Calculus and Analysis: A Combined Approach
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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 mathematical logic. Next, the book explores the concept of sets and maps, providing readers with a strong foundation for understanding and solving modern mathematical problems. Ensuring a complete presentation, topics are uniformly presented in chapters that consist of three parts:

  • Introductory Motivations presents historical mathematical problems or problems arising from applications that led to the development of mathematical solutions

  • Theory provides rigorous development of the essential parts of the machinery of analysis; proofs are intentionally detailed, but simplified as much as possible to aid reader comprehension

  • Examples and Problems promotes problem-solving skills through application-based exercises that emphasize theoretical mechanics, general relativity, and quantum mechanics

Calculus and Analysis: A Combined Approach is an excellent book for courses on calculus and mathematical analysis at the upper-undergraduate and graduate levels. It is also a valuable resource for engineers, physicists, mathematicians, and anyone working in the applied sciences who would like to master their understanding of basic tools in modern calculus and analysis.


Horst R. Beyer, PHD, is Professor of Mathematics at the Institute for Physics and Mathematics at the University of Michoacán (Mexico). Dr. Beyer has written numerous published articles in his areas of research interest, which include functional analysis, operator theory, partial differential equations, and semigroup theory.



0.1 Short Introduction.

0.2 Introduction.


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.


Index of Notation.

Index of Terminology.

Problem Solutions.