{"product_id":"mathematical-analysis-9780470107966","title":"Mathematical Analysis","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eA self-contained introduction to the fundamentals of mathematical analysis  Mathematical Analysis: A Concise Introduction presents the foundations of analysis and illustrates its role in mathematics.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"This highly original, interesting and very useful book includes over 900 exercises which are ranging in levels of difficulty, from conceptual questions and adaptations of proofs to proofs with and without hints.\" (\u003ci\u003eMathematical Reviews\u003c\/i\u003e, 2008h)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface xi\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart I: Analysis of Functions of a Single Real Variable\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 The Real Numbers 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Field Axioms 1\u003c\/p\u003e \u003cp\u003e1.2 Order Axioms 4\u003c\/p\u003e \u003cp\u003e1.3 Lowest Upper and Greatest Lower Bounds 8\u003c\/p\u003e \u003cp\u003e1.4 Natural Numbers, Integers, and Rational Numbers 11\u003c\/p\u003e \u003cp\u003e1.5 Recursion, Induction, Summations, and Products 17\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Sequences of Real Number V 25\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Limits 25\u003c\/p\u003e \u003cp\u003e2.2 Limit Laws 30\u003c\/p\u003e \u003cp\u003e2.3 Cauchy Sequences 36\u003c\/p\u003e \u003cp\u003e2.4 Bounded Sequences 40\u003c\/p\u003e \u003cp\u003e2.5 Infinite Limits 44\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Continuous Functions 49\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Limits of Functions 49\u003c\/p\u003e \u003cp\u003e3.2. Limit Laws 52\u003c\/p\u003e \u003cp\u003e3.3 One-Sided Limits and Infinite Limits 56\u003c\/p\u003e \u003cp\u003e3.4 Continuity 59\u003c\/p\u003e \u003cp\u003e3.5 Properties of Continuous Functions 66\u003c\/p\u003e \u003cp\u003e3.6 Limits at Infinity 69\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Differentiable Functions 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Differentiability 71\u003c\/p\u003e \u003cp\u003e4.2 Differentiation Rules 74\u003c\/p\u003e \u003cp\u003e4.3 Rolle's Theorem and the Mean Value Theorem 80\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 The Riemann Integral I 85\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Riemann Sums and the Integral 85\u003c\/p\u003e \u003cp\u003e5.2 Uniform Continuity and Integrability of Continuous Functions 91\u003c\/p\u003e \u003cp\u003e5.3 The Fundamental Theorem of Calculus 95\u003c\/p\u003e \u003cp\u003e5.4 The Darboux Integral 97\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Series of Real Numbers I 101\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Series as a Vehicle to Define Infinite Sums 101\u003c\/p\u003e \u003cp\u003e6.2 Absolute Convergence and Unconditional Convergence 108\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Some Set Theory 117\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 The Algebra of Sets 117\u003c\/p\u003e \u003cp\u003e7.2 Countable Sets 122\u003c\/p\u003e \u003cp\u003e7.3 Uncountable Sets 124\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 The Riemann Integral II 127\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Outer Lebesgue Measure 127\u003c\/p\u003e \u003cp\u003e8.2 Lebesgue's Criterion for Riemann Integrability 131\u003c\/p\u003e \u003cp\u003e8.3 More Integral Theorems 136\u003c\/p\u003e \u003cp\u003e8.4 Improper Riemann Integrals 140\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 The Lebesgue Integral 145\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Lebesgue Measurable Sets 147\u003c\/p\u003e \u003cp\u003e9.2 Lebesgue Measurable Functions 153\u003c\/p\u003e \u003cp\u003e9.3 Lebesgue Integration 158\u003c\/p\u003e \u003cp\u003e9.4 Lebesgue Integrals versus Riemann Integrals 165\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Series of Real Numbers II 169\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Limits Superior and Inferior 169\u003c\/p\u003e \u003cp\u003e10.2 The Root Test and the Ratio Test 172\u003c\/p\u003e \u003cp\u003e10.3 Power Series 175\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Sequences of Functions 179\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Notions of Convergence 179\u003c\/p\u003e \u003cp\u003e11.2 Uniform Convergence 182\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Transcendental Functions 189\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 The Exponential Function 189\u003c\/p\u003e \u003cp\u003e12.2 Sine and Cosine 193\u003c\/p\u003e \u003cp\u003e12.3 L.' Hôpital's Rule 199\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Numerical Methods 203\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Approximation with Taylor Polynomials 204\u003c\/p\u003e \u003cp\u003e13.2 Newton's Method 208\u003c\/p\u003e \u003cp\u003e13.3 Numerical Integration 214\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II: Analysis in Abstract Spaces\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Integration on Measure Spaces 225\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Measure Spaces 225\u003c\/p\u003e \u003cp\u003e14.2 Outer Measures 230\u003c\/p\u003e \u003cp\u003e14.3 Measurable Functions 234\u003c\/p\u003e \u003cp\u003e14.4 Integration of Measurable Functions 235\u003c\/p\u003e \u003cp\u003e14.5 Monotone and Dominated Convergence 238\u003c\/p\u003e \u003cp\u003e14.6 Convergence in Mean, in Measure, and Almost Everywhere 242\u003c\/p\u003e \u003cp\u003e14.7 Product Ϭ-Algebras 245\u003c\/p\u003e \u003cp\u003e14.8 Product Measures and Fubini's Theorem 251\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 The Abstract Venues for Analysis 255\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Abstraction I: Vector Spaces 255\u003c\/p\u003e \u003cp\u003e15.2 Representation of Elements; Bases and Dimension 259\u003c\/p\u003e \u003cp\u003e15.3 Identification of Spaces: Isomorphism 262\u003c\/p\u003e \u003cp\u003e15.4 Abstraction II: Inner Product Spaces 264\u003c\/p\u003e \u003cp\u003e15.5 Nicer Representations: Orthonormal Sets 267\u003c\/p\u003e \u003cp\u003e15.6 Abstraction III: Norrned Spaces 269\u003c\/p\u003e \u003cp\u003e15.7 Abstraction IV: Metric Spaces 275\u003c\/p\u003e \u003cp\u003e15.8 \u003ci\u003eL\u003csup\u003eP\u003c\/sup\u003e\u003c\/i\u003e Spaces 278\u003c\/p\u003e \u003cp\u003e15.9 Another Number Field: Complex Numbers 281\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 The Topology of Metric Spaces 287\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Convergence of Sequences 287\u003c\/p\u003e \u003cp\u003e16.2 Completeness 291\u003c\/p\u003e \u003cp\u003e16.3 Continuous Functions 296\u003c\/p\u003e \u003cp\u003e16.4 Open and Closed Sets 301\u003c\/p\u003e \u003cp\u003e16.5 Compactness 309\u003c\/p\u003e \u003cp\u003e16.6 The Normed Topology of R\u003ci\u003e\u003csup\u003ed\u003c\/sup\u003e\u003c\/i\u003e 316\u003c\/p\u003e \u003cp\u003e16.7 Dense Subspaces 322\u003c\/p\u003e \u003cp\u003e16.8 Connectedness 330\u003c\/p\u003e \u003cp\u003e16.9 Locally Compact Spaces 333\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Differentiation in Normed Spaces 341\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Continuous Linear Functions 342\u003c\/p\u003e \u003cp\u003e17.2 Matrix Representation of Linear Functions 348\u003c\/p\u003e \u003cp\u003e17.3 Differentiability 353\u003c\/p\u003e \u003cp\u003e17.4 The Mean Value Theorem 360\u003c\/p\u003e \u003cp\u003e17.5 How Partial Derivatives Fit In 362\u003c\/p\u003e \u003cp\u003e17.6 Multilinear Functions (Tensors) 369\u003c\/p\u003e \u003cp\u003e17.7 Higher Derivatives 373\u003c\/p\u003e \u003cp\u003e17.8 The. Implicit Function Theorem 380\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Measure, Topology, and Differentiation 385\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Lebesgue Measurable Sets in R\u003ci\u003e\u003csup\u003ed\u003c\/sup\u003e\u003c\/i\u003e 385\u003c\/p\u003e \u003cp\u003e18.2 C\u003csup\u003eꝏ\u003c\/sup\u003e and Approximation of Integrable Functions 391\u003c\/p\u003e \u003cp\u003e18.3 Tensor Algebra and Determinants 397\u003c\/p\u003e \u003cp\u003e18.4 Multidimensional Substitution 407\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Introduction to Differential Geometry 421\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 Manifolds 421\u003c\/p\u003e \u003cp\u003e19.2 Tangent Spaces and Differentiable Functions 427\u003c\/p\u003e \u003cp\u003e19.3 Differential Forms, Integrals Over the Unit Cube 434\u003c\/p\u003e \u003cp\u003e19.4 \u003ci\u003ek\u003c\/i\u003e-Forms and Integrals Over \u003ci\u003ek\u003c\/i\u003e-Chains 443\u003c\/p\u003e \u003cp\u003e19.5 Integration on Manifolds 452\u003c\/p\u003e \u003cp\u003e19.6 Stokes' Theorem 458\u003c\/p\u003e \u003cp\u003e\u003cb\u003e20 Hilbert Spaces 463\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e20.1 Orthonormal Bases 463\u003c\/p\u003e \u003cp\u003e20.2 Fourier Series 467\u003c\/p\u003e \u003cp\u003e20.3 The Riesz Representation Theorem 475\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart III: Applied Analysis\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e21 Physics Background 483\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21.1 Harmonic Oscillators 484\u003c\/p\u003e \u003cp\u003e21.2 Heat and Diffusion 486\u003c\/p\u003e \u003cp\u003e21.3 Separation of Variables, Fourier Series, and Ordinary Differential Equa-tions 490\u003c\/p\u003e \u003cp\u003e21.4 Maxwell's Equations 493\u003c\/p\u003e \u003cp\u003e21.5 The Navier Stokes Equation for the Conservation of Mass 496\u003c\/p\u003e \u003cp\u003e\u003cb\u003e22 Ordinary Differential Equations 505\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e22.1 Burwell Space Valued Differential Equations 505\u003c\/p\u003e \u003cp\u003e22.2 An Existence and Uniqueness Theorem 508\u003c\/p\u003e \u003cp\u003e22.3 Linear Differential Equations 510\u003c\/p\u003e \u003cp\u003e\u003cb\u003e23 The Finite Element Method 513\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e23.1 Ritz-Galerkin Approximation 513\u003c\/p\u003e \u003cp\u003e23.2 Wealth Differentiable Functions 518\u003c\/p\u003e \u003cp\u003e23,3 Sobolev Spaces 524\u003c\/p\u003e \u003cp\u003e23.4 Elliptic Differential Operators 532\u003c\/p\u003e \u003cp\u003e23.5 Finite Elements 536\u003c\/p\u003e \u003cp\u003eConclusion and Outlook 544\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendices\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eA Logic 545\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 Statements 545\u003c\/p\u003e \u003cp\u003eA.2 Negations 546\u003c\/p\u003e \u003cp\u003e\u003cb\u003eB Set Theory 547\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB. 1 The Zermelo-Fraenkel Axioms 547\u003c\/p\u003e \u003cp\u003eB.2 Relations and Functions 548\u003c\/p\u003e \u003cp\u003e\u003cb\u003eC Natural Numbers, Integers, and Rational Numbers 549\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eC.1 The Natural Numbers 549\u003c\/p\u003e \u003cp\u003eC.2 The Integers 550\u003c\/p\u003e \u003cp\u003eC.3 The Rational Numbers 550\u003c\/p\u003e \u003cp\u003eBibliography 551\u003c\/p\u003e \u003cp\u003eIndex 553\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49402287128919,"sku":"9780470107966","price":95.36,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780470107966.jpg?v=1730479952","url":"https:\/\/bookcurl.com\/products\/mathematical-analysis-9780470107966","provider":"Book Curl","version":"1.0","type":"link"}