Description
Book SynopsisA NEW APPROACH TO CALCULUS THAT BETTER ENABLES STUDENTS TO PROGRESS TO MORE ADVANCED COURSES AND APPLICATIONS Calculus and Analysis: A Combined Approach bridges the gap between mathematical thinking skills and advanced calculus topics by providing an introduction to the key theory for understanding and working with applications in engineering and the sciences. Through a modern approach that utilizes fully calculated problems, the book addresses the importance of calculus and analysis in the applied sciences, with a focus on differential equations.
Differing from the common classical approach to the topic, this book presents a modern perspective on calculus that follows motivations from Otto Toeplitz''s famous genetic model. The result is an introduction that leads to great simplifications and provides a focused treatment commonly found in the applied sciences, particularly differential equations. The author begins with a short introduction to elementary mathema
Table of Contents
Preface.
0.1 Short Introduction.
0.2 Introduction.
Acknowledgments.
1 Calculus I.
1.1 A Sketch of the Development of Rigor in Calculus and Analysis.
1.2 Basics.
1.3 Limits and Continuous Functions.
1.4 Differentiation.
1.5 Applications of Differentiation.
1.6 Riemann Integration.
2 Calculus II.
2.1 Techniques of Integration.
2.2 Improper Integrals.
2.3 Series of Real Numbers.
2.4 Series of Functions.
2.5 Analytical Geometry and Elementary Vector Calculus.
3 Calculus III.
3.1 Vector-Valued Functions of Several Variables.
3.2 Derivatives of Vector-Valued Functions of Several Variables.
3.3 Applications of Differentiation.
3.4 Integration of Functions of Several Variables.
3.5 Vector Calculus.
3.6 Generalizations of the Fundamental Theorem of Calculus.
Appendix A.
A.1 Construction of the Real-Number System.
A.2 The Lebesgue Criterion for Riemann Integrability.
A.3 Properties of the Determinant.
A.4 The Inverse Mapping Theorem.
References.
Index of Notation.
Index of Terminology.
Problem Solutions.
