The Calculus of Functions of Several Variables

by Dan Sloughter, Furman University


Chapter 1: Geometry of  \(\mathbb{R}^n\)

Introduction to \(\mathbb{R}^n\)

Angles and the dot product

The cross product

Lines, planes, and hyperplanes

Linear and affine functions

Operations with matrices

Chapter 2: Functions from \(\mathbb{R}\) to \(\mathbb{R}^n\)


Best affine approximations

Motion along a curve

Chapter 3:  Functions from \(\mathbb{R}^n\) to \(\mathbb{R}\)

Geometry, limits, and continuity

Directional derivatives and the gradient

Best affine approximations

Second-order approximations

Extreme values

Definite integrals

Change of variables in definite integrals

Chapter 4:  Functions from \(\mathbb{R}^m\) to \(\mathbb{R}^n\)

Geometry, limits, and continuity

Best affine approximations

Line integrals

Green’s theorem

Answers for selected problems are available here.

A concatenated version (single download) is available here. This file is approximately 2.1 MB.

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Dan Sloughter, Department of Mathematics, Furman University, Greenville, SC, 29613