." (The American Statistician, August 2008) "In this book on modern engineering statistics, Ryan does an excellent job of providing the appropriate statistical concepts and tools using engineering resources.... Highly recommended. Lower- and upper-division undergraduates" (CHOICE, April 2008) "This self-contained volume motivates an appreciation of statistical techniques within the context of engineering; many datasets that are used in the chapters and exercises are from engineering sources. This book is ideal for either a one- or two-semester course in engineering statistics." (Computing Reviews, April 2008)Table of ContentsPreface xvii 1. Methods of Collecting and Presenting Data 1 1.1 Observational Data and Data from Designed Experiments 3 1.2 Populations and Samples 5 1.3 Variables 6 1.4 Methods of Displaying Small Data Sets 7 1.5 Methods of Displaying Large Data Sets 16 1.6 Outliers 22 1.7 Other Methods 22 1.8 Extremely Large Data Sets: Data Mining 23 1.9 Graphical Methods: Recommendations 23 1.10 Summary 24 References 24 Exercises 25 2. Measures of Location and Dispersion 45 2.1 Estimating Location Parameters 46 2.2 Estimating Dispersion Parameters 50 2.3 Estimating Parameters from Grouped Data 55 2.4 Estimates from a Boxplot 57 2.5 Computing Sample Statistics with MINITAB 58 2.6 Summary 58 Reference 58 Exercises 58 3. Probability and Common Probability Distributions 68 3.1 Probability: From the Ethereal to the Concrete 68 3.3 Common Discrete Distributions 76 3.4 Common Continuous Distributions 92 3.5 General Distribution Fitting 106 3.6 How to Select a Distribution 107 3.7 Summary 108 References 109 Exercises 109 4. Point Estimation 121 4.1 Point Estimators and Point Estimates 121 4.2 Desirable Properties of Point Estimators 121 4.3 Distributions of Sampling Statistics 125 4.4 Methods of Obtaining Estimators 128 4.5 Estimating σθ 132 4.6 Estimating Parameters Without Data 133 4.7 Summary 133 References 134 Exercises 134 5. Confidence Intervals and Hypothesis Tests—One Sample 140 5.1 Confidence Interval for μ: Normal Distribution σ Not Estimated from Sample Data 140 5.2 Confidence Interval for μ: Normal Distribution σ Estimated from Sample Data 146 5.3 Hypothesis Tests for μ: Using Z and t 147 5.4 Confidence Intervals and Hypothesis Tests for a Proportion 157 5.5 Confidence Intervals and Hypothesis Tests for σ2 and σ 161 5.6 Confidence Intervals and Hypothesis Tests for the Poisson Mean 164 5.7 Confidence Intervals and Hypothesis Tests When Standard Error Expressions are Not Available 166 5.8 Type I and Type II Errors 168 5.9 Practical Significance and Narrow Intervals: The Role of n 172 5.10 Other Types of Confidence Intervals 173 5.11 Abstract of Main Procedures 174 5.12 Summary 175 Appendix: Derivation 176 References 176 Exercises 177 6. Confidence Intervals and Hypothesis Tests—Two Samples 189 6.1 Confidence Intervals and Hypothesis Tests for Means: Independent Samples 189 6.2 Confidence Intervals and Hypothesis Tests for Means: Dependent Samples 197 6.3 Confidence Intervals and Hypothesis Tests for Two Proportions 200 6.4 Confidence Intervals and Hypothesis Tests for Two Variances 202 6.5 Abstract of Procedures 204 6.6 Summary 205 References 205 Exercises 205 7. Tolerance Intervals and Prediction Intervals 214 7.1 Tolerance Intervals: Normality Assumed 215 7.2 Tolerance Intervals and Six Sigma 219 7.3 Distribution-Free Tolerance Intervals 219 7.4 Prediction Intervals 221 7.5 Choice Between Intervals 227 7.6 Summary 227 References 228 Exercises 229 8. Simple Linear Regression Correlation and Calibration 232 8.1 Introduction 232 8.2 Simple Linear Regression 232 8.3 Correlation 254 8.4 Miscellaneous Uses of Regression 256 8.5 Summary 264 References 264 Exercises 265 9. Multiple Regression 276 9.1 How Do We Start? 277 9.2 Interpreting Regression Coefficients 278 9.3 Example with Fixed Regressors 279 9.4 Example with Random Regressors 281 9.5 Example of Section 8.2.4 Extended 291 9.6 Selecting Regression Variables 293 9.7 Transformations 299 9.8 Indicator Variables 300 9.9 Regression Graphics 300 9.10 Logistic Regression and Nonlinear Regression Models 301 9.11 Regression with Matrix Algebra 302 9.12 Summary 302 References 303 Exercises 304 10. Mechanistic Models 314 10.1 Mechanistic Models 315 10.2 Empirical–Mechanistic Models 316 10.3 Additional Examples 324 10.4 Software 325 10.5 Summary 326 References 326 Exercises 327 11. Control Charts and Quality Improvement 330 11.1 Basic Control Chart Principles 330 11.2 Stages of Control Chart Usage 331 11.3 Assumptions and Methods of Determining Control Limits 334 11.4 Control Chart Properties 335 11.5 Types of Charts 336 11.6 Shewhart Charts for Controlling a Process Mean and Variability (Without Subgrouping) 336 11.7 Shewhart Charts for Controlling a Process Mean and Variability (With Subgrouping) 344 11.8 Important Use of Control Charts for Measurement Data 349 11.9 Shewhart Control Charts for Nonconformities and Nonconforming Units 349 11.10 Alternatives to Shewhart Charts 356 11.11 Finding Assignable Causes 359 11.12 Multivariate Charts 362 11.13 Case Study 362 11.14 Engineering Process Control 364 11.15 Process Capability 365 11.16 Improving Quality with Designed Experiments 366 11.17 Six Sigma 367 11.18 Acceptance Sampling 368 11.19 Measurement Error 368 11.20 Summary 368 References 369 Exercises 370 12. Design and Analysis of Experiments 382 12.1 Processes Must be in Statistical Control 383 12.2 One-Factor Experiments 384 12.3 One Treatment Factor and at Least One Blocking Factor 392 12.4 More Than One Factor 395 12.5 Factorial Designs 396 12.6 Crossed and Nested Designs 405 12.7 Fixed and Random Factors 406 12.8 ANOM for Factorial Designs 407 12.9 Fractional Factorials 409 12.10 Split-Plot Designs 413 12.11 Response Surface Designs 414 12.12 Raw Form Analysis Versus Coded Form Analysis 415 12.13 Supersaturated Designs 416 12.14 Hard-to-Change Factors 416 12.15 One-Factor-at-a-Time Designs 417 12.16 Multiple Responses 418 12.17 Taguchi Methods of Design 419 12.18 Multi-Vari Chart 420 12.19 Design of Experiments for Binary Data 420 12.20 Evolutionary Operation (EVOP) 421 12.21 Measurement Error 422 12.22 Analysis of Covariance 422 12.23 Summary of MINITAB and Design-Expert® Capabilities for Design of Experiments 422 12.24 Training for Experimental Design Use 423 12.25 Summary 423 Appendix A Computing Formulas 424 Appendix B Relationship Between Effect Estimates and Regression Coefficients 426 References 426 Exercises 428 13. Measurement System Appraisal 441 13.1 Terminology 442 13.2 Components of Measurement Variability 443 13.3 Graphical Methods 449 13.4 Bias and Calibration 449 13.5 Propagation of Error 454 13.6 Software 455 13.7 Summary 456 References 456 Exercises 457 14. Reliability Analysis and Life Testing 460 14.1 Basic Reliability Concepts 461 14.2 Nonrepairable and Repairable Populations 463 14.3 Accelerated Testing 463 14.4 Types of Reliability Data 466 14.5 Statistical Terms and Reliability Models 467 14.6 Reliability Engineering 473 14.7 Example 474 14.8 Improving Reliability with Designed Experiments 474 14.9 Confidence Intervals 477 14.10 Sample Size Determination 478 14.11 Reliability Growth and Demonstration Testing 479 14.12 Early Determination of Product Reliability 480 14.13 Software 480 14.14 Summary 481 References 481 Exercises 482 15. Analysis of Categorical Data 487 15.1 Contingency Tables 487 15.2 Design of Experiments: Categorical Response Variable 497 15.3 Goodness-of-Fit Tests 498 15.4 Summary 500 References 500 Exercises 501 16. Distribution-Free Procedures 507 16.1 Introduction 507 16.2 One-Sample Procedures 508 16.3 Two-Sample Procedures 512 16.4 Nonparametric Analysis of Variance 514 16.5 Exact Versus Approximate Tests 519 16.6 Nonparametric Regression 519 16.7 Nonparametric Prediction Intervals and Tolerance Intervals 521 16.8 Summary 521 References 521 Exercises 522 17. Tying It All Together 525 17.1 Review of Book 525 17.2 The Future 527 17.3 Engineering Applications of Statistical Methods 528 Reference 528 Exercises 528 Answers to Selected Excercises 533 Appendix: Statistical Tables 562 Table A Random Numbers 562 Table B Normal Distribution 564 Table C t-Distribution 566 Table D F-Distribution 567 Table E Factors for Calculating Two-Sided 99% Statistical Intervals for a Normal Population to Contain at Least 100p% of the Population 570 Table F Control Chart Constants 571 Author Index 573 Subject Index 579

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Book SynopsisThe increasing sophistication of buildings and bridges demands new analytical techniques. Reliability-based design is a well established technique in the structural and mechanical engineering communities that is now gaining momentum among geotechnical engineers.Trade Review"The publication presents an examination of the theories and methodologies available for risk assessment in geotechnical engineering, spanning the full range from established single-variable and “first order” methods to the most recent, advanced numerical developments. In response to the growing application of LRFD methodologies in geotechnical design, coupled with increased demand for risk assessments by clients ranging from regulatory agencies to insurance companies, the authors have introduced an innovative reliability-based risk assessment method, the Random Finite Element Method (RFEM). The authors have spent more than fifteen years developing this statistically based method for modeling the real spatial variability of soils and rocks." (MCEER, Information Service, January 5, 2009)Table of ContentsPreface. Acknowledgements. PART 1: THEORY. Chapter 1: Review of Probability Theory. 1.1 Introduction. 1.2 Basic Set Theory. 1.3 Probability. 1.4 Conditional Probability. 1.5 Random Variables and Probability Distributions. 1.6 Measures of Central Tendency, Variability, and Association. 1.7 Linear Combinations of Random Variables. 1.8 Functions of Random Variables. 1.9 Common Discrete Probability Distributions. 1.10 Common Continuous Probability Distributions. 1.11 Extreme-Value Distributions. Chapter2: Discrete random Processes. 2.1 Introduction. 2.2 Discrete-Time, Discrete-State Markov Chains. 2.3 Continuous-Time Markov Chains. 2.4 Queueing Models. Chapter 3: Random Fields. 3.1 Introduction. 3.2 Covariance Function. 3.3 Spectral Density Function. 3.4 Variance Function. 3.5 Correlation Length. 3.6 Some Common Models. 3.7 Random Fields in Higher Dimensions. Chapter 4: Best Estimates, Excursions, and Averages. 4.1 Best Linear Unbiased Estimation. 4.2 Threshold Excursions in One Dimension. 4.3 Threshold Excursions in Two Dimensions. 4.4 Averages. Chapter 5: Estimation. 5.1 Introduction. 5.2 Choosing a Distribution. 5.3 Estimation in Presence of Correlation. 5.4 Advanced Estimation Techniques. Chapter 6: Simulation. 6.1 Introduction. 6.2 Random-Number Generators. 6.3 Generating Nonuniform Random Variables. 6.4 Generating Random Fields. 6.5 Conditional Simulation of Random Fields. 6.6 Monte carlo Simulation. Chapter 7: Reliability-Based Design. 7.1 Acceptable Risk. 7.2 Assessing Risk. 7.3 Background to Design Methodologies. 7.4 Load and Resistance Factor Design. 7.5 Going Beyond Calibration. 7.6 Risk-Based Decision making. PART 2: PRACTICE. Chapter 8: Groundwater Modeling. 8.1 Introduction. 8.2 Finite-Element Model. 8.3 One-Dimensional Flow. 8.4 Simple Two-Dimensional Flow. 8.5 Two-Dimensional Flow Beneath Water-Retaining Structures. 8.6 Three-Dimensional Flow. 8.7 Three Dimensional Exit Gradient Analysis. Chapter 9: Flow Through Earth Dams. 9.1 Statistics of Flow Through Earth Dams. 9.2 Extreme Hydraulic Gradient Statistics. Chapter 10: Settlement of Shallow Foundations. 10.1 Introduction. 10.2 Two-Dimensional Probabilistic Foundation Settlement. 10.3 Three-Dimensional Probabilistic Foundation Settlement. 10.4 Strip Footing Risk Assessment. 10.5 Resistance Factors for Shallow-Foundation Settlement Design. Chapter 11: Bearing Capacity. 11.1 Strip Footings on c-ø Soils. 11.2 Load and Resistance Factor Design of Shallow Foundations. 11.3 Summary. Chapter 12: Deep Foundations. 12.1 Introduction. 12.2 Random Finite-Element Method. 12.3 Monte Carlo Estimation of Pile Capacity. 12.4 Summary. Chapter 13: Slope Stability. 13.1 Introduction. 13.2 Probabilistic Slope Stability Analysis. 13.3 Slope Stability Reliability Model. Chapter 14: Earth Pressure. 14.1 Introduction. 14.2 Passive Earth Pressures. 14.3 Active Earth Pressures: Retaining Wall Reliability. Chapter 15: Mine Pillar Capacity. 15.1 Introduction. 15.2 Literature. 15.3 Parametric Studies. 15.4 Probabilistic Interpretation. 15.5 Summary. Chapter 16: Liquefaction. 16.1 Introduction. 16.2 Model Size: Soil Liquefaction. 16.3 Monte Carlo Analysis and Results. 16.4 Summary PART 3: APPENDIXES. APPENDIX A: PROBABILITY TABLES. A.1 Normal Distribution. A.2 Inverse Student t-Distribution. A.3 Inverse Chi-Square Distribution APPENDIX B: NUMERICAL INTEGRATION. B.1 Gaussian Quadrature. APPENDIX C. COMPUTING VARIANCES AND CONVARIANCES OF LOCAL AVERAGES. C.1 One-Dimensional Case. C.2 Two-Dimensional Case C.3 Three-Dimensional Case. Index.

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Book Synopsis* This textbook has been in constant use since 1980, and this edition has been rewritten to be even cleaner and clearer and new features have been introduced. * The authors continue to provide real-world, technical applications that promote intuitive reader learning.Table of Contents1 Review of Numerical Computation 1 1–1 The Real Numbers 2 1–2 Addition and Subtraction 9 1–3 Multiplication 15 1–4 Division 19 1–5 Powers and Roots 23 1–6 Combined Operations 29 1–7 Scientific Notation and Engineering Notation 32 1–8 Units of Measurement 41 1–9 Percentage 51 Chapter 1 Review Problems 59 2 Introduction to Algebra 62 2–1 Algebraic Expressions 63 2–2 Adding and Subtracting Polynomials 67 2–3 Laws of Exponents 72 2–4 Multiplying a Monomial by a Monomial 80 2–5 Multiplying a Monomial and a Multinomial 83 2–6 Multiplying a Binomial by a Binomial 86 2–7 Multiplying a Multinomial by a Multinomial 88 2–8 Raising a Multinomial to a Power 90 2–9 Dividing a Monomial by a Monomial 92 2–10 Dividing a Polynomial by a Monomial 95 2–11 Dividing a Polynomial by a Polynomial 98 Chapter 2 Review Problems 101 3 Simple Equations and Word Problems 103 3–1 Solving a Simple Equation 104 3–2 Solving Word Problems 113 3–3 Uniform Motion Applications 118 3–4 Money Problems 121 3–5 Applications Involving Mixtures 123 3–6 Statics Applications 127 3–7 Applications to Work, Fluid Flow, and Energy Flow 129 Chapter 3 Review Problems 133 4 Functions 136 4–1 Functions and Relations 137 4–2 More on Functions 144 Chapter 4 Review Problems 154 5 Graphs 156 5–1 Rectangular Coordinates 157 5–2 Graphing an Equation 161 5–3 Graphing a Function by Calculator 164 5–4 The Straight Line 167 5–5 Solving an Equation Graphically 172 Chapter 5 Review Problems 173 6 Geometry 175 6–1 Straight Lines and Angles 176 6–2 Triangles 180 6–3 Quadrilaterals 187 6–4 The Circle 190 6–5 Polyhedra 196 6–6 Cylinder, Cone, and Sphere 201 Chapter 6 Review Problems 205 7 Right Triangles and Vectors 207 7–1 The Trigonometric Functions 208 7–2 Solution of Right Triangles 212 7–3 Applications of the Right Triangle 216 7–4 Angles in Standard Position 221 7–5 Introduction to Vectors 222 7–6 Applications of Vectors 226 Chapter 7 Review Problems 229 8 Oblique Triangles and Vectors 231 8–1 Trigonometric Functions of Any Angle 232 8–2 Finding the Angle When the Trigonometric Function Is Known 236 8–3 Law of Sines 240 8–4 Law of Cosines 246 8–5 Applications 251 8–6 Non-Perpendicular Vectors 255 Chapter 8 Review Problems 260 9 Systems of Linear Equations 263 9–1 Systems of Two Linear Equations 264 9–2 Applications 270 9–3 Other Systems of Equations 279 9–4 Systems of Three Equations 284 Chapter 9 Review Problems 290 10 Matrices and Determinants 292 10–1 Introduction to Matrices 293 10–2 Solving Systems of Equations by the Unit Matrix Method 297 10–3 Second-Order Determinants 302 10–4 Higher-Order Determinants 308 Chapter 10 Review Problems 316 11 Factoring and Fractions 319 11–1 Common Factors 320 11–2 Difference of Two Squares 323 11–3 Factoring Trinomials 326 11–4 Other Factorable Expressions 333 11–5 Simplifying Fractions 335 11–6 Multiplying and Dividing Fractions 340 11–7 Adding and Subtracting Fractions 344 11–8 Complex Fractions 349 11–9 Fractional Equations 352 11–10 Literal Equations and Formulas 355 Chapter 11 Review Problems 360 12 Quadratic Equations 363 12–1 Solving a Quadratic Equation Graphically and by Calculator 364 12–2 Solving a Quadratic by Formula 368 12–3 Applications 372 Chapter 12 Review Problems 377 13 Exponents and Radicals 379 13–1 Integral Exponents 380 13–2 Simplification of Radicals 385 13–3 Operations with Radicals 392 13–4 Radical Equations 398 Chapter 13 Review Problems 403 14 Radian Measure, Arc Length, and Rotation 405 14–1 Radian Measure 406 14–2 Arc Length 413 14–3 Uniform Circular Motion 416 Chapter 14 Review Problems 420 15 Trigonometric, Parametric, and Polar Graphs 422 15–1 Graphing the Sine Wave by Calculator 423 15–2 Manual Graphing of the Sine Wave 430 15–3 The Sine Wave as a Function of Time 435 15–4 Graphs of the Other Trigonometric Functions 441 15–5 Graphing a Parametric Equation 448 15–6 Graphing in Polar Coordinates 452 Chapter 15 Review Problems 459 16 Trigonometric Identities and Equations 461 16–1 Fundamental Identities 462 16–2 Sum or Difference of Two Angles 469 16–3 Functions of Double Angles and Half-Angles 474 16–4 Evaluating a Trigonometric Expression 481 16–5 Solving a Trigonometric Equation 484 Chapter 16 Review Problems 489 17 Ratio, Proportion, and Variation 491 17–1 Ratio and Proportion 492 17–2 Similar Figures 497 17–3 Direct Variation 501 17–4 The Power Function 505 17–5 Inverse Variation 509 17–6 Functions of More Than One Variable 513 Chapter 17 Review Problems 518 18 Exponential and Logarithmic Functions 521 18–1 The Exponential Function 522 18–2 Logarithms 532 18–3 Properties of Logarithms 539 18–4 Solving an Exponential Equation 547 18–5 Solving a Logarithmic Equation 554 Chapter 18 Review Problems 560 19 Complex Numbers 562 19–1 Complex Numbers in Rectangular Form 563 19–2 Complex Numbers in Polar Form 568 19–3 Complex Numbers on the Calculator 572 19–4 Vector Operations Using Complex Numbers 575 19–5 Alternating Current Applications 578 Chapter 19 Review Problems 584 20 Sequences, Series, and the Binomial Theorem 586 20–1 Sequences and Series 587 20–2 Arithmetic and Harmonic Progressions 593 20–3 Geometric Progressions 600 20–4 Infinite Geometric Progressions 604 20–5 The Binomial Theorem 607 Chapter 20 Review Problems 614 21 Introduction to Statistics and Probability 617 21–1 Definitions and Terminology 618 21–2 Frequency Distributions 622 21–3 Numerical Description of Data 628 21–4 Introduction to Probability 638 21–5 The Normal Curve 648 21–6 Standard Errors 654 21–7 Process Control 661 21–8 Regression 669 Chapter 21 Review Problems 674 22 Analytic Geometry 679 22–1 The Straight Line 680 22–2 Equation of a Straight Line 687 22–3 The Circle 694 22–4 The Parabola 702 22–5 The Ellipse 713 22–6 The Hyperbola 725 Chapter 22 Review Problems 733 Appendices A Summary of Facts and Formulas A-0 B Conversion Factors A-0 C Table of Integrals A-0 Indexes Applications Index I-0 Index to Writing Questions I-0 Index to Projects I-0 General Index I-0

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Book SynopsisThis is a practical book on how to apply statistical methods successfully. The Authors have deliberately kept formulae to a minimum to enable the reader to concentrate on how to use the methods and to understand what the methods are for. Each method is introduced and used in a real situation from industry or research. Each chapter features situations based on the authors' experience and looks at statistical methods for analysing data and, where appropriate, discusses the assumptions of these methods. Key features: Provides a practical hands-on manual for workplace applications. Introduces a broad range of statistical methods from confidence intervals to trend analysis. Combines realistic case studies and examples with a practical approach to statistical analysis. Features examples drawn from a wide range of industries including chemicals, petrochemicals, nuclear power, food and pharmaceuticals. Includes a supporting Trade Review"Overall, the book could be a clear introduction to a set of useful tools either in self study or used as an aid for instruction for those with no previous exposure." (The American Statistician, 1 February 2011) Table of ContentsPreface. 1 Samples and populations. Introduction. What a lottery! No can do. Nobody is listening to me. How clean is my river? Discussion. 2 What is the true mean? Introduction. Presenting data. Averages. Measures of variability. Relative standard deviation . Degrees of freedom. Confidence interval for the population mean. Sample sizes. How much moisture is in the raw material? Problems. 3 Exploratory data analysis. Introduction. Histograms: is the process capable of meeting specifications? Box plots: how long before the lights go out? The box plot in practice. Problems. 4 Significance testing. Introduction. The one-sample t -test. The significance testing procedure. Confidence intervals as an alternative to significance testing. Confidence interval for the population standard deviation. F-test for ratio of standard deviations. Problems. 5 The normal distribution. Introduction. Properties of the normal distribution. Example. Setting the process mean. Checking for normality. Uses of the normal distribution. Problems. 6 Tolerance intervals. Introduction. Example. Confidence intervals and tolerance intervals. 7 Outliers. Introduction. Grubbs’ test. Warning. 8 Significance tests for comparing two means. Introduction. Example: watching paint lose its gloss. The two-sample t -test for independent samples. An alternative approach: a confidence intervals for the difference between population means. Sample size to estimate the difference between two means. A production example. Confidence intervals for the difference between the two suppliers. Sample size to estimate the difference between two means. Conclusions. Problems. 9 Significance tests for comparing paired measurements. Introduction. Comparing two fabrics. The wrong way. The paired sample t -test. Presenting the results of significance tests. One-sided significance tests. Problems. 10 Regression and correlation. Introduction. Obtaining the best straight line. Confidence intervals for the regression statistics. Extrapolation of the regression line. Correlation coefficient. Is there a significant relationship between the variables? How good a fit is the line to the data? Assumptions. Problems. 11 The binomial distribution. Introduction. Example. An exact binomial test. A quality assurance example. What is the effect of the batch size? Problems. 12 The Poisson distribution. Introduction. Fitting a Poisson distribution. Are the defects random? The Poisson distribution. Poisson dispersion test. Confidence intervals for a Poisson count. A significance test for two Poisson counts. How many black specks are in the batch? How many pathogens are there in the batch? Problems. 13 The chi-squared test for contingency tables. Introduction. Two-sample test for percentages. Comparing several percentages. Where are the differences? Assumptions. Problems. 14 Non-parametric statistics. Introduction. Descriptive statistics. A test for two independent samples: Wilcoxon–Mann–Whitney test. A test for paired data: Wilcoxon matched-pairs sign test. What type of data can be used? Example: cracking shoes. Problems. 15 Analysis of variance: Components of variability. Introduction. Overall variability. Analysis of variance. A practical example. Terminology. Calculations. Significance test. Variation less than chance? When should the above methods not be used? Between- and within-batch variability. How many batches and how many prawns should be sampled? Problems. 16 Cusum analysis for detecting process changes. Introduction. Analysing past data. Intensity. Localised standard deviation. Significance test. Yield. Conclusions from the analysis. Problem. 17 Rounding of results. Introduction. Choosing the rounding scale. Reporting purposes: deciding the amount of rounding. Reporting purposes: rounding of means and standard deviations. Recording the original data and using means and standard deviations in statistical analysis. References. Solutions to Problems. Statistical Tables. Index.

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Book SynopsisMost books discuss the theory and computational procedures of finite elements (FE). In the past this was necessary, but today''s software packages make FE accessible to users who knows nothing to the theory or of how FE works. People are now using FE software packages as black boxes'', without knowing the dangers of poor modeling, the need to verify that results are reasonable, or that worthless results can be convincingly displayed. Therefore, it is important to understand the physics of the problem, how elements behave, the assumptions and restrictions of FE implementations, and the need to assess the correctness of computed results.Table of ContentsBars and Beams: Linear Static Analysis. Plane Problems. Isoparametric Elements and Solution Techniques. Modeling, Errors, and Accuracy in Linear Analysis. Solids and Solids of Revolution. Plates and Shells. Thermal Analysis. Vibration and Dynamics. Nonlinearity in Stress Analysis. References. Index.

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Book SynopsisCovers a range of limit theorems useful in mathematical statistics, along with methods of proof and techniques of application. This book emphasizes the manipulation of "probability" theorems to obtain "statistical" theorems.Trade Review"...even today it still provides a really good introduction into asymptotic statistics..."(Zentralblatt Math, Vol. 1001, No.01, 2003)Table of Contents1 Preliminary Tools and Foundations 1 1.1 Preliminary Notation and Definitions 1 1.2 Modes of Convergence of a Sequence of Random Variables 6 1.3 Relationships Among the Modes of Convergence 9 1.4 Convergence of Moments; Uniform Integrability 13 1.5 Further Discussion of Convergence in Distribution 16 1.6 Operations on Sequences to Produce Specified Convergence Properties 22 1.7 Convergence Properties of Transformed Sequences 24 1.8 Basic Probability Limit Theorems: The WLLN and SLLN 26 1.9 Basic Probability Limit Theorems: The CLT 28 1.10 Basic Probability Limit Theorems: The LIL 35 1.11 Stochastic Process Formulation of the CLT 37 1.12 Taylor’s Theorem; Differentials 43 1.13 Conditions for Determination of a Distribution by Its Moments 45 1.14 Conditions for Existence of Moments of a Distribution 46 1.15 Asymptotic Aspects of Statistical Inference Procedures 47 1.P Problems 52 2 The Basic Sample Statistics 55 2.1 The Sample Distribution Function 56 2.2 The Sample Moments 66 2.3 The Sample Quantiles 74 2.4 The Order Statistics 87 2.5 Asymptotic Representation Theory for Sample Quantiles Order Statistics and Sample Distribution Functions 91 2.6 Confidence Intervals for Quantiles 102 2.7 Asymptotic Multivariate Normality of Cell Frequency Vectors 107 2.8 Stochastic Processes Associated with a Sample 109 2.P Problems 113 3 Transformations of Given Statistics 117 3.1 Functions of Asymptotically Normal Statistics: Univariate Case 118 3.2 Examples and Applications 120 3.3 Functions of Asymptotically Normal Vectors 122 3.4 Further Examples and Applications 125 3.5 Quadratic Forms in Asymptotically Multivariate Normal Vectors 128 3.6 Functions of Order Statistics 134 3.P Problems 136 4 Asymptotic Theory in Parametric Inference 138 4.1 Asymptotic Optimality in Estimation 138 4.2 Estimation by the Method of Maximum Likelihood 143 4.3 Other Approaches toward Estimation 150 4.4 Hypothesis Testing by Likelihood Methods 151 4.5 Estimation via Product-Multinomial Data 160 4.6 Hypothesis Testing via Product-Multinomial Data 165 4.P Problems 169 5 U-Statistics 171 5.1 Basic Description of U-Statistics 172 5.2 The Variance and Other Moments of a U-Statistic 181 5.3 The Projection of a U-Statistic on the Basie Observations 187 5.4 Almost Sure Behavior of U-Statistics 190 5.5 Asymptotic Distribution Theory of U-Statistics 192 5.6 Probability Inequalities and Deviation Probabilities for U-Statistics 199 5.7 Complements 203 5.P Problems 207 6 Von Mises Differentiable Statistical Functions 210 6.1 Statistics Considered as Functions of the Sample Distribution Function 211 6.2 Reduction to a Differential Approximation 214 6.3 Methodology for Analysis of the Differential Approximation 221 6.4 Asymptotic Properties of Differentiable Statistical Functions 225 6.5 Examples 231 6.6 Complements 238 6.P Problems 241 7 M-Estimates 243 7.1 Basic Formulation and Examples 243 7.2 Asymptotic Properties of M-Estimates 248 7.3 Complements 257 7.P Problems 260 8 L-Estimates 8.1 Basic Formulation and Examples 262 8.2 Asymptotic Properties of L-Estimates 271 8.P Problems 290 9 R-Estimates 9.1 Basic Formulation and Examples 292 9.2 Asymptotic Normality of Simple Linear Rank Statistics 295 9.3 Complements 311 9.P Problems 312 10 Asymptotic Relative Efficiency 10.1 Approaches toward Comparison of Test Procedures 314 10.2 The Pitman Approach 316 10.3 The Chernoff Index 325 10.4 Bahadur’s “Stochastic Comparison,” 332 10.5 The Hodges-Lehmann Asymptotic Relative Efficiency 341 10.6 Hoeffding’s Investigation (Multinomial Distributions) 342 10.7 The Rubin‒Sethuraman “Bayes Risk” Efficiency 347 I0.P Problems 348 Appendix 351 References 553 Author Index 365 Subject Index 369

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Book SynopsisMaster the essential statistical skills used in social and behavioral sciences Essentials of Statistics for the Social and Behavioral Sciences distills the overwhelming amount of material covered in introductory statistics courses into a handy, practical resource for students and professionals.Table of ContentsSeries Preface. One. Descriptive Statistics. Two. Introduction to Null Hypothesis Testing. Three. The Two-Group t II Test. Four. Correlation and Regression. Five. One-Way ANOVA and Multiple Comparisons. Six. Power Analysis. Seven. Factorial ANOVA. Eight. Repeated-Measures ANOVA. Nine. Nonparametric Statistics. Appendix A: Statistical Tables. Appendix B: Answers to Putting it into Practice Exercises. References. Annotated Bibliography. Index. Acknowledgments. About the Authors.

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Book SynopsisA thorough guide to the fundamentals--and how to use them--of finite element analysis for elastic structures For elastic structures, the finite element method is an invaluable tool which is used most effectively only when one understands completely each of its facets.Table of ContentsFinite Element Method Prerequisites. The Finite Element Method. Element Stiffness Equations by Direct Methods. Global Stiffness Equations. Element Stiffness Equations by Displaced State Virtual Work Applications. General Approach to Element Stiffness Equations. Plane Stress and Plane Strain. Plane Stress Structural Triangular Finite Elements. Isoparametric Plane Stress Structural Quadrilateral Finite Elements. Flat Plate Flexural Finite Elements. Axisymmetric Structural Finite Elements. Structural Finite Elements in Perspective. Appendix. Answers to Selected Problems. Index.

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Book SynopsisWith the growing complexity of engineered systems, reliability has increased in importance throughout the twentieth century. Initially developed to meet practical needs, reliability theory has become an applied mathematical discipline that permits a priori evaluations of various reliability indices at the design stages.Table of ContentsFundamentals. Reliability Indexes. Unrepairable Systems. Load-Strength Reliability Models. Distributions with Monotone Intensity Functions. Repairable Systems. Repairable Duplicated Systems. Analysis of Performance Effectiveness. Two-Pole Networks. Optimal Redundancy. Optimal Technical Diagnosis. Additional Optimization Problems in Reliability Theory. Heuristic Methods in Reliability. Index.

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Book SynopsisFirst published in 1986, this unique reference to clinical experimentation remains just as relevant today. Focusing on the principles of design and analysis of studies on human subjects, this book utilizes and integrates both modern and classical designs.Table of ContentsReliability of Measurement. Simple Linear Regression Analysis. The Parallel Groups Design. Special Cases of the Parallel Groups Study. Blocking to Control for Prognostic Variables. Stratification to Control for Prognostic Variables. Analysis of Covariance and the Study of Change. Repeated Measurements Studies. Latin and Greco-Latin Squares. The Crossover Study. Balanced Incomplete Block Designs. Factorial Experiments. Split-Plot Designs and Confounding. Appendix. Indexes.

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Book SynopsisAuthors Cook, Malkus, Plesha and Witt have revised Concepts and Applications of Finite Element Analysis, a text suited for both introductory and more advanced courses in Finite Element Analysis. The fourth edition of this market leading text provides students with up-to-date coverage and clear explanations of finite element analysis concepts and modeling procedures.Table of ContentsNotation. Introduction. One-Dimensional Elements, Computational Procedures. Basic Elements. Formulation Techniques: Variational Methods. Formulation Techniques: Galerkin and Other Weighted Residual Methods. Isoparametric Elements. Isoparametric Triangles and Tetrahedra. Coordinate Transformation and Selected Analysis Options. Error, Error Estimation, and Convergence. Modeling Considerations and Software Use. Finite Elements in Structural Dynamics and Vibrations. Heat Transfer and Selected Fluid Problems. Constaints: Penalty Forms, Locking, and Constraint Counting. Solid of Revolution. Plate Bending. Shells. Nonlinearity: An Introduction. Stress Stiffness and Buckling. Appendix A: Matrices: Selected Definition and Manipulations. Appendix B: Simultaneous Algebraic Equations. Appendix C: Eigenvalues and Eigenvectors. References. Index.

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Book SynopsisFrom the reviews of Numerical Solution of Partial Differential Equations in Science and Engineering: The book by Lapidus and Pinder is a very comprehensive, even exhaustive, survey of the subject . . . [It] is unique in that it covers equally finite difference and finite element methods. Burrelle''s The authors have selected an elementary (but not simplistic) mode of presentation. Many different computational schemes are described in great detail . . . Numerous practical examples and applications are described from beginning to the end, often with calculated results given. Mathematics of Computing This volume . . . devotes its considerable number of pages to lucid developments of the methods [for solving partial differential equations] . . . the writing is very polished and I found it a pleasure to read! Mathematics of Computation Of related interest . . . NUMERICAL ANALYSTable of ContentsFundamental Concepts. Basic Concepts in the Finite Difference and Finite Element Methods. Finite Elements on Irregular Subspaces. Parabolic Partial Differential Equations. Elliptic Partial Differential Equations. Hyperbolic Partial Differential Equations. Index.

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Book SynopsisA useful balance of theory, applications, and real-world examples The Finite Element Method for Engineers, Fourth Edition presents a clear, easy-to-understand explanation of finite element fundamentals and enables readers to use the method in research and in solving practical, real-life problems.Table of ContentsPART I. 1. Meet the Finite Element Method. 2. The Direct Approach: A Physical Interpretation. 3. The Mathematical Approach: A Variational Interpretation. 4. The Mathematical Approach: A Generalized Interpretation. 5. Elements and Interpolation Functions. PART II. 6. Elasticity Problems. 7. General Field Problems. 8. Heat Transfer Problems. 9. Fluid Mechanics Problems. 10. Boundary Conditions, Mesh Generation, and Other Practical Considerations 11. Finite Elements in Design.

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Book SynopsisExpert coverage of the design and implementation of state estimation algorithms for tracking and navigation Estimation with Applications to Tracking and Navigation treats the estimation of various quantities from inherently inaccurate remote observations.Table of ContentsPreface. Acronyms. Mathematical Notations. Introduction. Basic Concepts in Estimation. Linear Estimation in Static Systems. Linear Dynamic Systems with Random Inputs. State Estimation in Discrete-Time Linear Dynamic Systems. Estimation for Kinematic Models. Computational Aspects of Estimation. Extensions of Discrete-Time Linear Estimation. Continuous-Time Linear State Estimation. State Estimation for Nonlinear Dynamic Systems. Adaptive Estimation and Maneuvering Targets. Introduction to Navigation Applications. Bibliography. Index.

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Book SynopsisThis book provides an excellent introduction to the subject designed for readers from a variety of mathematical backgrounds. It features introductory properties of metric spaces and Banach spaces, and an appendix contains a summary of the concepts of set theory.Trade Review"...deserves to be on the bookshelf of everyone who wants to know about fixed-point theory in metric and Banach spaces and experts who want to read an insightful survey of some basic ideas..." (Mathematical Reviews, 2002b) "Clear, friendly exposition." (American Mathematical Monthly, August/September 2002)Table of ContentsPreface ix I Metric Spaces 1 Introduction 3 1.1 The real numbers R 3 1.2 Continuous mappings in R 5 1.3 The triangle inequality in R 7 1.4 The triangle inequality in R" 8 1.5 Brouwer's Fixed Point Theorem 10 Exercises 11 2 Metric Spaces 13 2.1 The metric topology 15 2.2 Examples of metric spaces 19 2.3 Completeness 26 2.4 Separability and connectedness 33 2.5 Metric convexity and convexity structures 35 Exercises 38 3 Metric Contraction Principles 41 3.1 Banach's Contraction Principle 41 3.2 Further extensions of Banach's Principle 46 3.3 The Caristi-Ekeland Principle 55 3.4 Equivalents of the Caristi-Ekeland Principle 58 3.5 Set-valued contractions 61 3.6 Generalized contractions 64 Exercises 67 4 Hyperconvex Spaces 71 4.1 Introduction 71 4.2 Hyperconvexity 77 4.3 Properties of hyperconvex spaces 80 4.4 A fixed point theorem 84 4.5 Intersections of hyperconvex spaces 87 4.6 Approximate fixed points 89 4.7 Isbell's hyperconvex hull 91 Exercises 98 5 "Normal" Structures in Metric Spaces 101 5.1 A fixed point theorem 101 5.2 Structure of the fixed point set 103 5.3 Uniform normal structure 106 5.4 Uniform relative normal structure 110 5.5 Quasi-normal structure 112 5.6 Stability and normal structure 115 5.7 Ultrametric spaces 116 5.8 Fixed point set structure—separable case 120 Exercises 123 II Banach Spaces 6 Banach Spaces: Introduction 127 6.1 The definition 127 6.2 Convexity 131 6.3 £2 revisited 132 6.4 The modulus of convexity 136 6.5 Uniform convexity of the tp spaces 138 6.6 The dual space: Hahn-Banach Theorem 142 6.7 The weak and weak* topologies 144 6.8 The spaces c, CQ, t\ and ^ 146 6.9 Some more general facts 148 6.10 The Schur property and £j 150 6.11 More on Schauder bases in Banach spaces 154 6.12 Uniform convexity and reflexivity 163 6.13 Banach lattices 165 Exercises 168 7 Continuous Mappings in Banach Spaces 171 7.1 Introduction 171 7.2 Brouwer's Theorem 173 7.3 Further comments on Brouwer's Theorem 176 7.4 Schauder's Theorem 179 7.5 Stability of Schauder's Theorem 180 7.6 Banach algebras: Stone Weierstrass Theorem 182 7.7 Leray-Schauder degree 183 7.8 Condensing mappings 187 7.9 Continuous mappings in hyperconvex spaces 191 Exercises 195 8 Metric Fixed Point Theory 197 8.1 Contraction mappings 197 8.2 Basic theorems for nonexpansive mappings 199 8.3 A closer look at ßë 205 8.4 Stability results in arbitrary spaces 207 8.5 The Goebel-Karlovitz Lemma 211 8.6 Orthogonal convexity 213 8.7 Structure of the fixed point set 215 8.8 Asymptotically regular mappings 219 8.9 Set-valued mappings 222 8.10 Fixed point theory in Banach lattices 225 Exercises 238 9 Banach Space Ultrapowers 243 9.1 Finite representability 243 9.2 Convergence of ultranets 248 9.3 The Banach space ultrapower X 249 9.4 Some properties of X 252 9.5 Extending mappings to X 255 9.6 Some fixed point theorems 257 9.7 Asymptotically nonexpansive mappings 262 9.8 The demiclosedness principle 263 9.9 Uniformly non-creasy spaces 264 Exercises 270 Appendix: Set Theory 273 A.l Mappings 273 A.2 Order relations and Zermelo's Theorem 274 A.3 Zorn's Lemma and the Axiom Of Choice 275 A.4 Nets and subnets 277 A.5 Tychonoff's Theorem 278 A.6 Cardinal numbers 280 A. 7 Ordinal numbers and transfinite induction 281 A.8 Zermelo's Fixed Point Theorem 284 A.9 A remark about constructive mathematics 286 Exercises 287 Bibliography 289 Index 301

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Book SynopsisGives you what you need to know - basic essential concepts - about calculus. Suitable for those looking for a readable alternative to the usual unwieldy calculus text, this title provides in a condensed format the material covered in the standard two-year calculus course. It also covers vectors, lines, and planes in space; and line integrals.Trade Review"...expands coverage to vectors and calculus of several variables...plenty of worked out problems..." (American Mathematical Monthly, August/September 2003) "...material included is well formulated and approachable...recommended." (Choice, Vol. 41, No. 1, September 2003)Table of ContentsAUTHOR'S MESSAGE TO THE READER vii ANNOTATED TABLE OF CONTENTS ix ACKNOWLEDGMENTS xv CHAPTER 1 Lines 1 CHAPTER 2 Parabolas, Ellipses, Hyperbolas 7 CHAPTER 3 Differentiation 13 CHAPTER 4 Differentiation Formulas 19 CHAPTER 5 The Chain Rule 25 CHAPTER 6 Trigonometric Functions 31 CHAPTER 7 Exponential Functions and Logarithms 39 CHAPTER 8 Inverse Functions 45 CHAPTER 9 Derivatives and Graphs 51 CHAPTER 10 Following the Tangent Line 57 CHAPTER 11 The Indefinite Integral 63 CHAPTER 12 The Definite Integral 69 CHAPTER 13 Work, Volume, and Force 75 CHAPTER 14 Parametric Equations 81 CHAPTER 15 Change of Variable 87 CHAPTER 16 Integrating Rational Functions 91 CHAPTER 17 Integration By Parts 97 CHAPTER 18 Trigonometric Integrals 101 CHAPTER 19 Trigonometric Substitution 107 CHAPTER 20 Numerical Integration 115 CHAPTER 21 Limits At oo; Sequences 119 CHAPTER 22 Improper Integrals 127 CHAPTER 23 Series 133 CHAPTER 24 Power Series 141 CHAPTER 25 Taylor Polynomials 149 CHAPTER 26 Taylor Series 155 CHAPTER 27 Separable Differential Equations 161 CHAPTER 28 First-Order Linear Equations 167 CHAPTER 29 Homogeneous Second-Order Linear Equations 173 CHAPTER 30 Nonhomogeneous Second-Order Equations 179 CHAPTER 31 Vectors 185 CHAPTER 32 The Dot Product 195 CHAPTER 33 Lines and Planes in Space 201 CHAPTER 34 Surfaces 211 CHAPTER 35 Partial Derivatives 217 CHAPTER 36 Tangent Plane and Differential Approximation CHAPTER 37 Chain Rules 227 CHAPTER 38 Gradient and Directional Derivatives 233 CHAPTER 39 Maxima and Minima 239 CHAPTER 40 Double Integrals 245 CHAPTER 41 Line Integrals 255 CHAPTER 42 Green's Theorem 259 CHAPTER 43 Exact Differentials 267 ANSWERS 273 INDEX 299 ABOUT THE AUTHOR 303

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Book SynopsisToday''s need-to-know optimization techniques, at your fingertips The use of optimization methods is familiar territory to academicians and researchers. Yet, in today''s world of deregulated electricity markets, it''s just as important for electric power professionals to have a solid grasp of these increasingly relied upon techniques. Making those techniques readily accessible is the hallmark of Optimization Principles: Practical Applications to the Operation and Markets of the Electric Power Industry. With deregulation, market rules and economic principles dictate that commodities be priced at the marginal value of their production. As a result, it''s necessary to work with ever-more-sophisticated algorithms using optimization techniques-either for the optimal dispatch of the system itself, or for pricing commodities and the settlement of markets. Succeeding in this new environment takes a good understanding of methods that involve linear and nonTrade Review"...an important contribution to the field of power system analysis...should provide the reader with a pleasant learning experience." (IEEE Power & Energy Magazine, November/December 2005)Table of ContentsPreface. 1. Introduction. PART I: MATHEMATICAL BACKGROUND. 2. Fundamentals of Matrix Algebra. PART II: LINEAR OPTIMIZATION. 3. Solution of Equations, Inequalities, and Linear Programs. 4. Solved Linear Program Problems. PART III: NONLINEAR OPTIMIZATION. 5. Mathematical Background to Nonlinear Programs. 6. Unconstrained Nonlinear Optimization. 7. Constrained Nonlinear Optimization. 8. Solved Nonlinear Optimization Problems. Appendix A: Basic Principles of Electricity. Appendix B: Network Equations. Appendix C: Relation Between Pseudo-Inverse and Least-Square Error Fit. Bibliography. Index. About the Author.

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Book SynopsisBayesian statistics uses information from past experience to infer the results of future events. With recent advances in computing power and the development of computer intensive methods for statistical estimation, Bayesian approaches to model estimation have become more feasible and popular.Trade Review"I recommend…highly to statisticians, [and] health researchers...among others to consider keeping on their bookshelf." (Journal of Statistical Computation and Simulation, April 2005) "…a great book…fills a critical gap in existing literature. It is an excellent book for anyone interested in Bayesian modeling…" (Journal of the American Statistical Association, March 2005) "It is certainly a fine choice as a supporting reference in either a first or second Bayesian methods course…” (Technometrics, May 2004) "...has a contemporary feel, with recent developments in financial time series modelling and epidemiology included..." (Short Book Reviews, Vol 23(3), December 2003)Table of ContentsPreface. The Basis for, and Advantages of, Bayesian Model Estimation via Repeated Sampling. Hierarchical Mixture Models. Regression Models. Analysis of Multi-Level Data. Models for Time Series. Analysis of Panel Data. Models for Spatial Outcomes and Geographical Association. Structural Equation and Latent Variable Models. Survival and Event History Models. Modelling and Establishing Causal Relations: Epidemiological Methods and Models. Index.

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Book SynopsisIn modern society, we are ever more aware of the environmentalissues we face, whether these relate to global warming, depletionof rivers and oceans, despoliation of forests, pollution of land,poor air quality, environmental health issues, etc. At the mostfundamental level it is necessary to monitor what is happening inthe environment - collecting data to describe the changingscene. More importantly, it is crucial to formally describe theenvironment with sound and validated models, and to analyse andinterpret the data we obtain in order to take action. Environmental Statistics provides a broad overview of thestatistical methodology used in the study of the environment,written in an accessible style by a leading authority on thesubject. It serves as both a textbook for students of environmentalstatistics, as well as a comprehensive source of reference foranyone working in statistical investigation of environmentalissues. * Provides broad coverage of the methodology used in tTrade Review"Inspired by the Encyclopedia of Statistical Sciences, SecondEdition (ESS2e), this volume presents a concise, well-rounded focuson the statistical concepts and applications that are essential forunderstanding gathered data in the fields of engineering, qualitycontrol, and the physical sciences. The book successfully upholdsthe goals of ESS2e by combining both previously-published and newlydeveloped contributions written by over 100 leading academics,researchers, and practitioner in a comprehensive, approachableformat. The result is a succinct reference that unveils modern,cutting-edge approaches to acquiring and analyzing data acrossdiverse subject areas within these three disciplines, includingoperations research, chemistry, physics, the earth sciences,electrical engineering, and quality assurance." (Finwin, 7September 2011) "In this book, Vic Barnett, a distinguished environmentalstatistician, provides an overview of statistical methods that havebeen used on such problems in the environmental sciences."(Journal of the American Statistical Association, September2006) "...combines sound fundamentals and their applications."(European Journal of Soil Science, No.56, April 2005) "Many tables, graphs and figures illustrate the environmentalapplications of the statistical methods that are described."(Journal of the Royal Statistical Society, Series A,Vol.168, No.2, March 2005) "...well written...methods are illustrated with interestingexamples...a comprehensive reference source for anyone working onenvironmental issues..." (Short Book Reviews, Vol.24, No.3,December 2004) "Statisticians should enjoy the book. The author is an extremelyknowledgeable statistician, and he is writing about an applicationdomain that he clearly knows." (Technometrics, November2004) "An excellent book. Highly recommended." (Choice, July2004) "...this provides an excellent sketch of the current state ofdevelopment for new statistical methodologies...a valuableresource..." (Statistics in Medicine, 15th August 2005)Table of ContentsPreface. Chapter 1: Introduction. 1.1 Tomorrow is too Late! 1.2 Environmental Statistics. 1.3 Some Examples. 1.3.1 ‘Getting it all together’. 1.3.2 ‘In time and space’. 1.3.3 ‘Keep it simple’. 1.3.4 ‘How much can we take?’ 1.3.5 ‘Over the top’. 1.4 Fundamentals. 1.5 Bibliography. PART I: EXTREMAL STRESSES: EXTREMES, OUTLIERS, ROBUSTNESS. Chapter 2: Ordering and Extremes: Applications, models, inference. 2.1 Ordering the Sample. 2.1.1 Order statistics. 2.2 Order-based Inference. 2.3 Extremes and Extremal Processes. 2.3.1 Practical study and empirical models; generalized extreme-value distributions. 2.4 Peaks over Thresholds and the Generalized Pareto Distribution. Chapter 3: Outliers and Robustness. 3.1 What is an Outlier? 3.2 Outlier Aims and Objectives. 3.3 Outlier-Generating Models. 3.3.1 Discordancy and models for outlier generation. 3.3.2 Tests of discordancy for specific distributions. 3.4 Multiple Outliers: Masking and Swamping. 3.5 Accommodation: Outlier-Robust Methods. 3.6 A Possible New Approach to Outliers. 3.7 Multivariate Outliers. 3.8 Detecting Multivariate Outliers. 3.8.1 Principles. 3.8.2 Informal methods. 3.9 Tests of Discordancy. 3.10 Accommodation. 3.11 Outliers in linear models. 3.12 Robustness in General. PART II: COLLECTING ENVIRONMENTAL DATA: SAMPLING AND MONITORING. Chapter 4: Finite-Population Sampling. 4.1 A Probabilistic Sampling Scheme. 4.2 Simple Random Sampling. 4.2.1 Estimating the mean, &Xmacr;. 4.2.2 Estimating the variance, S2. 4.2.3 Choice of sample size, n. 4.2.4 Estimating the population total, XT. 4.2.5 Estimating a proportion, P. 4.3 Ratios and Ratio Estimators. 4.3.1 The estimation of a ratio. 4.3.2 Ratio estimator of a population total or mean. 4.4 Stratified (simple) Random Sampling. 4.4.1 Comparing the simple random sample mean and the stratified sample mean. 4.4.2 Choice of sample sizes. 4.4.3 Comparison of proportional allocation and optimum allocation. 4.4.4 Optimum allocation for estimating proportions. 4.5 Developments of Survey Sampling. Chapter 5: Inaccessible and Sensitive Data. 5.1 Encountered Data. 5.2 Length-Biased or Size-Biased Sampling and Weighted Distributions. 5.2.1 Weighted distribution methods. 5.3 Composite Sampling. 5.3.1 Attribute Sampling. 5.3.2 Continuous variables. 5.3.3 Estimating mean and variance. 5.4 Ranked-Set Sampling. 5.4.1 The ranked-set sample mean. 5.4.2 Optimal estimation. 5.4.3 Ranked-set sampling for normal and exponential distributions. 5.4.4 Imperfect ordering. Chapter 6: Sampling in the Wild. 6.1 Quadrat Sampling. 6.2 Recapture Sampling. 6.2.1 The Petersen and Chapman estimators. 6.2.2 Capture–recapture methods in open populations. 6.3 Transect Sampling. 6.3.1 The simplest case: strip transects. 6.3.2 Using a detectability function. 6.3.3 Estimating f (y). 6.3.4 Modifications of approach. 6.3.5 Point transects or variable circular plots. 6.4 Adaptive Sampling. 6.4.1 Simple models for adaptive sampling. Part III: EXAMINING ENVIRONMENTAL EFFECTS: STIMULUS–RESPONSE RELATIONSHIPS. Chapter 7: Relationship: regression-type models and methods. 7.1 Linear Models. 7.1.1 The linear model. 7.1.2 The extended linear model. 7.1.3 The normal linear model. 7.2 Transformations. 7.2.1 Looking at the data. 7.2.2 Simple transformations. 7.2.3 General transformations. 7.3 The Generalized Linear Model. Chapter 8: Special Relationship Models, Including Quantal Response and Repeated Measures. 8.1 Toxicology Concerns. 8.2 Quantal Response. 8.3 Bioassay. 8.4 Repeated Measures. Part IV: STANDARDS AND REGULATIONS. Chapter 9: Environmental Standards. 9.1 Introduction. 9.2 The Statistically Verifiable Ideal Standard. 9.2.1 Other sampling methods. 9.3 Guard Point Standards. 9.4 Standards Along the Cause–Effect Chain. Part V: A MANY-DIMENSIONAL ENVIRONMENT: SPATIAL AND TEMPORAL PROCESSES. Chapter 10: Time-Series Methods. 10.1 Space and Time Effects. 10.2 Time Series. 10.3 Basic Issues. 10.4 Descriptive Methods. 10.4.1 Estimating or eliminating trend. 10.4.2 Periodicities. 10.4.3 Stationary time series. 10.5 Time-Domain Models and Methods. 10.6 Frequency-Domain Models and Methods. 10.6.1 Properties of the spectral representation. 10.6.2 Outliers in time series. 10.7 Point Processes. 10.7.1 The Poisson process. 10.7.2 Other point processes. Chapter 11: Spatial Methods for Environmental Processes. 11.1 Spatial Point Process Models and Methods. 11.2 The General Spatial Process. 11.2.1 Predication, interpolation and kriging. 11.2.2 Estimation of the variogram. 11.2.3 Other forms of kriging. 11.3 More about Standards Over Space and Time. 11.4 Relationship. 11.5 More about Spatial Models. 11.5.1 Types of spatial model. 11.5.2 Harmonic analysis of spatial processes. 11.6 Spatial Sampling and Spatial Design. 11.6.1 Spatial sampling. 11.6.2 Spatial design. 11.7 Spatial-Temporal Models and Methods. References. Index.

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Book SynopsisContact mechanics is a specialist area in engineering mechanics. It deals with non standard mechanics which frequently appear in real technical applications. Examples include the simulation of car crashes, human joints, car tyres, rubber seals and metal forming processes.Table of ContentsPreface. Introduction. Introduction to Contact Mechanics. Continuum Solid Mechanics and Weak Forms. Contact Kinematics. Constitutive Equations for Contact Interfaces. Contact Boundary Value Problem and Weak Form. Discretization of the Continuum. Discretization, Small Deformation Contact. Discretization, Large Deformation Contact. Solution Algorithms. Thermo-mechanical Contact. Beam Contact. Adaptive Finite Element Methods for Contact Problems. Computation of Critical Points with Contact Constraints. Appendix A: Gauss Integration Rules. Appendix B: Convective Coordinates. Appendix C: Parameter Identification for Friction Materials. References. Index.

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Book SynopsisThe classic introduction to engineering optimization theory and practice--now expanded and updated Engineering optimization helps engineers zero in on the most effective, efficient solutions to problems. This text provides a practical, real-world understanding of engineering optimization. Rather than belaboring underlying proofs and mathematical derivations, it emphasizes optimization methodology, focusing on techniques and stratagems relevant to engineering applications in design, operations, and analysis. It surveys diverse optimization methods, ranging from those applicable to the minimization of a single-variable function to those most suitable for large-scale, nonlinear constrained problems. New material covered includes the duality theory, interior point methods for solving LP problems, the generalized Lagrange multiplier method and generalization of convex functions, and goal programming for solving multi-objective optimization problems. A practical, hands-on referTable of ContentsPreface. 1 Introduction to Optimization. 1.1 Requirements for the Application of Optimization Methods. 1.2 Applications of Optimization in Engineering. 1.3 Structure of Optimization Problems. 1.4 Scope of This Book. References. 2 Functions of a Single Variable. 2.1 Properties of Single-Variable Functions. 2.2 Optimality Criteria. 2.3 Region Elimination Methods. 2.4 Polynomial Approximation or Point Estimation Methods. 2.5 Methods Requiring Derivatives. 2.6 Comparison of Methods. 2.7 Summary. References. Problems. 3 Functions of Several Variables. 3.1 Optimality Criteria. 3.2 Direct-Search Methods. 3.3 Gradient-Based Methods. 3.4 Comparison of Methods and Numerical Results. 3.5 Summary. References. Problems. 4 Linear Programming. 4.1 Formulation of Linear Programming Models. 4.2 Graphical Solution of Linear Programs in Two Variables. 4.3 Linear Program in Standard Form. 4.5 Computer Solution of Linear Programs. 4.5.1 Computer Codes. 4.6 Sensitivity Analysis in Linear Programming. 4.7 Applications. 4.8 Additional Topics in Linear Programming. 4.9 Summary. References. Problems. 5 Constrained Optimality Criteria. 5.1 Equality-Constrained Problems. 5.2 Lagrange Multipliers. 5.3 Economic Interpretation of Lagrange Multipliers. 5.4 Kuhn-Tucker Conditions. 5.5 Kuhn-Tucker Theorems. 5.6 Saddlepoint Conditions. 5.7 Second-Order Optimality Conditions. 5.8 Generalized Lagrange Multiplier Method. 5.9 Generalization of Convex Functions. 5.10 Summary. References. Problems. 6 Transformation Methods. 6.1 Penalty Concept. 6.2 Algorithms, Codes, and Other Contributions. 6.3 Method of Multipliers. 6.4 Summary. References. Problems. 7 Constrained Direct Search. 7.1 Problem Preparation. 7.2 Adaptations of Unconstrained Search Methods. 7.3 Random-Search Methods. 7.4 Summary. References. Problems. 8 Linearization Methods for Constrained Problems. 8.1 Direct Use of Successive Linear Programs. 8.2 Separable Programming. 8.3 Summary. References. Problems. 9 Direction Generation Methods Based on Linearization. 9.1 Method of Feasible Directions. 9.2 Simplex Extensions for Linearly Constrained Problems. 9.3 Generalized Reduced Gradient Method. 9.4 Design Application. 9.5 Summary. References. Problems. 10 Quadratic Approximation Methods for Constrained Problems. 10.1 Direct Quadratic Approximation. 10.2 Quadratic Approximation of the Lagrangian Function. 10.3 Variable Metric Methods for Constrained Optimization. 10.4 Discussion. 10.5 Summary. References. Problems. 11 Structured Problems and Algorithms. 11.1 Integer Programming. 11.2 Quadratic Programming. 11.3 Complementary Pivot Problems. 11.4 Goal Programming. 11.5 Summary. References. Problems. 12 Comparison of Constrained Optimization Methods. 12.1 Software Availability. 12.2 A Comparison Philosophy. 12.3 Brief History of Classical Comparative Experiments. 12.4 Summary. References. 13 Strategies for Optimization Studies. 13.1 Model Formulation. 13.2 Problem Implementation. 13.3 Solution Evaluation. 13.4 Summary. References. Problems. 14 Engineering Case Studies. 14.1 Optimal Location of Coal-Blending Plants by Mixed-Integer Programming. 14.2 Optimization of an Ethylene Glycol-Ethylene Oxide Process. 14.3 Optimal Design of a Compressed Air Energy Storage System. 14.4 Summary. References. Appendix A Review of Linear Algebra. A.1 Set Theory. A.2 Vectors. A.3 Matrices. A.3.1 Matrix Operations. A.3.2 Determinant of a Square Matrix. A.3.3 Inverse of a Matrix. A.3.4 Condition of a Matrix. A.3.5 Sparse Matrix. A.4 Quadratic Forms. A.4.1 Principal Minor. A.4.2 Completing the Square. A.5 Convex Sets. Appendix B Convex and Concave Functions. Appendix C Gauss-Jordan Elimination Scheme. Author Index. Subject Index.

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Book SynopsisThis comprehensive reference presents the latest techniques for numerical analysis of forming operations. This is the perfect tool for those who wish to investigate new analytical methods for forming.Table of ContentsThe Tensile Test and Basic Material Behavior. Tensors, Matrices, Notation. Stress. Strain. Standard Mechanical Principles. Elasticity. Plasticity. Crystal-Based Plasticity. Friction. Classical Forming Analysis. Index.

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Book SynopsisNumerical Computation of Internal and External Flows Volume 2: Computational Methods for Inviscid and Viscous Flows C. Hirsch, Vrije Universiteit Brussel, Brussels, Belgium This second volume deals with the applications of computational methods to the problems of fluid dynamics.Table of ContentsPreface xv Nomenclature xix Part V: The Numerical Computation of Potential Flows 1 Chapter 13 The Mathematical Formulations of the Potential Flow Model 4 13.1 Conservative Form of the Potential Equation 4 13.2 The Non-conservative Form of the Isentropic Potential Flow Model 6 13.2.1 Small-perturbation potential equation 7 13.3 The Mathematical Properties of the Potential Equation 9 13.3.1 Unsteady potential flow 9 13.3.2 Steady potential flow 9 13.4 Boundary Conditions 14 13.4.1 Solid wall boundary condition 14 13.4.2 Far field conditions 15 13.4.3 Cascade and channel flows 17 13.4.4 Circulation and Kutta condition 18 13.5 Integral or Weak Formulation of the Potential Model 18 13.5.1 Bateman variational principle 19 13.5.2 Analysis of some properties of the variational integral 20 Chapter 14 The Discretization of the Subsonic Potential Equation 26 14.1 Finite Difference Formulation 27 14.1.1 Numerical estimation of the density 29 14.1.2 Curvilinear mesh 31 14.1.3 Consistency of the discretization of metric coefficients 34 14.1.4 Boundary conditions—curved solid wall 36 14.2 Finite Volume Formulation 38 14.2.1 Jameson and Caughey’s finite volume method 39 14.3 Finite Element Formulation 42 14.3.1 The finite element—Galerkin method 43 14.3.2 Least squares or optimal control approach 47 14.4 Iteration Scheme for the Density 47 Chapter 15 The Computation of Stationary Transonic Potential Flows 57 15.1 The Treatment of the Supersonic Region: Artificial Viscosity—Density and Flux Upwinding 61 15.1.1 Artificial viscosity—non-conservative potential equation 62 15.1.2 Artificial viscosity—conservative potential equation 66 15.1.3 Artificial compressibility 67 15.1.4 Artificial flux or flux upwinding 70 15.2 Iteration Schemes for Potential Flow Computations 77 15.2.1 Line relaxation schemes 77 15.2.2 Guidelines for resolution of the discretized potential equation 81 15.2.3 The alternating direction implicit method—approximate factorization schemes 88 15.2.4 Other techniques—multigrid methods 98 15.3 Non-uniqueness and Non-isentropic Potential Models 104 15.3.1 Isentropic shocks 105 15.3.2 Non-uniqueness and breakdown of the transonic potential flow model 105 15.3.3 Non-isentropic potential models 112 15.4 Conclusions 117 Part VI: The Numerical Solution of the System of Euler Equations 125 Chapter 16 The Mathematical Formulation of the System of Euler Equations 132 16.1 The Conservative Formulation of the Euler Equations 132 16.1.1 Integral conservative formulation of the Euler equations 133 16.1.2 Differential conservative formulation 134 16.1.3 Cartesian system of coordinates 134 16.1.4 Discontinuities and Rankine-Hugoniot relations—entropy condition 135 16.2 The Quasi-linear Formulation of the Euler Equations 138 16.2.l The Jacobian matrices for conservative variables 138 16.2.2 The Jacobian matrices for primitive variables 145 16.2.3 Transformation matrices between conservative and non-conservative variables 147 16.3 The Characteristic Formulation of the Euler Equations—Eigenvalues and Compatibility Relations 150 16.3.1 General properties of characteristics 151 16.3.2 Diagonalization of the Jacobian matrices 153 16.3.3 Compatibility equations 154 16.4 Characteristic Variables and Eigenvalues for One-dimensional Flows 157 16.4.1 Eigenvalues and eigenvectors of Jacobian matrix 158 16.4.2 Characteristic variables 162 16.4.3 Characteristics in the xt-plane—shocks and contact discontinuities 168 16.4.4 Physical boundary conditions 171 16.4.5 Characteristics and simple wave solutions 173 16.5 Eigenvalues and Compatibility Relations in Multidimensional Flows 176 16.5.1 Jacobian eigenvalues and eigenvectors in primitive variables 177 16.5.2 Diagonalization of the conservative Jacobians 180 16.5.3 Mach cone and compatibility relations 184 16.5.4 Boundary conditions 191 16.6 Some Simple Exact Reference Solutions for One-dimensional Inviscid Flows 196 16.6.1 The linear wave equation 196 16.6.2 The inviscid Burgers equation 196 16.6.3 The shock tube problem or Riemann problem 204 16.6.4 The quasi-one-dimensional nozzle flow 211 Chapter 17 The Lax–Wendroff Family of Space-centred Schemes 224 17.1 The Space-centred Explicit Schemes of First Order 226 17.1.1 The one-dimensional Lax–Friedrichs scheme 226 17.1.2 The two-dimensional Lax–Friedrichs scheme 229 17.1.3 Corrected viscosity scheme 233 17.2 The Space-centred Explicit Schemes of Second Order 234 17.2.1 The basic one-dimensional Lax–Wendroff scheme 234 17.2.2 The two-step Lax–Wendroff schemes in one dimension 238 17.2.3 Lerat and Peyret’s family of non-linear two-step Lax–Wendroff schemes 246 17.2.4 One-step Lax–Wendroff schemes in two dimensions 251 17.2.5 Two-step Lax–Wendroff schemes in two dimensions 258 17.3 The Concept of Artificial Dissipation or Artificial Viscosity 272 17.3.1 General form of artificial dissipation terms 273 17.3.2 Von Neumann–Richtmyer artificial viscosity 274 17.3.3 Higher-order artificial viscosities 279 17.4 Lerat’s Implicit Schemes of Lax–Wendroff Type 283 17.4.1 Analysis for linear systems in one dimension 285 17.4.2 Construction of the family of schemes 288 17.4.3 Extension to non-linear systems in conservation form 292 17.4.4 Extension to multi-dimensional flows 296 17.5 Summary 296 Chapter 18 The Central Schemes with Independent Time Integration 307 18.1 The Central Second-order Implicit Schemes of Beam and Warming in One Dimension 309 18.1.1 The basic Beam and Warming schemes 310 18.1.2 Addition of artificial viscosity 315 18.2 The Multidimensional Implicit Beam and Warming Schemes 326 18.2.1 The diagonal variant of Pulliam and Chaussee 328 18.3 Jameson’s Multistage Method 334 18.3.1 Time integration 334 18.3.2 Convergence acceleration to steady state 335 Chapter 19 The Treatment of Boundary Conditions 344 19.1 One-dimensional Boundary Treatment for Euler Equations 345 19.1.1 Characteristic boundary conditions 346 19.1.2 Compatibility relations 347 19.1.3 Characteristic boundary conditions as a function of conservative and primitive variables 349 19.1.4 Extrapolation methods 353 19.1.5 Practical implementation methods for numerical boundary conditions 357 19.1.6 Nonreflecting boundary conditions 369 19.2 Multidimensional Boundary Treatment 372 19.2.1 Physical and numerical boundary conditions 372 19.2.2 Multidimensional compatibility relations 376 19.2.3 Farfield treatment for steadystate flows 377 19.2.4 Solid wall boundary 379 19.2.5 Nonreflective boundary conditions 384 19.3 The Far-field Boundary Corrections 385 19.4 The Kutta Condition 395 19.5 Summary 401 Chapter 20 Upwind Schemes for the Euler Equations 408 20.1 The Basic Principles of Upwind Schemes 409 20.2 One-dimensional Flux Vector Splitting 415 20.2.1 Steger and Warming flux vector splitting 415 20.2.2 Properties of split flux vectors 417 20.2.3 Van Leer’s flux splitting 420 20.2.4 Non-reflective boundary conditions and split fluxes 425 20.3 One-dimensional Upwind Discretizations Based on Flux Vector Splitting 426 20.3.1 First-order explicit upwind schemes 426 20.3.2 Stability conditions for first-order flux vector splitting schemes 428 20.3.3 Non-conservative firstorder upwind schemes 438 20.4 Multi-dimensional Flux Vector Splitting 438 20.4.1 Steger and Warming flux splitting 440 20.4.2 Van Leer flux splitting 440 20.4.3 Arbitrary meshes 441 20.5 The Godunov-type Schemes 443 20.5.1 The basic Godunov scheme 444 20.5.2 Osher’s approximate Riemann solver 453 20.5.3 Roe’s approximate Riemann solver 460 20.5.4 Other Godunov-type methods 469 20.5.5 Summary 472 20.6 First-order Implicit Upwind Schemes 473 20.7 Multi-dimensional First-order Upwind Schemes 475 Chapter 21 Second-order Upwind and High-resolution Schemes 493 21.1 General Formulation of Higher-order Upwind Schemes 494 21.1.1 Higher-order projection stages-variable extrapolation or MUSCL approach 495 21.1.2 Numerical flux for higher-order upwind schemes 498 21.1.3 Second-order space- and time-accurate upwind schemes based on variable extrapolation 499 21.1.4 Linearized analysis of second-order upwind schemes 502 21.1.5 Numerical flux for higher-order upwind schemes—flux extrapolation 504 21.1.6 Implicit second-order upwind schemes 512 21.1.7 Implicit second-order upwind schemes in two dimensions 514 21.1.8 Summary 516 21.2 The Definition of High-resolution Schemes 517 21.2.1 The generalized entropy condition for inviscid equations 519 21.2.2 Monotonicity condition 525 21.2.3 Total variation diminishing (TVD)schemes 528 21.3 Second-order TVD Semi-discretized Schemes with Limiters 536 21.3.1 Definition of limiters for the linear convection equation 537 21.3.2 General definition of flux limiters 550 21.3.3 Limiters for variable extrapolation—MUSCL—method 552 21.4 Timeintegration Methods for TVD Schemes 556 21.4.1 Explicit TVD schemes of first-order accuracy in time 557 21.4.2 Implicit TVD schemes 558 21.4.3 Explicit second-order TVD schemes 560 21.4.4 TVD schemes and artificial dissipation 564 21.4.5 TVD limiters and the entropy condition 568 21.5 Extension to Non-linear Systems and to Multi-dimensions 570 21.6 Conclusions to Part VI 583 Part VII: The Numerical Solution of the Navier-Stokes Equations 595 Chapter 22 The Properties of the System of Navier–Stokes Equations 597 22.1 Mathematical Formulation of the Navier–Stokes Equations 597 22.1.1 Conservative form of the Navier–Stokes equations 597 22.1.2 Integral form of the Navier–Stokes equations 599 22.1.3 Shock waves and contact layers 600 22.1.4 Mathematical properties and boundary conditions 601 22.2 Reynolds-averaged Navier–Stokes Equations 603 22.2.1 Turbulent-averaged energy equation 604 22.3 Turbulence Models 606 22.3.1 Algebraic models 608 22.3.2 One- and two-equation models—k–ε models 613 22.3.3 Algebraic Reynolds stress models 615 22.4 Some Exact One-dimensional Solutions 618 22.4.1 Solutions to the linear convection-diffusion equation 618 22.4.2 Solutions to Burgers equation 620 22.4.3 Other simple test cases 621 Chapter 23 Discretization Methods for the Navier–Stokes Equations 624 23.1 Discretization of Viscous and Heat Conduction Terms 625 23.2 Time-dependent Methods for Compressible Navier–Stokes Equations 627 23.2.1 First-order explicit central schemes 628 23.2.2 One-step Lax–Wendroff schemes 629 23.2.3 Two-step Lax–Wendroff schemes 630 23.2.4 Central schemes with separate space and time discretization 636 23.2.5 Upwind schemes 648 23.3 Discretization of the Incompressible Navier–Stokes Equations 654 23.3.1 Incompressible Navier–Stokes equations 654 23.3.2 Pseudo-compressibility method 656 23.3.3 Pressure correction methods 661 23.3.4 Selection of the space discretization 666 23.4 Conclusions to Part VII 674 Index 685

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Book SynopsisChange-point problems arise in a variety of experimental andmathematical sciences, as well as in engineering and healthsciences. This rigorously researched text provides a comprehensivereview of recent probabilistic methods for detecting various typesof possible changes in the distribution of chronologically orderedobservations. Further developing the already well-establishedtheory of weighted approximations and weak convergence, the authorsprovide a thorough survey of parametric and non-parametric methods,regression and time series models together with sequential methods.All but the most basic models are carefully developed with detailedproofs, and illustrated by using a number of data sets. Contains athorough survey of: * The Likelihood Approach * Non-Parametric Methods * Linear Models * Dependent Observations This book is undoubtedly of interest to all probabilists andstatisticians, experimental and health scientists, engineers, andessential for those wTrade Review"This book is suitable for Ph.D. students who wish to establish a solid grounding in the field, and researchers who need a reliable reference to precisely formulated results and their proofs. The book contains a very extensive list of references reading into the late 1990's." (Mathematical Reviews, 2011)Table of ContentsThe Likelihood Approach. Nonparametric Methods. Linear Models. Dependent Observations. Appendix. References. Indexes.

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Book SynopsisThe concept of a system as an entity in its own right has emergedwith increasing force in the past few decades in, for example, theareas of electrical and control engineering, economics, ecology,urban structures, automaton theory, operational research andindustry. The more definite concept of a large-scale system isimplicit in these applications, but is particularly evident infields such as the study of communication networks, computernetworks and neural networks. The Wiley-Interscience Series inSystems and Optimization has been established to serve the needs ofresearchers in these rapidly developing fields. It is intended forworks concerned with developments in quantitative systems theory,applications of such theory in areas of interest, or associatedmethodology. This is the first book-length treatment of risk-sensitive control,with many new results. The quadratic cost function of the standardLQG (linear/quadratic/Gaussian) treatment is replaced by theexponential of a quadratTable of ContentsBASICS. Deterministic Models. Stochastic Models. BEYOND. Risk-Sensitive and H infinity Criteria. Time-Integral Methods and Optimal Stationary Policies. Near-Determinism and Large Deviation Theory. Appendices. References. Index.

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Book SynopsisClearly written and free of statistical jargon, this invaluable guide concentrates on the practicalities of statistical analysis for anyone involved with agricultural research. Each section starts with the key points, giving a quick reference to the contents and plenty of examples using a reala data.Table of ContentsAcknowledgements INTRODUCTION Notation A little history Population versus samples PLANNING Formulating the idea Defining objectives Defining the population Formulating hypotheses Hypothesis testing Anticipating treatment differences DESIGN Variables Choosing the treatments Constraints Replication Blocking Randomization Covariants Confounding TRIAL STRUCTURE Considerations Single-treatment factor designs Multi-treatment factor designs Some other designs DATA ENTRY AND EXPLORATION Data entry Data Data checking Data exploration ANALYTICAL TECHNIQUES Parametric techniques Non-parametric techniques Comparison of parametric and non-parametric techniques OTHER STATISTICAL TECHNIQUES Multivariate analysis Time series analysis ASPECTS OF COMPUTING APPENDICES Glossary of Statistical Terms Analysis of Variance Formulae INDEX

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Book SynopsisMatrices are used in many fields such as statistics, econometrics, mathematics, natural sciences and engineering. They provide a concise, simple method for describing long and complicated computations. This is a comprehensive handbook and dictionary of terms for matrix theory.Table of ContentsDefinitions, Notations, Terminology. Rules for Matrix Operations. Matrix Valued Functions of a Matrix. Trace, Determinant and Rank of a Matrix. Eigenvalues and Singular Values. Matrix Decompositions and Canonical Forms. Vectorization Operators. Vector and Matrix Norms. Properties of Special Matrices. Vector and Matrix Derivatives. Polynomials, Power Series and Matrices. Appendix. References. Index.

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Book SynopsisThe finite element method and the boundary element method are two computational methods available for designing structures ranging from aircraft and ships to dams and tunnels. This text presents the mathematical basis of the joint use of both methods and their computer implementation.Table of ContentsBasic principle and domains of application. I. BOUNDARY INTEGRAL EQUATIONS FOR STATIC PROBLEMS : Integral Equations and Representations for the Poisson Equation; Numerical Solution using Boundary Elements; Integral Equations and Representations for Elastostatics; Integral Representations of Gradients and Stresses on the Boundary; Some Classical Mathematical Results II. BOUNDARY INTEGRAL EQUATIONS FOR WAVE AND EVOLUTION PROBLEMS: Waves and Elastodynamics in Time Domain; Waves and Elastodynamics in Frequency Domain; Diffusion, Fluid Flow. III. ADVANCED TOPICS : Variational Boundary Integral Formulations; Exploitation of Geometrical Symmetry; Domain Derivative and Boundary Integral Eequations. IV. ADDITIONAL TOPICS IN SOLID MECHANICS : Boundary Integral Equations for Cracked Solids; Initial Strain or Stress: Inclusions, Elastoplasticity. APPENDICES : Tangential Differential Operators and Integration by Parts; Interpolation Functions and Numerical Integration. Bibliography. Index.

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Book SynopsisThis book presents ordinary differential equations based on Lie group analysis and related invariance principles. The author provides students and teachers with a text for one-semester undergraduate and graduate courses that spans a variety of topics, from the basic theory through to applications.Trade Review"…this is the first self-contained university text on ordinary differential equations…" (Zentralblatt Math, Vol.1047, No.22, 2004)Table of ContentsIntroduction to Differential Equations. Transformation Groups. Lie Group Analysis of Ordinary Differential Equations. Brief on Lie Algebras. First Order Differential Equations. Integration of Second Order Equations. Basic Theory of Linear Equations. Nonlinear Second Order Equations. Integration of Third Order Equations. Nonlinear Superposition Principle. Index.

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Book SynopsisThis text shows how stochastic geometry can be applied to real structural problems in materials science and technology. It pays particular attention to describing spatial sizes and shapes of grains and particles, developments in stochastic geometry, and relevant computer simulation techniques.Trade Review"...provides many examples...comprehensive discussions...an introduction to the analysis of two-dimensional and three-dimensional microscopic images...references are comprehensive..." (Short Book Reviews, Vol. 21, No. 2, August 2001) "There is no book I know in our own field that deals with the subject in anything like the depth and breadth as this one does." (European Journal of Soil Science, No. 52 2001) "It can be expected that this unusually careful work will soon be acknowledged as an authoritative treatment, and certainly it will remain a major reference of applied stereology in the next two decades at least. Scientific and technical libraries should have multiple copies available." (Ceramics, Vol.45 No.3, 2001) "...an ideal textbook for a one-semester course...also an excellent reference book..." (Technometrics, February 2002)Table of ContentsDedication to Günter Bach. Preface. Series Preface. Acknowledgements. List of Notation. List of Source Codes. Introduction. Methodological Tools. Statistical Estimation of Basic Characteristics. Basic Characteristics and Digitalization. Covariance and Spectral Density. Size Distribution of Spherical Particles. Nonspherical Particles of Constant Shape. Size-Shape Distribution of Particles. Arrangement of Objects. Single-Phase Polyhedral Microstructures. Appendix A: Characteristics of Geometric Objects. Appendix B: Software Utilities. References. Index.

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Book SynopsisThis volume presents a unified framework for systems engineering and a systematic and rigorous source for a comprehensive description of the utilization of Monte Carlo methods in practical engineering problems. The author suggests that efficiency can be improved through such an integrated approach.Table of Contents1. Introduction - Probability and statistics 2. Basic concepts in system engineering 3. Basic concepts in Monte Carlo methods 4. Additional applications 5. Elements of uncertainty and uncertainty analysis 6. System transport 7. Realization of system transport Appendix

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Book SynopsisThis text examines synthetic and dynamical properties of fuzzy control systems in a quantitative manner. It includes fuzzy dynamical systems, controllability and sensitivity analysis and how these affect parameters in membership functions, fuzzification, defuzzification and inferencing.Trade Review"Design and control engineers will value the advanced control techniques, new design and analysis tools presented. Post-graduates...a useful reference." (Engineering Design, July 2000) "...a good read...it boldly tackles the stability issue of fuzzy control systems..." (Measurement and Control, October 2000) "Design and control engineers will value the advanced control techniques and new design and analysis tools presented. Postgraduates studying fuzzy control will find this book a useful reference...." (European Power Electronics & Drives Journal September 2001)Table of ContentsMODELING. Information Granularity in the Analysis and Design of Fuzzy Controllers. Fuzzy Modeling for Predictive Control. Adaptive and Learning Schemes for Fuzzy Modeling. Fuzzy System Identification with General Parameter Radial Basis Function Neural Network. ANALYSIS. Lyapunov Stability Analysis of Fuzzy Dynamic Systems. Passivity and Stability of Fuzzy Control Systems. Frequency Domain Analysis of MIMO Fuzzy Control Systems. Analytical Study of Structure of a Mamdani Fuzzy Controller with Three Input Variables. An Approach to the Analysis of Robust Stability of Fuzzy Control Systems. Fuzzy Control Systems Stability Analysis with Application to Aircraft Systems. SYNTHESIS. Observer-Based Controller Synthesis for Model-Based Fuzzy Systems via Linear Matrix Inequalities. LMI-Based Fuzzy Control: Fuzzy Regulator and Fuzzy Observer Design via LMIs. A Framework for the Synthesis of PDC-Type Takagi-Sugano Fuzzy Control Systems: An LMI Approach. On Adaptive Fuzzy Logic Control on Non-linear Systems--Synthesis and Analysis. Stabilization of Direct Adaptive Fuzzy Control Systems: Two Approaches. Gain Scheduling Based Control of a Class of TSK Systems. Output Tracking Using Fuzzy Neural Networks. Fuzzy Life-Extending Control of Mechanical Systems. Epilogue. Index.

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Book SynopsisThis work is a guide to the principles behind sensitivity analysis. It suggests suitable methods for particular types of problem, which allows a greater understanding of the entire causal assessment chain. This makes the impact of source uncertainties and framing assumptions more transparent.Trade Review"The book has a fair price...I think this is a book that everyone who does modeling should buy. It can readily be read piecemeal...so it is ideal for leisurely self-study..." (Technometrics Vol. 42, No. 4 May 2001) "...this book will prove helpful in the solution of many modeling problems." (La Doc Sti, September 2000) "...presents many different sensitivity analysis methodologies and demonstrates their usefulness in scientific research." (Zentralblatt MATH, Vol. 961, 2001/11)Table of ContentsWhat is Sensitivity Analysis. Hitchhiker's Guide to Sensitivity Analysis. METHODS. Designs of Experiments. Screening Methods. Local Methods. Sampling-Based Methods. Reliability Algorithms: FORM and SORM Methods. Variance-Based Methods. Managing the Tyranny of Parameters in Mathematical Modelling of Physical Systems. Bayesian Sensitivity Analysis. Graphical Methods. APPLICATIONS. Practical Experience in Applying Sensitivity and Uncertainty Analysis. Scenario and Parametric Sensitivity and Uncertainty Analysis in Nuclear Waste Disposal Risk Assessment: The Case of GESAMAC. Sensitivity Analysis for Signal Extraction in Economic Time Series. A Dataless Precalibration Analysis in Solid State Physics. Appplication of First-Order (FORM) and Second-Order (SORM) Reliability Methods: Analysis and Interpretation of Sensitivity Measures Related to Groundwater Pressure Decreases and Resulting Ground Subsidence. One-at-a-Time and Mini-Global Analyses for Characterizing Model Sensitivity in the Nonlinear Ozone Predictions from the US EPA Regional Acid Deposition Model (RADM). Comparing Different Sensitivity Analysis Methods on a Chemical Reactions Model. An Application of Sensitivity Analysis to Fish Population Dynamics. Global Sensitivity Analysis: A Quality Assurance Tool in Environmental Policy Modelling. CONCLUSIONS. Assuring the Quality of Models Designed for Predictive Tasks. Fortune and Future of Sensitivity Analysis. References. Appendix. Index.

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Book SynopsisAt present, most books on ecological modelling rely on very complex mathematics, resulting in students and researchers shying away from investigating the potential uses of ecological models and their methods of construction. This new book aims to open up this exciting area to a much wider audience.Trade Review"Teachers of courses on ecological modelling will find [this book] a useful source-book at a competitive price."Table of ContentsIntroduction: Themes Of Ecological Modelling. Probability Of Population Extinction. Looking For Cycles: The Dynamics Of Predators And Their Prey. Population Dynamics Of Species With Complex Life-Histories. Dynamics Of Ecological Communities. Spatial Models And Thresholds. Disease And Biological Control. Answers To Questions. Glossary Of Symbols And Terms. References. Index

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Book SynopsisIntroduces the distributed control of robotic networks. This book presents a set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity. It analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation.Trade Review"This book covers its subject very thoroughly. The framework the authors have established is very elegant and, if it catches on, this book could be the primary reference for this approach. I don't know of any other book that covers this set of topics."—Richard M. Murray, California Institute of Technology"The authors do an excellent job of clearly describing the problems and presenting rigorous, provably correct algorithms with complexity bounds for each problem. The authors also do a fantastic job of providing the mathematical insight necessary for such complex problems."—Ali Jadbabaie, University of Pennsylvania"The order of presentation makes much sense, and the book thoroughly covers what it sets out to cover. The algorithms and results are presented using a clear mathematical and computer science formalism, which allows a uniform presentation. The formalism used and the way of presenting the algorithms may be helpful for structuring the presentation of new algorithms in the future."—Vincent Blondel, Université catholique de LouvainTable of ContentsPreface ix Chapter 1. An introduction to distributed algorithms 1 1.1 Elementary concepts and notation 1 1.2 Matrix theory 6 1.3 Dynamical systems and stability theory 12 1.4 Graph theory 20 1.5 Distributed algorithms on synchronous networks 37 1.6 Linear distributed algorithms 52 1.7 Notes 66 1.8 Proofs 69 1.9 Exercises 85 Chapter 2. Geometric models and optimization 95 2.1 Basic geometric notions 95 2.2 Proximity graphs 104 2.3 Geometric optimization problems and multicenter functions 111 2.4 Notes 124 2.5 Proofs 125 2.6 Exercises 133 Chapter 3. Robotic network models and complexity notions 139 3.1 A model for synchronous robotic networks 139 3.2 Robotic networks with relative sensing 151 3.3 Coordination tasks and complexity notions 158 3.4 Complexity of direction agreement and equidistance 165 3.5 Notes 166 3.6 Proofs 169 3.7 Exercises 176 Chapter 4. Connectivity maintenance and rendezvous 179 4.1 Problem statement 180 4.2 Connectivity maintenance algorithms 182 4.3 Rendezvous algorithms 191 4.4 Simulation results 200 4.5 Notes 201 4.6 Proofs 204 4.7 Exercises 215 Chapter 5. Deployment 219 5.1 Problem statement 220 5.2 Deployment algorithms 222 5.3 Simulation results 233 5.4 Notes 237 5.5 Proofs 239 5.6 Exercises 245 Chapter 6. Boundary estimation and tracking 247 6.1 Event-driven asynchronous robotic networks 248 6.2 Problem statement 252 6.3 Estimate update and cyclic balancing law 256 6.4 Simulation results 266 6.5 Notes 268 6.6 Proofs 270 6.7 Exercises 275 Bibliography 279 Algorithm Index 305 Subject Index 307 Symbol Index 313

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Book SynopsisCo-published with Oxford University Press. This highly technical and thought-provoking book stresses the development of mathematical foundations for the application of the electromagnetic model to problems of research and technology.Table of ContentsPreface. Linear Analysis. The Green's Function Method. The Spectral Representation Method. Electromagnetic Sources. Electromagnetic Boundary Value Problems. Index.

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Book SynopsisSPATIAL ERROR ANALYSIS is an all-in-one sourcebook on error measurements in one-, two-, and three-dimensional spaces. This book features exhaustive, systematic coverage of error measurement relationships, techniques, and solutions used to solve general, correlated cases. It is packed with 62 figures and 24 tables. MATLAB-based M-files* for practical applications created especially for this volume are available on the Web at ftp://ftp.mathworks.com/pub/books/hsu. Solutions to two- and three-dimensional problems are presented without relying on equal standard deviations from each channel. They also make no assumption that the random variables of interest are independent or uncorrelated. * MATLAB (developed by MathWorks, Inc.) must be purchased separately. Sponsored by: IEEE Aerospace and Electronic Systems Society.Table of ContentsPreface. List of Figures. List of Tables. Introduction. Prameter Estimation from Samples. One-Dimensional Error Analysis. Two-Dimensional Error Analysis. Three-Dimensional Error Analysis. Maximum Likelihood Estimation of Radial Error PDF. Position Location Problems. Risk Analysis. Appendix A: Probability Density Functions. Appendix B: Method of Confidence Intervals. Appendix C: Function of N Random Variables. Appendix D: GPS Dilution of Precisions. Appendix E: Listing of Author-Generated M-files. Bibliography. Index. About the Author.

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Book SynopsisThis book is an adventure into the computer analysis of three dimensional composite structures using the finite element method (FEM). Once the basic philosophy of the method is understood, the reader may expand its application and modify the computer programs to suit particular needs.Trade Review`The book is highly recommended as a reference text for advanced undergraduate students, as a graduate course on the FE analysis of composites, and as a reference work for both researchers in laboratories and practising engineers in industry.' Zentralblatt MATH, 906 Table of ContentsPreface. 1. Some Results from Continuum Mechanics. 2. A Brief History of FEM. 3. Natural Modes for Finite Elements. 4. Composites. 5. Composite Beam Element. 6. Composite Plate and Shell Element. 7. Computational Statistics. 8. Nonlinear Analysis of Anisotropic Shells. 9. Programming Aspects. Appendices: A. Geometry of the Bema Element in Space. B. Contents of the Floppy Disk. Bibliography. Index.

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Book SynopsisThis text is intended for the undergraduate course in math methods, with an audience of physics and engineering majors. As a required course in most departments, the text relies heavily on explained examples, real-world applications and student engagement.Trade Review"[Mathematical Methods in Engineering and Physics] is my book of choice for teaching undergraduates...I honestly never thought that I could be so enchanted by the heat equation before seeing how Felder and Felder effectively have students derive it as part of honing their intuition for how to think about partial differential equations." - Christine Aidala, PhD, Associate Professor of Physics at University of Michigan for the American Journal of Physics Table of ContentsPreface xi 1 Introduction to Ordinary Differential Equations 1 1.1 Motivating Exercise: The Simple Harmonic Oscillator 2 1.2 Overview of Differential Equations 3 1.3 Arbitrary Constants 15 1.4 Slope Fields and Equilibrium 25 1.5 Separation of Variables 34 1.6 Guess and Check, and Linear Superposition 39 1.7 Coupled Equations (see felderbooks.com) 1.8 Differential Equations on a Computer (see felderbooks.com) 1.9 Additional Problems (see felderbooks.com) 2 Taylor Series and Series Convergence 50 2.1 Motivating Exercise: Vibrations in a Crystal 51 2.2 Linear Approximations 52 2.3 Maclaurin Series 60 2.4 Taylor Series 70 2.5 Finding One Taylor Series from Another 76 2.6 Sequences and Series 80 2.7 Tests for Series Convergence 92 2.8 Asymptotic Expansions (see felderbooks.com) 2.9 Additional Problems (see felderbooks.com) 3 Complex Numbers 104 3.1 Motivating Exercise: The Underdamped Harmonic Oscillator 104 3.2 Complex Numbers 105 3.3 The Complex Plane 113 3.4 Euler’s Formula I—The Complex Exponential Function 117 3.5 Euler’s Formula II—Modeling Oscillations 126 3.6 Special Application: Electric Circuits (see felderbooks.com) 3.7 Additional Problems (see felderbooks.com) 4 Partial Derivatives 136 4.1 Motivating Exercise: The Wave Equation 136 4.2 Partial Derivatives 137 4.3 The Chain Rule 145 4.4 Implicit Differentiation 153 4.5 Directional Derivatives 158 4.6 The Gradient 163 4.7 Tangent Plane Approximations and Power Series (see felderbooks.com) 4.8 Optimization and the Gradient 172 4.9 Lagrange Multipliers 181 4.10 Special Application: Thermodynamics (see felderbooks.com) 4.11 Additional Problems (see felderbooks.com) 5 Integrals in Two or More Dimensions 188 5.1 Motivating Exercise: Newton’s Problem (or) The Gravitational Field of a Sphere 188 5.2 Setting Up Integrals 189 5.3 Cartesian Double Integrals over a Rectangular Region 204 5.4 Cartesian Double Integrals over a Non-Rectangular Region 211 5.5 Triple Integrals in Cartesian Coordinates 216 5.6 Double Integrals in Polar Coordinates 221 5.7 Cylindrical and Spherical Coordinates 229 5.8 Line Integrals 240 5.9 Parametrically Expressed Surfaces 249 5.10 Surface Integrals 253 5.11 Special Application: Gravitational Forces (see felderbooks.com) 5.12 Additional Problems (see felderbooks.com) 6 Linear Algebra I 266 6.1 The Motivating Example on which We’re Going to Base the Whole Chapter: The Three-Spring Problem 266 6.2 Matrices: The Easy Stuff 276 6.3 Matrix Times Column 280 6.4 Basis Vectors 286 6.5 Matrix Times Matrix 294 6.6 The Identity and Inverse Matrices 303 6.7 Linear Dependence and the Determinant 312 6.8 Eigenvectors and Eigenvalues 325 6.9 Putting It Together: Revisiting the Three-Spring Problem 336 6.10 Additional Problems (see felderbooks.com) 7 Linear Algebra II 346 7.1 Geometric Transformations 347 7.2 Tensors 358 7.3 Vector Spaces and Complex Vectors 369 7.4 Row Reduction (see felderbooks.com) 7.5 Linear Programming and the Simplex Method (see felderbooks.com) 7.6 Additional Problems (see felderbooks.com) 8 Vector Calculus 378 8.1 Motivating Exercise: Flowing Fluids 378 8.2 Scalar and Vector Fields 379 8.3 Potential in One Dimension 387 8.4 From Potential to Gradient 396 8.5 From Gradient to Potential: The Gradient Theorem 402 8.6 Divergence, Curl, and Laplacian 407 8.7 Divergence and Curl II—The Math Behind the Pictures 416 8.8 Vectors in Curvilinear Coordinates 419 8.9 The Divergence Theorem 426 8.10 Stokes’ Theorem 432 8.11 Conservative Vector Fields 437 8.12 Additional Problems (see felderbooks.com) 9 Fourier Series and Transforms 445 9.1 Motivating Exercise: Discovering Extrasolar Planets 445 9.2 Introduction to Fourier Series 447 9.3 Deriving the Formula for a Fourier Series 457 9.4 Different Periods and Finite Domains 459 9.5 Fourier Series with Complex Exponentials 467 9.6 Fourier Transforms 472 9.7 Discrete Fourier Transforms (see felderbooks.com) 9.8 Multivariate Fourier Series (see felderbooks.com) 9.9 Additional Problems (see felderbooks.com) 10 Methods of Solving Ordinary Differential Equations 484 10.1 Motivating Exercise: A Damped, Driven Oscillator 485 10.2 Guess and Check 485 10.3 Phase Portraits (see felderbooks.com) 10.4 Linear First-Order Differential Equations (see felderbooks.com) 10.5 Exact Differential Equations (see felderbooks.com) 10.6 Linearly Independent Solutions and the Wronskian (see felderbooks.com) 10.7 Variable Substitution 494 10.8 Three Special Cases of Variable Substitution 505 10.9 Reduction of Order and Variation of Parameters (see felderbooks.com) 10.10 Heaviside, Dirac, and Laplace 512 10.11 Using Laplace Transforms to Solve Differential Equations 522 10.12 Green’s Functions 531 10.13 Additional Problems (see felderbooks.com) 11 Partial Differential Equations 541 11.1 Motivating Exercise: The Heat Equation 542 11.2 Overview of Partial Differential Equations 544 11.3 Normal Modes 555 11.4 Separation of Variables—The Basic Method 567 11.5 Separation of Variables—More than Two Variables 580 11.6 Separation of Variables—Polar Coordinates and Bessel Functions 589 11.7 Separation of Variables—Spherical Coordinates and Legendre Polynomials 607 11.8 Inhomogeneous Boundary Conditions 616 11.9 The Method of Eigenfunction Expansion 623 11.10 The Method of Fourier Transforms 636 11.11 The Method of Laplace Transforms 646 11.12 Additional Problems (see felderbooks.com) 12 Special Functions and ODE Series Solutions 652 12.1 Motivating Exercise: The Circular Drum 652 12.2 Some Handy Summation Tricks 654 12.3 A Few Special Functions 658 12.4 Solving Differential Equations with Power Series 666 12.5 Legendre Polynomials 673 12.6 The Method of Frobenius 682 12.7 Bessel Functions 688 12.8 Sturm-Liouville Theory and Series Expansions 697 12.9 Proof of the Orthgonality of Sturm-Liouville Eigenfunctions (see felderbooks.com) 12.10 Special Application: The Quantum Harmonic Oscillator and Ladder Operators (see felderbooks.com) 12.11 Additional Problems (see felderbooks.com) 13 Calculus with Complex Numbers 708 13.1 Motivating Exercise: Laplace’s Equation 709 13.2 Functions of Complex Numbers 710 13.3 Derivatives, Analytic Functions, and Laplace’s Equation 716 13.4 Contour Integration 726 13.5 Some Uses of Contour Integration 733 13.6 Integrating Along Branch Cuts and Through Poles (see felderbooks.com) 13.7 Complex Power Series 742 13.8 Mapping Curves and Regions 747 13.9 Conformal Mapping and Laplace’s Equation 754 13.10 Special Application: Fluid Flow (see felderbooks.com) 13.11 Additional Problems (see felderbooks.com) Appendix A Different Types of Differential Equations 765 Appendix B Taylor Series 768 Appendix C Summary of Tests for Series Convergence 770 Appendix D Curvilinear Coordinates 772 Appendix E Matrices 774 Appendix F Vector Calculus 777 Appendix G Fourier Series and Transforms 779 Appendix H Laplace Transforms 782 Appendix I Summary: Which PDE Technique Do I Use? 787 Appendix J Some Common Differential Equations and Their Solutions 790 Appendix K Special Functions 798 Appendix L Answers to “Check Yourself” in Exercises 801 Appendix M Answers to Odd-Numbered Problems (see felderbooks.com) Index 805

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Book SynopsisAlgebraic Identification and Estimation Methods in Feedback Control Systems presents a model-based algebraic approach to online parameter and state estimation in uncertain dynamic feedback control systems. This approach evades the mathematical intricacies of the traditional stochastic approach, proposing a direct model-based scheme with several easy-to-implement computational advantages. The approach can be used with continuous and discrete, linear and nonlinear, mono-variable and multi-variable systems. The estimators based on this approach are not of asymptotic nature, and do not require any statistical knowledge of the corrupting noises to achieve good performance in a noisy environment. These estimators are fast, robust to structured perturbations, and easy to combine with classical or sophisticated control laws. This book uses module theory, differential algebra, and operational calculus in an easy-to-understand manner and also details how to apply these in the coTable of ContentsSeries Preface xiii Preface xv 1 Introduction 1 1.1 Feedback Control of Dynamic Systems 2 1.1.1 Feedback 2 1.1.2 Why Do We Need Feedback? 3 1.2 The Parameter Identification Problem 3 1.2.1 Identifying a System 4 1.3 A Brief Survey on Parameter Identification 4 1.4 The State Estimation Problem 5 1.4.1 Observers 6 1.4.2 Reconstructing the State via Time Derivative Estimation 7 1.5 Algebraic Methods in Control Theory: Differences from Existing Methodologies 8 1.6 Outline of the Book 9 References 12 2 Algebraic Parameter Identification in Linear Systems 15 2.1 Introduction 15 2.1.1 The Parameter-Estimation Problem in Linear Systems 16 2.2 Introductory Examples 17 2.2.1 Dragging an Unknown Mass in Open Loop 17 2.2.2 A Perturbed First-Order System 24 2.2.3 The Visual Servoing Problem 30 2.2.4 Balancing of the Plane Rotor 35 2.2.5 On the Control of the Linear Motor 38 2.2.6 Double-Bridge Buck Converter 42 2.2.7 Closed-Loop Behavior 43 2.2.8 Control of an unknown variable gain motor 47 2.2.9 Identifying Classical Controller Parameters 50 2.3 A Case Study Introducing a “Sentinel” Criterion 53 2.3.1 A Suspension System Model 54 2.4 Remarks 67 References 68 3 Algebraic Parameter Identification in Nonlinear Systems 71 3.1 Introduction 71 3.2 Algebraic Parameter Identification for Nonlinear Systems 72 3.2.1 Controlling an Uncertain Pendulum 74 3.2.2 A Block-Driving Problem 80 3.2.3 The Fully Actuated Rigid Body 84 3.2.4 Parameter Identification Under Sliding Motions 90 3.2.5 Control of an Uncertain Inverted Pendulum Driven by a DC Motor 92 3.2.6 Identification and Control of a Convey Crane 96 3.2.7 Identification of a Magnetic Levitation System 103 3.3 An Alternative Construction of the System of Linear Equations 105 3.3.1 Genesio–Tesi Chaotic System 107 3.3.2 The Ueda Oscillator 108 3.3.3 Identification and Control of an Uncertain Brushless DC Motor 112 3.3.4 Parameter Identification and Self-tuned Control for the Inertia Wheel Pendulum 119 3.3.5 Algebraic Parameter Identification for Induction Motors 128 3.3.6 A Criterion to Determine the Estimator Convergence: The Error Index 136 3.4 Remarks 141 References 141 4 Algebraic Parameter Identification in Discrete-Time Systems 145 4.1 Introduction 145 4.2 Algebraic Parameter Identification in Discrete-Time Systems 145 4.2.1 Main Purpose of the Chapter 146 4.2.2 Problem Formulation and Assumptions 147 4.2.3 An Introductory Example 148 4.2.4 Samuelson’s Model of the National Economy 150 4.2.5 Heating of a Slab from Two Boundary Points 155 4.2.6 An Exact Backward Shift Reconstructor 157 4.3 A Nonlinear Filtering Scheme 160 4.3.1 Hénon System 161 4.3.2 A Hard Disk Drive 164 4.3.3 The Visual Servo Tracking Problem 166 4.3.4 A Shape Control Problem in a Rolling Mill 170 4.3.5 Algebraic Frequency Identification of a Sinusoidal Signal by Means of Exact Discretization 175 4.4 Algebraic Identification in Fast-Sampled Linear Systems 178 4.4.1 The Delta-Operator Approach: A Theoretical Framework 179 4.4.2 Delta-Transform Properties 181 4.4.3 A DC Motor Example 181 4.5 Remarks 188 References 188 5 State and Parameter Estimation in Linear Systems 191 5.1 Introduction 191 5.1.1 Signal Time Derivation Through the “Algebraic Derivative Method” 192 5.1.2 Observability of Nonlinear Systems 192 5.2 Fast State Estimation 193 5.2.1 An Elementary Second-Order Example 193 5.2.2 An Elementary Third-Order Example 194 5.2.3 A Control System Example 198 5.2.4 Control of a Perturbed Third-Order System 201 5.2.5 A Sinusoid Estimation Problem 203 5.2.6 Identification of Gravitational Wave Parameters 205 5.2.7 A Power Electronics Example 210 5.2.8 A Hydraulic Press 213 5.2.9 Identification and Control of a Plotter 218 5.3 Recovering Chaotically Encrypted Signals 222 5.3.1 State Estimation for a Lorenz System 227 5.3.2 State Estimation for Chen’s System 229 5.3.3 State Estimation for Chua’s Circuit 231 5.3.4 State Estimation for Rossler’s System 232 5.3.5 State Estimation for the Hysteretic Circuit 234 5.3.6 Simultaneous Chaotic Encoding–Decoding with Singularity Avoidance 239 5.3.7 Discussion 240 5.4 Remarks 241 References 242 6 Control of Nonlinear Systems via Output Feedback 245 6.1 Introduction 245 6.2 Time-Derivative Calculations 246 6.2.1 An Introductory Example 247 6.2.2 Identifying a Switching Input 253 6.3 The Nonlinear Systems Case 255 6.3.1 Control of a Synchronous Generator 256 6.3.2 Control of a Multi-variable Nonlinear System 261 6.3.3 Experimental Results on a Mechanical System 267 6.4 Remarks 278 References 279 7 Miscellaneous Applications 281 7.1 Introduction 281 7.1.1 The Separately Excited DC Motor 282 7.1.2 Justification of the ETEDPOF Controller 285 7.1.3 A Sensorless Scheme Based on Fast Adaptive Observation 287 7.1.4 Control of the Boost Converter 292 7.2 Alternative Elimination of Initial Conditions 298 7.2.1 A Bounded Exponential Function 299 7.2.2 Correspondence in the Frequency Domain 300 7.2.3 A System of Second Order 301 7.3 Other Functions of Time for Parameter Estimation 304 7.3.1 A Mechanical System Example 304 7.3.2 A Derivative Approach to Demodulation 310 7.3.3 Time Derivatives via Parameter Identification 312 7.3.4 Example 314 7.4 An Algebraic Denoising Scheme 318 7.4.1 Example 321 7.4.2 Numerical Results 322 7.5 Remarks 325 References 326 Appendix A Parameter Identification in Linear Continuous Systems: A Module Approach 329 A.1 Generalities on Linear Systems Identification 329 A.1.1 Example 330 A.1.2 Some Definitions and Results 330 A.1.3 Linear Identifiability 331 A.1.4 Structured Perturbations 333 A.1.5 The Frequency Domain Alternative 337 References 338 Appendix B Parameter Identification in Linear Discrete Systems: A Module Approach 339 B.1 A Short Review of Module Theory over Principal Ideal Rings 339 B.1.1 Systems 340 B.1.2 Perturbations 340 B.1.3 Dynamics and Input–Output Systems 341 B.1.4 Transfer Matrices 341 B.1.5 Identifiability 342 B.1.6 An Algebraic Setting for Identifiability 342 B.1.7 Linear identifiability of transfer functions 344 B.1.8 Linear Identification of Perturbed Systems 345 B.1.9 Persistent Trajectories 347 References 348 Appendix C Simultaneous State and Parameter Estimation: An Algebraic Approach 349 C.1 Rings, Fields and Extensions 349 C.2 Nonlinear Systems 350 C.2.1 Differential Flatness 351 C.2.2 Observability and Identifiability 352 C.2.3 Observability 352 C.2.4 Identifiable Parameters 352 C.2.5 Determinable Variables 352 C.3 Numerical Differentiation 353 C.3.1 Polynomial Time Signals 353 C.3.2 Analytic Time Signals 353 C.3.3 Noisy Signals 354 References 354 Appendix D Generalized Proportional Integral Control 357 D.1 Generalities on GPI Control 357 D.2 Generalization to MIMO Linear Systems 365 References 368 Index 369

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Book SynopsisTheory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the model most suitable for the problems they are interested in. The book is targeted at graduate students and researchers who plan to investigate multiphase flows using LBMs. Throughout the text most of the popular multiphase LBMs are analyzed both theoretically and through numerical simulation. The authors present many of the mathematical derivations of the models in greater detail tTable of ContentsPreface xi About the companion website xiii 1 Introduction 1 1.1 History of the Lattice Boltzmann method 2 1.2 The Lattice Boltzmann method 3 1.3 Multiphase LBM 6 1.3.1 Color-gradient model 7 1.3.2 Shan–Chen model 7 1.3.3 Free-energy model 8 1.3.4 Interface tracking model 9 1.4 Comparison of models 9 1.5 Units in this book and parameter conversion 11 1.6 Appendix: Einstein summation convention 14 1.6.1 Kronecker δ function 15 1.6.2 Lattice tensors 15 1.7 Use of the Fortran code in the book 16 2 Single-component multiphase Shan–Chen-type model 18 2.1 Introduction 18 2.1.1 "Equilibrium" velocity in the SC model 20 2.1.2 Inter-particle forces in the SC SCMP LBM 20 2.2 Typical equations of state 21 2.2.1 Parameters in EOS 27 2.3 Thermodynamic consistency 28 2.3.1 The SCMP LBM EOS 29 2.3.2 Incorporating other EOS into the SC model 31 2.4 Analytical surface tension 32 2.4.1 Inter-particle Force Model A 32 2.4.2 Inter-particle Force Model B 33 2.5 Contact angle 34 2.6 Capillary rise 36 2.7 Parallel flow and relative permeabilities 39 2.8 Forcing term in the SC model 40 2.8.1 Schemes to incorporate the body force 42 2.8.2 Scheme overview 44 2.8.3 Theoretical analysis 44 2.8.4 Numerical results and discussion 46 2.9 Multirange pseudopotential (Inter-particle Force Model B) 55 2.10 Conclusions 58 2.11 Appendix A: Analytical solution for layered multiphase flow in a channel 58 2.12 Appendix B: FORTRAN code to simulate single component multiphase droplet contacting a wall as shown in Figure 2.7(c) 60 3 Shan and Chen-type multi-component multiphase models 71 3.1 Multi-component multiphase SC LBM 71 3.1.1 Fluid–fluid cohesion and fluid–solid adhesion 73 3.2 Derivation of the pressure 73 3.2.1 Pressure in popular papers (2D) 74 3.2.2 Pressure in popular papers (3D) 75 3.3 Determining Gc and the surface tension 76 3.4 Contact angle 78 3.4.1 Application of Young's equation to MCMP LBM 79 3.4.2 Contact angle measurement 79 3.4.3 Verification of proposed equation 80 3.5 Flow through capillary tubes 83 3.6 Layered two-phase flow in a 2D channel 85 3.7 Pressure or velocity boundary conditions 87 3.7.1 Boundary conditions for 2D simulations 87 3.7.2 Boundary conditions for 3D simulations 89 3.8 Displacement in a 3D porous medium 91 4 Rothman–Keller multiphase Lattice Boltzmann model 94 4.1 Introduction 94 4.2 RK color-gradient model 96 4.3 Theoretical analysis (Chapman–Enskog expansion) 99 4.3.1 Discussion of above formulae 103 4.4 Layered two-phase flow in a 2D channel 103 4.4.1 Cases of two fluids with identical densities 104 4.4.2 Cases of two fluids with different densities 106 4.5 Interfacial tension and isotropy of the RK model 110 4.5.1 Interfacial tension 110 4.5.2 Isotropy 110 4.6 Drainage and capillary filling 111 4.7 MRT RK model 113 4.8 Contact angle 114 4.8.1 Spurious currents 115 4.9 Tests of inlet/outlet boundary conditions 117 4.10 Immiscible displacements in porous media 118 4.11 Appendix A 121 4.12 Appendix B 122 5 Free-energy-based multiphase Lattice Boltzmann model 136 5.1 Swift free-energy based single-component multiphase LBM 136 5.1.1 Derivation of the coefficients in the equilibrium distribution function 138 5.2 Chapman–Enskog expansion 143 5.3 Issue of Galilean invariance 146 5.4 Phase separation 149 5.5 Contact angle 154 5.5.1 How to specify a desired contact angle 154 5.5.2 Numerical verification 155 5.6 Swift free-energy-based multi-component multiphase LBM 158 5.7 Appendix 158 6 Inamuro's multiphase Lattice Boltzmann model 167 6.1 Introduction 167 6.1.1 Inamuro's method 167 6.1.2 Comment on the presentation 169 6.1.3 Chapman–Enskog expansion analysis 170 6.1.4 Cahn–Hilliard equation (equation for order parameter) 173 6.1.5 Poisson equation 174 6.2 Droplet collision 175 6.3 Appendix 178 7 He–Chen–Zhang multiphase Lattice Boltzmann model 196 7.1 Introduction 196 7.2 HCZ model 196 7.3 Chapman–Enskog analysis 199 7.3.1 N–S equations 199 7.3.2 CH equation 202 7.4 Surface tension and phase separation 202 7.5 Layered two-phase flow in a channel 204 7.6 Rayleigh–Taylor instability 205 7.7 Contact angle 210 7.8 Capillary rise 213 7.9 Geometric scheme to specify the contact angle and its hysteresis 215 7.9.1 Examples of droplet slipping in shear flows 218 7.10 Oscillation of an initially ellipsoidal droplet 219 7.11 Appendix A 222 7.12 Appendix B: 2D code 223 7.13 Appendix C: 3D code 238 8 Axisymmetric multiphase HCZ model 253 8.1 Introduction 253 8.2 Methods 253 8.2.1 Macroscopic governing equations 253 8.2.2 Axisymmetric HCZ LBM (Premnath and Abraham 2005a) 255 8.2.3 MRT version of the axisymmetric LBM (McCracken and Abraham 2005) 256 8.2.4 Axisymmetric boundary conditions 258 8.3 The Laplace law 258 8.4 Oscillation of an initially ellipsoidal droplet 259 8.5 Cylindrical liquid column break 263 8.6 Droplet collision 265 8.6.1 Effect of gradient and Laplacian calculation 267 8.6.2 Effect of BGK and MRT 274 8.7 A revised axisymmetric HCZ model (Huang et al. 2014) 276 8.7.1 MRT collision 276 8.7.2 Calculation of the surface tension 277 8.7.3 Mass correction 278 8.8 Bubble rise 279 8.8.1 Numerical validation 281 8.8.2 Surface-tension calculation effect 283 8.8.3 Terminal bubble shape 284 8.8.4 Wake behind the bubble 284 8.9 Conclusion 286 8.10 Appendix A: Chapman–Enskog analysis 288 8.10.1 Preparation for derivation 288 8.10.2 Mass conservation 289 8.10.3 Momentum conservation 289 8.10.4 CH equation 291 9 Extensions of the HCZ model for high-density ratio two-phase flows 292 9.1 Introduction 292 9.2 Model I (Lee and Lin 2005) 293 9.2.1 Stress and potential form of intermolecular forcing terms 293 9.2.2 Model description 294 9.2.3 Implementation 297 9.2.4 Directional derivative 298 9.2.5 Droplet splashing on a thin liquid film 299 9.3 Model II (Amaya-Bower and Lee 2010) 301 9.3.1 Implementation 302 9.4 Model III (Lee and Liu 2010) 304 9.5 Model IV 305 9.6 Numerical tests for different models 306 9.6.1 A drop inside a box with periodic boundary conditions 306 9.6.2 Layered two-phase flows in a channel 311 9.6.3 Galilean invariance 313 9.7 Conclusions 316 9.8 Appendix A: Analytical solutions for layered two-phase flow in a channel 317 9.9 Appendix B: 2D code based on Amaya-Bower and Lee (2010) 319 10 Axisymmetric high-density ratio two-phase LBMs (extension of the HCZ model) 334 10.1 Introduction 334 10.2 The model based on Lee and Lin (2005) 334 10.2.1 The equilibrium distribution functions I 336 10.2.2 The equilibrium distribution functions II 336 10.2.3 Source terms 337 10.2.4 Stress and potential form of intermolecular forcing terms 337 10.2.5 Chapman–Enskog analysis 338 10.2.6 Implementation 340 10.2.7 Droplet splashing on a thin liquid film 342 10.2.8 Head-on droplet collision 342 10.3 Axisymmetric model based on Lee and Liu (2010) 345 10.3.1 Implementation 347 10.3.2 Head-on droplet collision 348 10.3.3 Bubble rise 353 Index 371

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Book SynopsisAn introduction to mathematical techniques used in engineering with an emphasis on applications in linear circuits and systems This book provides an integrated approach to learning the necessary mathematical tools specifically used for linear circuits and systems.Table of ContentsPreface xiii Notation and Bibliography xvii About the Companion Website xix 1 Overview and Background 1 1.1 Introduction 1 1.2 Mathematical Models 3 1.3 Frequency Content 12 1.4 Functions and Properties 16 1.5 Derivatives and Integrals 22 1.6 Sine, Cosine, and 𝜋 33 1.7 Napier’s Constant e and Logarithms 38 PART I CIRCUITS, MATRICES, AND COMPLEX NUMBERS 51 2 Circuits and Mechanical Systems 53 2.1 Introduction 53 2.2 Voltage, Current, and Power 54 2.3 Circuit Elements 60 2.4 Basic Circuit Laws 67 2.4.1 Mesh-Current and Node-Voltage Analysis 69 2.4.2 Equivalent Resistive Circuits 71 2.4.3 RC and RL Circuits 75 2.4.4 Series RLC Circuit 78 2.4.5 Diode Circuits 82 2.5 Mechanical Systems 85 2.5.1 Simple Pendulum 86 2.5.2 Mass on a Spring 92 2.5.3 Electrical and Mechanical Analogs 95 3 Linear Equations and Matrices 105 3.1 Introduction 105 3.2 Vector Spaces 106 3.3 System of Linear Equations 108 3.4 Matrix Properties and Special Matrices 113 3.5 Determinant 122 3.6 Matrix Subspaces 128 3.7 Gaussian Elimination 135 3.7.1 LU and LDU Decompositions 146 3.7.2 Basis Vectors 148 3.7.3 General Solution of 𝐀𝐲 = 𝐱 151 3.8 Eigendecomposition 152 3.9 MATLAB Functions 156 4 Complex Numbers and Functions 163 4.1 Introduction 163 4.2 Imaginary Numbers 165 4.3 Complex Numbers 167 4.4 Two Coordinates 169 4.5 Polar Coordinates 171 4.6 Euler’s Formula 175 4.7 Matrix Representation 182 4.8 Complex Exponential Rotation 183 4.9 Constant Angular Velocity 189 4.10 Quaternions 192 PART II SIGNALS, SYSTEMS, AND TRANSFORMS 203 5 Signals, Generalized Functions, and Fourier Series 205 5.1 Introduction 205 5.2 Energy and Power Signals 206 5.3 Step and Ramp Functions 208 5.4 Rectangle and Triangle Functions 211 5.5 Exponential Function 214 5.6 Sinusoidal Functions 217 5.7 Dirac Delta Function 220 5.8 Generalized Functions 223 5.9 Unit Doublet 233 5.10 Complex Functions and Singularities 240 5.11 Cauchy Principal Value 242 5.12 Even and Odd Functions 245 5.13 Correlation Functions 248 5.14 Fourier Series 251 5.15 Phasor Representation 261 5.16 Phasors and Linear Circuits 265 6 Differential Equation Models for Linear Systems 275 6.1 Introduction 275 6.2 Differential Equations 276 6.3 General Forms of The Solution 278 6.4 First-Order Linear ODE 280 6.4.1 Homogeneous Solution 283 6.4.2 Nonhomogeneous Solution 285 6.4.3 Step Response 287 6.4.4 Exponential Input 287 6.4.5 Sinusoidal Input 289 6.4.6 Impulse Response 290 6.5 Second-Order Linear ODE 294 6.5.1 Homogeneous Solution 296 6.5.2 Damping Ratio 304 6.5.3 Initial Conditions 306 6.5.4 Nonhomogeneous Solution 307 6.6 Second-Order ODE Responses 311 6.6.1 Step Response 311 6.6.2 Step Response (Alternative Method) 313 6.6.3 Impulse Response 319 6.7 Convolution 319 6.8 System of ODEs 323 7 Laplace Transforms and Linear Systems 335 7.1 Introduction 335 7.2 Solving ODEs Using Phasors 336 7.3 Eigenfunctions 339 7.4 Laplace Transform 340 7.5 Laplace Transforms and Generalized Functions 347 7.6 Laplace Transform Properties 352 7.7 Initial and Final Value Theorems 364 7.8 Poles and Zeros 367 7.9 Laplace Transform Pairs 372 7.9.1 Constant Function 372 7.9.2 Rectangle Function 373 7.9.3 Triangle Function 374 7.9.4 Ramped Exponential Function 376 7.9.5 Sinusoidal Functions 376 7.10 Transforms and Polynomials 377 7.11 Solving Linear ODEs 380 7.12 Impulse Response and Transfer Function 382 7.13 Partial Fraction Expansion 387 7.13.1 Distinct Real Poles 388 7.13.2 Distinct Complex Poles 391 7.13.3 Repeated Real Poles 396 7.13.4 Repeated Complex Poles 402 7.14 Laplace Transforms and Linear Circuits 409 8 Fourier Transforms and Frequency Responses 423 8.1 Introduction 423 8.2 Fourier Transform 425 8.3 Magnitude and Phase 435 8.4 Fourier Transforms and Generalized Functions 437 8.5 Fourier Transform Properties 442 8.6 Amplitude Modulation 449 8.7 Frequency Response 453 8.7.1 First-Order Low-Pass Filter 455 8.7.2 First-Order High-Pass Filter 459 8.7.3 Second-Order Band-Pass Filter 460 8.7.4 Second-Order Band-Reject Filter 463 8.8 Frequency Response of Second-Order Filters 466 8.9 Frequency Response of Series RLC Circuit 475 8.10 Butterworth Filters 478 8.10.1 Low-Pass Filter 481 8.10.2 High-Pass Filter 484 8.10.3 Band-Pass Filter 487 8.10.4 Band-Reject Filter 490 APPENDICES 499 Introduction to Appendices 500 A Extended Summaries of Functions and Transforms 501 A.1 Functions and Notation 501 A.2 Laplace Transform 502 A.3 Fourier Transform 504 A.4 Magnitude and Phase 506 A.5 Impulsive Functions 511 A.5.1 Dirac Delta Function (Shifted) 511 A.5.2 Unit Doublet (Shifted) 514 A.6 Piecewise Linear Functions 514 A.6.1 Unit Step Function 514 A.6.2 Signum Function 517 A.6.3 Constant Function (Two-Sided) 517 A.6.4 Ramp Function 521 A.6.5 Absolute Value Function (Two-Sided Ramp) 523 A.6.6 Rectangle Function 524 A.6.7 Triangle Function 528 A.7 Exponential Functions 529 A.7.1 Exponential Function (Right-Sided) 529 A.7.2 Exponential Function (Ramped) 531 A.7.3 Exponential Function (Two-Sided) 533 A.7.4 Gaussian Function 537 A.8 Sinusoidal Functions 539 A.8.1 Cosine Function (Two-Sided) 539 A.8.2 Cosine Function (Right-Sided) 541 A.8.3 Cosine Function (ExponentiallyWeighted) 544 A.8.4 Cosine Function (ExponentiallyWeighted and Ramped) 544 A.8.5 Sine Function (Two-Sided) 549 A.8.6 Sine Function (Right-Sided) 550 A.8.7 Sine Function (ExponentiallyWeighted) 553 A.8.8 Sine Function (ExponentiallyWeighted and Ramped) 554 B Inverse Laplace Transforms 559 B.1 Improper Rational Function 559 B.2 Unbounded System 562 B.3 Double Integrator and Feedback 563 C Identities, Derivatives, and Integrals 565 C.1 Trigonometric Identities 565 C.2 Summations 566 C.3 Miscellaneous 567 C.4 Completing the Square 567 C.5 Quadratic and Cubic Formulas 568 C.6 Derivatives 571 C.7 Indefinite Integrals 573 C.8 Definite Integrals 574 D Set Theory 577 D.1 Sets and Subsets 577 D.2 Set Operations 579 E Series Expansions 583 E.1 Taylor Series 583 E.2 Maclaurin Series 585 E.3 Laurent Series 588 F Lambert W-Function 593 F.1 LambertW-Function 593 F.2 Nonlinear Diode Circuit 597 F.3 System of Nonlinear Equations 598 Glossary 601 Bibliography 609 Index 615

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Book SynopsisA comprehensive guide to using energy principles and variational methods for solving problems in solid mechanics This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates. It begins with a review of the basic equations of mechanics, the concepts of work and energy, and key topics from variational calculus. It presents virtual work and energy principles, energy methods of solid and structural mechanics, Hamilton's principle for dynamical systems, and classical variational methods of approximation. And it takes a more unified approach than that found in most solid mechanics books, to introduce the finite element method. Featuring more than 200 illustrations and tables, this Third Edition has been extensively reorganized and contains much new Table of ContentsAbout the Author xvii About the Companion Website xix Preface to the Third Edition xxi Preface to the Second Edition xxiii Preface to the First Edition xxv 1. Introduction and Mathematical Preliminaries 1 1.1 Introduction 1 1.1.1 Preliminary Comments 1 1.1.2 The Role of Energy Methods and Variational Principles 1 1.1.3 A Brief Review of Historical Developments 2 1.1.4 Preview 4 1.2 Vectors 5 1.2.1 Introduction 5 1.2.2 Definition of a Vector 6 1.2.3 Scalar and Vector Products 8 1.2.4 Components of a Vector 12 1.2.5 Summation Convention 13 1.2.6 Vector Calculus 17 1.2.7 Gradient, Divergence, and Curl Theorems 22 1.3 Tensors 26 1.3.1 Second-Order Tensors 26 1.3.2 General Properties of a Dyadic 29 1.3.3 Nonion Form and Matrix Representation of a Dyad 30 1.3.4 Eigenvectors Associated with Dyads 34 1.4 Summary 39 Problems 40 2. Review of Equations of Solid Mechanics 47 2.1 Introduction 47 2.1.1 Classification of Equations 47 2.1.2 Descriptions of Motion 48 2.2 Balance of Linear and Angular Momenta 50 2.2.1 Equations of Motion 50 2.2.2 Symmetry of Stress Tensors 54 2.3 Kinematics of Deformation 56 2.3.1 Green-Lagrange Strain Tensor 56 2.3.2 Strain Compatibility Equations 62 2.4 Constitutive Equations 65 2.4.1 Introduction 65 2.4.2 Generalized Hooke's Law 66 2.4.3 Plane Stress-Reduced Constitutive Relations 68 2.4.4 Thermoelastic Constitutive Relations 70 2.5 Theories of Straight Beams 71 2.5.1 Introduction 71 2.5.2 The Bernoulli-Euler Beam Theory 73 2.5.3 The Timoshenko Beam Theory 76 2.5.4 The von Ka’rma’n Theory of Beams 81 2.5.4.1 Preliminary Discussion 81 2.5.4.2 The Bernoulli-Euler Beam Theory 82 2.5.4.3 The Timoshenko Beam Theory 84 2.6 Summary 85 Problems 88 3. Work, Energy, and Variational Calculus 97 3.1 Concepts of Work and Energy 97 3.1.1 Preliminary Comments 97 3.1.2 External and Internal Work Done 98 3.2 Strain Energy and Complementary Strain Energy 102 3.2.1 General Development 102 3.2.2 Expressions for Strain Energy and Complementary Strain Energy Densities of Isotropic Linear Elastic Solids 107 3.2.2.1 Stain energy density 107 3.2.2.2 Complementary stain energy density 108 3.2.3 Strain Energy and Complementary Strain Energy for Trusses 109 3.2.4 Strain Energy and Complementary Strain Energy for Torsional Members 114 3.2.5 Strain Energy and Complementary Strain Energy for Beams 117 3.2.5.1 The Bernoulli-Euler Beam Theory 117 3.2.5.2 The Timoshenko Beam Theory 119 3.3 Total Potential Energy and Total Complementary Energy 123 3.3.1 Introduction 123 3.3.2 Total Potential Energy of Beams 124 3.3.3 Total Complementary Energy of Beams 125 3.4 Virtual Work 126 3.4.1 Virtual Displacements 126 3.4.2 Virtual Forces 131 3.5 Calculus of Variations 135 3.5.1 The Variational Operator 135 3.5.2 Functionals 138 3.5.3 The First Variation of a Functional 139 3.5.4 Fundamental Lemma of Variational Calculus 140 3.5.5 Extremum of a Functional 141 3.5.6 The Euler Equations 143 3.5.7 Natural and Essential Boundary Conditions 146 3.5.8 Minimization of Functionals with Equality Constraints 151 3.5.8.1 The Lagrange Multiplier Method 151 3.5.8.2 The Penalty Function Method 153 3.6 Summary 156 Problems 159 4. Virtual Work and Energy Principles of Mechanics 167 4.1 Introduction 167 4.2 The Principle of Virtual Displacements 167 4.2.1 Rigid Bodies 167 4.2.2 Deformable Solids 168 4.2.3 Unit Dummy-Displacement Method 172 4.3 The Principle of Minimum Total Potential Energy and Castigliano's Theorem I 179 4.3.1 The Principle of Minimum Total Potential Energy179 4.3.2 Castigliano's Theorem I 188 4.4 The Principle of Virtual Forces 196 4.4.1 Deformable Solids 196 4.4.2 Unit Dummy-Load Method 198 4.5 Principle of Minimum Total Complementary Potential Energy and Castigliano's Theorem II 204 4.5.1 The Principle of the Minimum total Complementary Potential Energy 204 4.5.2 Castigliano's Theorem II 206 4.6 Clapeyron's, Betti's, and Maxwell's Theorems 217 4.6.1 Principle of Superposition for Linear Problems 217 4.6.2 Clapeyron's Theorem 220 4.6.3 Types of Elasticity Problems and Uniqueness of Solutions 224 4.6.4 Betti's Reciprocity Theorem 226 4.6.5 Maxwell's Reciprocity Theorem 230 4.7 Summary 232 Problems 235 5. Dynamical Systems: Hamilton's Principle 243 5.1 Introduction 243 5.2 Hamilton's Principle for Discrete Systems 243 5.3 Hamilton's Principle for a Continuum 249 5.4 Hamilton's Principle for Constrained Systems 255 5.5 Rayleigh's Method 260 5.6 Summary 262 Problems 263 6. Direct Variational Methods 269 6.1 Introduction 269 6.2 Concepts from Functional Analysis 270 6.2.1 General Introduction 270 6.2.2 Linear Vector Spaces 271 6.2.3 Normed and Inner Product Spaces 276 6.2.3.1 Norm 276 6.2.3.2 Inner product 279 6.2.3.3 Orthogonality 280 6.2.4 Transformations, and Linear and Bilinear Forms 281 6.2.5 Minimum of a Quadratic Functional 282 6.3 The Ritz Method 287 6.3.1 Introduction 287 6.3.2 Description of the Method 288 6.3.3 Properties of Approximation Functions 293 6.3.3.1 Preliminary Comments 293 6.3.3.2 Boundary Conditions 293 6.3.3.3 Convergence 294 6.3.3.4 Completeness 294 6.3.3.5 Requirements on ɸ0 and ɸi 295 6.3.4 General Features of the Ritz Method 299 6.3.5 Examples 300 6.3.6 The Ritz Method for General Boundary-Value Problems 323 6.3.6.1 Preliminary Comments 323 6.3.6.2 Weak Forms 323 6.3.6.3 Model Equation 1 324 6.3.6.4 Model Equation 2 328 6.3.6.5 Model Equation 3 330 6.3.6.6 Ritz Approximations 332 6.4 Weighted-Residual Methods 337 6.4.1 Introduction 337 6.4.2 The General Method of Weighted Residuals 339 6.4.3 The Galerkin Method 44 6.4.4 The Least-Squares Method 349 6.4.5 The Collocation Method 356 6.4.6 The Subdomain Method 359 6.4.7 Eigenvalue and Time-Dependent Problems 361 6.4.7.1 Eigenvalue Problems 361 6.4.7.2 Time-Dependent Problems 362 6.5 Summary 381 Problems 383 7. Theory and Analysis of Plates 391 7.1 Introduction 391 7.1.1 General Comments 391 7.1.2 An Overview of Plate Theories 393 7.1.2.1 The Classical Plate Theory 394 7.1.2.2 The First-Order Plate Theory 395 7.1.2.3 The Third-Order Plate Theory 396 7.1.2.4 Stress-Based Theories 397 7.2 The Classical Plate Theory 398 7.2.1 Governing Equations of Circular Plates 398 7.2.2 Analysis of Circular Plates 405 7.2.2.1 Analytical Solutions For Bending 405 7.2.2.2 Analytical Solutions For Buckling 411 7.2.2.3 Variational Solutions 414 7.2.3 Governing Equations in Rectangular Coordinates 427 7.2.4 Navier Solutions of Rectangular Plates 435 7.2.4.1 Bending 438 7.2.4.2 Natural Vibration 443 7.2.4.3 Buckling Analysis 445 7.2.4.4 Transient Analysis 447 7.2.5 Lévy Solutions of Rectangular Plates 449 7.2.6 Variational Solutions: Bending 454 7.2.7 Variational Solutions: Natural Vibration 470 7.2.8 Variational Solutions: Buckling 475 7.2.8.1 Rectangular Plates Simply Supported along Two Opposite Sides and Compressed in the Direction Perpendicular to Those Sides 475 7.2.8.2 Formulation for Rectangular Plates with Arbitrary Boundary Conditions 478 7.3 The First-Order Shear Deformation Plate Theory 486 7.3.1 Equations of Circular Plates 486 7.3.2 Exact Solutions of Axisymmetric Circular Plates 488 7.3.3 Equations of Plates in Rectangular Coordinates 492 7.3.4 Exact Solutions of Rectangular Plates 496 7.3.4.1 Bending Analysis 498 7.3.4.2 Natural Vibration 501 7.3.4.3 Buckling Analysis 502 7.3.5 Variational Solutions of Circular and Rectangular Plates 503 7.3.5.1 Axisymmetric Circular Plates 503 7.3.5.2 Rectangular Plates 505 7.4 Relationships Between Bending Solutions of Classical and Shear Deformation Theories 507 7.4.1 Beams 507 7.4.1.1 Governing Equations 508 7.4.1.2 Relationships Between BET and TBT 508 7.4.2 Circular Plates 512 7.4.3 Rectangular Plates 516 7.5 Summary 521 Problems 521 8. The Finite Element Method 527 8.1 Introduction 527 8.2 Finite Element Analysis of Straight Bars 529 8.2.1 Governing Equation 529 8.2.2 Representation of the Domain by Finite Elements 530 8.2.3 Weak Form over an Element 531 8.2.4 Approximation over an Element 532 8.2.5 Finite Element Equations 537 8.2.5.1 Linear Element 538 8.2.5.2 Quadratic Element 539 8.2.6 Assembly (Connectivity) of Elements 539 8.2.7 Imposition of Boundary Conditions 542 8.2.8 Postprocessing 543 8.3 Finite Element Analysis of the Bernoulli-Euler Beam Theory 549 8.3.1 Governing Equation 549 8.3.2 Weak Form over an Element 549 8.3.3 Derivation of the Approximation Functions 550 8.3.4 Finite Element Model 552 8.3.5 Assembly of Element Equations 553 8.3.6 Imposition of Boundary Conditions 555 8.4 Finite Element Analysis of the Timoshenko Beam Theory 558 8.4.1 Governing Equations 558 8.4.2 Weak Forms 558 8.4.3 Finite Element Models 559 8.4.4 Reduced Integration Element (RIE) 559 8.4.5 Consistent Interpolation Element (CIE) 561 8.4.6 Superconvergent Element (SCE) 562 8.5 Finite Element Analysis of the Classical Plate Theory 565 8.5.1 Introduction 565 8.5.2 General Formulation 566 8.5.3 Conforming and Nonconforming Plate Elements 568 8.5.4 Fully Discretized Finite Element Models 569 8.5.4.1 Static Bending 569 8.5.4.2 Buckling 569 8.5.4.3 Natural Vibration 570 8.5.4.4 Transient Response 570 8.6 Finite Element Analysis of the First-Order Shear Deformation Plate Theory 574 8.6.1 Governing Equations and Weak Forms 574 8.6.2 Finite Element Approximations 576 8.6.3 Finite Element Model 577 8.6.4 Numerical Integration 579 8.6.5 Numerical Examples 582 8.6.5.1 Isotropic Plates 582 8.6.5.2 Laminated Plates 584 8.7 Summary 587 Problems 588 9. Mixed Variational and Finite Element Formulations 595 9.1 Introduction 595 9.1.1 General Comments 595 9.1.2 Mixed Variational Principles 595 9.1.3 Extremum and Stationary Behavior of Functionals 597 9.2 Stationary Variational Principles 599 9.2.1 Minimum Total Potential Energy 599 9.2.2 The Hellinger-Reissner Variational Principle 601 9.2.3 The Reissner Variational Principle 605 9.3 Variational Solutions Based on Mixed Formulations 606 9.4 Mixed Finite Element Models of Beams 610 9.4.1 The Bernoulli-Euler Beam Theory 610 9.4.1.1 Governing Equations And Weak Forms 610 9.4.1.2 Weak-Form Mixed Finite Element Model 610 9.4.1.3 Weighted-Residual Finite Element Models 613 9.4.2 The Timoshenko Beam Theory 615 9.4.2.1 Governing Equations 615 9.4.2.2 General Finite Element Model 615 9.4.2.3 ASD-LLCC Element 617 9.4.2.4 ASD-QLCC Element 617 9.4.2.5 ASD-HQLC Element 618 9.5 Mixed Finite Element Analysis of the Classical Plate Theory 620 9.5.1 Preliminary Comments 620 9.5.2 Mixed Model I 620 9.5.2.1 Governing Equations 620 9.5.2.2 Weak Forms 621 9.5.2.3 Finite Element Model 622 9.5.3 Mixed Model II 625 9.5.3.1 Governing Equations 625 9.5.3.2 Weak Forms 625 9.5.3.3 Finite Element Model 626 9.6 Summary 630 Problems 631 10. Analysis of Functionally Graded Beams and Plates 635 10.1 Introduction 635 10.2 Functionally Graded Beams 638 10.2.1 The Bernoulli-Euler Beam Theory 638 10.2.1.1 Displacement and strain fields 638 10.2.1.2 Equations of motion and boundary conditions 638 10.2.2 The Timoshenko Beam Theory 639 10.2.2.1 Displacement and strain fields 639 10.2.2.2 Equations of motion and boundary conditions 640 10.2.3 Equations of Motion in terms of Generalized Displacements 641 10.2.3.1 Constitutive Equations 641 10.2.3.2 Stress Resultants of BET 641 10.2.3.3 Stress Resultants of TBT 642 10.2.3.4 Equations of Motion of the BET 642 10.2.3.5 Equations of Motion of the TBT 642 10.2.4 Stiffiness Coefficients643 10.3 Functionally Graded Circular Plates 645 10.3.1 Introduction 645 10.3.2 Classical Plate Theory 646 10.3.2.1 Displacement and Strain Fields 646 10.3.2.2 Equations of Motion 646 10.3.3 First-Order Shear Deformation Theory 647 10.3.3.1 Displacement and Strain Fields 647 10.3.3.2 Equations of Motion 648 10.3.4 Plate Constitutive Relations 649 10.3.4.1 Classical Plate Theory 649 10.3.4.2 First-Order Plate Theory 649 10.4 A General Third-Order Plate Theory 650 10.4.1 Introduction 650 10.4.2 Displacements and Strains 651 10.4.3 Equations of Motion 653 10.4.4 Constitutive Relations 657 10.4.5 Specialization to Other Theories 658 10.4.5.1 A General Third-Order Plate Theory with Traction-Free Top and Bottom Surfaces 658 10.4.5.2 The Reddy Third-Order Plate Theory 661 10.4.5.3 The First-Order Plate Theory 663 10.4.5.4 The Classical Plate Theory 664 10.5 Navier's Solutions 664 10.5.1 Preliminary Comments 664 10.5.2 Analysis of Beams 665 10.5.2.1 Bernoulli-Euler Beams 665 10.5.2.2 Timoshenko Beams 667 10.5.2.3 Numerical Results 669 10.5.3 Analysis of Plates 671 10.5.3.1 Boundary Conditions 672 10.5.3.2 Expansions of Generalized Displacements 672 10.5.3.3 Bending Analysis 673 10.5.3.4 Free Vibration Analysis 676 10.5.3.5 Buckling Analysis 677 10.5.3.6 Numerical Results 679 10.6 Finite Element Models 681 10.6.1 Bending of Beams 681 10.6.1.1 Bernoulli-Euler Beam Theory 681 10.6.1.2 Timoshenko Beam Theory 683 10.6.2 Axisymmetric Bending of Circular Plates 684 10.6.2.1 Classical Plate Theory 681 10.6.2.2 First-Order Shear Deformation Plate Theory 686 10.6.3 Solution of Nonlinear Equations 688 10.6.3.1 Times approximation 688 10.6.3.2 Newton's Iteration Approach 688 10.6.3.3 Tangent Stiffiness Coefficients for the BET 690 10.6.3.4 Tangent Stiffiness Coefficients for the TBT 692 10.6.3.5 Tangent Stiffiness Coefficients for the CPT 693 10.6.3.6 Tangent Stiffiness Coefficients for the FSDT 693 10.6.4 Numerical Results for Beams and Circular Plates 694 10.6.4.1 Beams 694 10.6.4.2 Circular Plates 697 10.7 Summary 699 Problems 700 References 701 Answers to Most Problems 711 Index 723

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Book SynopsisCovers multivariable calculus, starting from the basics and leading up to the three theorems of Green, Gauss, and Stokes, but always with an eye on practical applications. Written for a wide spectrum of undergraduate students by an experienced author, this book provides a very practical approach to advanced calculusstarting from the basics and leading up to the theorems of Green, Gauss, and Stokes. It explains, clearly and concisely, partial differentiation, multiple integration, vectors and vector calculus, and provides end-of-chapter exercises along with their solutions to aid the readers' understanding. Written in an approachable style and filled with numerous illustrative examples throughout, Two and Three Dimensional Calculus: with Applications in Science and Engineering assumes no prior knowledge of partial differentiation or vectors and explains difficult concepts with easy to follow examples. Rather than concentrating on mathematical structures, tTable of ContentsPreface xi 1 Revision of One-Dimensional Calculus 1 1.1 Limits and Convergence 1 1.2 Differentiation 3 1.2.1 Rules for Differentiation 5 1.2.2 Mean Value Theorem 7 1.2.3 Taylor’s Series 8 1.2.4 Maxima and Minima 12 1.2.5 Numerical Differentiation 13 1.3 Integration 16 Exercises 22 2 Partial Differentiation 25 2.1 Introduction 25 2.2 Differentials 29 2.2.1 Small Errors 30 2.3 Total Derivative 33 2.4 Chain Rule 36 2.4.1 Leibniz Rule 39 2.4.2 Chain Rule in n Dimensions 41 2.4.3 Implicit Functions 42 2.5 Jacobian 43 2.6 Higher Derivatives 46 2.6.1 Higher Differentials 49 2.7 Taylor’sTheorem 50 2.8 Conjugate Functions 52 2.9 Case Study:Thermodynamics 54 Exercises 58 3 Maxima and Minima 61 3.1 Introduction 61 3.2 Maxima, Minima and Saddle Points 63 3.3 Lagrange Multipliers 74 3.3.1 Generalisations 77 3.4 Optimisation 81 3.4.1 Hill Climbing Techniques 81 Exercises 85 4 Vector Algebra 89 4.1 Introduction 89 4.2 Vector Addition 90 4.3 Components 92 4.4 Scalar Product 94 4.5 Vector Product 97 4.5.1 Scalar Triple Product 102 4.5.2 Vector Triple Product 105 Exercises 106 5 Vector Differentiation 109 5.1 Introduction 109 5.2 Differential Geometry 111 5.2.1 Space Curves 112 5.2.2 Surfaces 120 5.3 Mechanics 129 Exercises 135 6 Gradient, Divergence, and Curl 139 6.1 Introduction 139 6.2 Gradient 139 6.3 Divergence 143 6.4 Curl 145 6.5 Vector Identities 146 6.6 Conjugate Functions 151 Exercises 154 7 Curvilinear Co-ordinates 157 7.1 Introduction 157 7.2 Curved Axes and Scale Factors 157 7.3 Curvilinear Gradient, Divergence, and Curl 161 7.3.1 Gradient 161 7.3.2 Divergence 163 7.3.3 Curl 165 7.4 Further Results and Tensors 166 7.4.1 Tensor Notation 166 7.4.2 Covariance and Contravariance 168 Exercises 171 8 PathIntegrals 173 8.1 Introduction 173 8.2 Integration Along a Curve 173 8.3 Practical Applications 181 Exercises 186 9 Multiple Integrals 191 9.1 Introduction 191 9.2 The Double Integral 191 9.2.1 Rotation and Translation 199 9.2.2 Change of Order of Integration 201 9.2.3 Plane Polar Co-ordinates 203 9.2.4 Applications of Double Integration 208 9.3 Triple Integration 213 9.3.1 Cylindrical and Spherical Polar Co-ordinates 219 9.3.2 Applications of Triple Integration 227 Exercises 233 10 Surface Integrals 241 10.1 Introduction 241 10.2 Green’s Theorem in the Plane 242 10.3 Integration over a Curved Surface 246 10.4 Applications of Surface Integration 253 Exercises 256 11 Integral Theorems 259 11.1 Introduction 259 11.2 Stokes’ Theorem 260 11.3 Gauss’ DivergenceTheorem 268 11.3.1 Green’s Second Identity 275 11.4 Co-ordinate-Free Definitions 277 11.5 Applications of Integral Theorems 279 11.5.1 Electromagnetic Theory 279 11.5.1.1 Maxwell’s Equations 279 11.5.2 Fluid Mechanics 283 11.5.3 ElasticityTheory 287 11.5.4 Heat Transfer 297 Exercises 298 12 Solutions and Answers to Exercises 301 References 375 Index 377

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