Advanced Calculus: An Introduction to Classical Analysis (Dover Books on Mathematics)

A course in analysis that focuses on the functions of a real variable, this text is geared toward upper-level undergraduate students. It introduces the basic concepts in their simplest setting and illustrates its teachings with numerous examples, practical theorems, and coherent proofs.
Starting with the structure of the system of real and complex numbers, the text deals at length with the convergence of sequences and series and explores the functions of a real variable and of several variables. Subsequent chapters offer a brief and self-contained introduction to vectors that covers important aspects, including gradients, divergence, and rotation. An entire chapter is devoted to the reversal of order in limiting processes, and the treatment concludes with an examination of Fourier series.