Calculus And Analysis - a Combined Approach

This book provides a strong orientation towards calculations and up-to-date mathematical methods to discuss the development of calculus. In addition, special care was taken to include material that is needed early on in the applied sciences, particularly differential equations. Readers are provided with basic tools that are essential to the understanding of modern mathematics. The book begins with an introduction to the concepts of sets and maps as well as a brief review of proofs. Each chapter contains historical mathematical problems and/or problems from applications that lead to the development of the discussed mathematics. Also, wherever possible, motivation is taken from applications. This is suitable, in particular, for readers from the natural sciences and engineering. This includes introductions to sections on improper integrals that use motivation from the mechanics of periodic motion where improper integrals occur naturally in the analysis. A large number of examples and problems consider basic problems related to theoretical mechanics, general relativity, and quantum mechanics. For cases where prototypical problems seemed unavailable, pure historical sketches of the development are used for the purpose of motivation. The book also contains theoretical sections in each chapter that provide a discussion on the rigorous development of essential parts of the machinery of analysis. The proofs are intentionally very detailed and have been simplified as much as possible in aid in reader comprehension. The example and problems found in the book are mostly calculation-based, as is appropriate for calculus courses suitable for students in the applied sciences. Examples appear throughout as fully worked problems, and as a result, they do not only exemplify the theory, but also teach problem solving skills.