## Multivariable Calculus

### George Cain & James Herod

This is a textbook for a course in multivariable calculus. It has been used for the past few years here at Georgia Tech. The notes are available as Adobe Acrobat documents. If you do not have an Adobe Acrobat Reader, you may down-load a copy, free of charge, from Adobe.

Chapter One - Euclidean Three Space
1.1 Introduction
1.2 Coordinates in Three-Space
1.3 Some Geometry
1.4 Some More Geometry--Level Sets

Chapter Two - Vectors--Algebra and Geometry
2.1 Vectors
2.2 Scalar Product
2.3 Vector Product

Chapter Three - Vector Functions
3.1 Relations and Functions
3.2 Vector Functions
3.3 Limits and Continuity

Chapter Four - Derivatives
4.1 Derivatives
4.2 Geometry of Space Curves--Curvature
4.3 Geometry of Space Curves--Torsion
4.4 Motion

Chapter Five - More Dimensions
5.1 The space Rn
5.2 Functions

Chapter Six - Linear Functions and Matrices
6.1 Matrices
6.2 Matrix Algebra

Chapter Seven - Continuity, Derivatives, and All That
7.1 Limits and Continuity
7.2 Derivatives
7.3 The Chain Rule

Chapter Eight - f:Rn-› R
8.1 Introduction
8.2 The Directional Derivative
8.3 Surface Normals
8.4 Maxima and Minima
8.5 Least Squares
8.6 More Maxima and Minima
8.7 Even More Maxima and Minima

Chapter Nine - The Taylor Polynomial
9.1 Introduction
9.2 The Taylor Polynomial
9.3 Error
Supplementary material for Taylor polynomial in several variables.

Chapter Ten - Sequences, Series, and All That
10.1 Introduction
10.2 Sequences
10.3 Series
10.4 More Series
10.5 Even More Series
10.6 A Final Remark

Chapter Eleven - Taylor Series
11.1 Power Series
11.2 Limit of a Power Series
11.3 Taylor Series

Chapter Twelve - Integration
12.1 Introduction
12.2 Two Dimensions

Chapter Thirteen - More Integration
13.1 Some Applications
13.2 Polar Coordinates
13.3 Three Dimensions

Chapter Fourteen - One Dimension Again
14.1 Scalar Line Integrals
14.2 Vector Line Integrals
14.3 Path Independence

Chapter Fifteen - Surfaces Revisited
15.1 Vector Description of Surfaces
15.2 Integration

Chapter Sixteen - Integrating Vector Functions
16.1 Introduction
16.2 Flux

Chapter Seventeen - Gauss and Green
17.1 Gauss's Theorem
17.2 Green's Theorem
17.3 A Pleasing Application

Chapter Eighteen - Stokes
18.1 Stokes's Theorem
18.2 Path Independence Revisited

Chapter Ninteen - Some Physics
19.1 Fluid Mechanics
19.2 Electrostatics

20 March 2000