Advanced Calculus: An Introduction to Modern Analysis

Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra
Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis.

Mathematical Analysis: An Introduction to Functions of Several Variables may be used in the classroom setting for advanced undergraduate and graduate students or as a self-study. It is also a valuable reference for researchers in most mathematical disciplines. An appendix highlights mathematicians and scientists who have made important contributions in the development of theories in the subject.

There are two major changes in the Fourth Edition of Differential Equations and Their Applications. The first concerns the computer programs in this text. In keeping with recent trends in computer science, we have replaced all the APL programs with Pascal and C programs. The Pascal programs appear in the text in place of the APL programs, where they are followed by the Fortran programs, while the C programs appear in Appendix C.

With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones–the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes–the text treats other important topics in differential analysis, such as Morse's lemma and the Poincaré lemma.

This innovative textbook bridges the gap between undergraduate analysis and graduate measure theory by guiding students from the classical foundations of analysis to more modern topics like metric spaces and Lebesgue integration. Designed for a two-semester introduction to real analysis, the text gives special attention to metric spaces and topology to familiarize students with the level of abstraction and mathematical rigor needed for graduate study in real analysis. Fitting in between analysis textbooks that are too formal or too casual, From Classical to Modern Analysis is a comprehensive, yet straightforward, resource for studying real analysis.