This book covers the standard material for a one-semester course in multivariable calculus. The topics include curves, differentiability, partial derivatives, multiple integrals, vector fields, line and surface integrals, and the theorems of Green, Stokes, and Gauss. Roughly speaking, the book is organized into three main parts corresponding to the type of function being studied: vector-valued functions of one variable, real-valued functions of many variables, and finally, the general case of vector-valued functions of many variables.
As is always the case, the most productive way for students to learn is by doing problems, and the book is written to get to the exercises as quickly as possible. The presentation is geared towards students who enjoy learning mathematics for its own sake. As a result, there is a priority placed on understanding why things are true and recognizing that they should be acknowledged when details are sketched or omitted. Otherwise, the level of rigor is fairly normal. Matrices are introduced and used freely. Prior experience with linear algebra is helpful but not required.