LECTURE NOTES OF WILLIAM CHEN

# MULTIVARIABLE AND VECTOR ANALYSIS

This set of notes has been organized in such a way to create a single volume suitable for an introduction to some of the basic ideas in multivariable and vector analysis. As a prerequisite, the reader is expected to have a reasonable understanding of first year calculus.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

## SECTION A --- MULTIVARIABLE ANALYSIS

### Chapter 1 : FUNCTIONS OF SEVERAL VARIABLES >>

• Basic Definitions
• Open Sets
• Limits and Continuity
• Limits and Continuity: Proofs

### Chapter 2 : DIFFERENTIATION >>

• Partial Derivatives
• Total Derivatives
• Consequences of Differentiability
• Conditions for Differentiability
• Properties of the Derivative

### Chapter 3 : IMPLICIT AND INVERSE FUNCTION THEOREMS >>

• Implicit Function Theorem
• Inverse Function Theorem

### Chapter 4 : HIGHER ORDER DERIVATIVES >>

• Iterated Partial Derivatives
• Taylor's Theorem
• Stationary Points
• Functions of Two Variables
• Constrained Maxima and Minima

### Chapter 5 : DOUBLE AND TRIPLE INTEGRALS >>

• Introduction
• Double Integrals over Rectangles
• Conditions for Integrability
• Double Integrals over Special Regions
• Fubini's Theorem
• Mean Value Theorem
• Triple Integrals

### Chapter 6 : CHANGE OF VARIABLES >>

• Introduction
• Planar Transformations
• The Jacobian
• Triple Integrals

## SECTION B --- VECTOR ANALYSIS

### Chapter 7 : PATHS >>

• Introduction
• Differentiable Paths
• Arc Length

### Chapter 8 : VECTOR FIELDS >>

• Introduction
• Divergence of a Vector Field
• Curl of a Vector Field
• Basic Identities of Vector Analysis

### Chapter 9 : INTEGRALS OVER PATHS >>

• Integrals of Scalar Functions over Paths
• Line Integrals
• Equivalent Paths
• Simple Curves

• Introduction
• Surface Area

### Chapter 11 : INTEGRALS OVER SURFACES >>

• Integrals of Scalar Functions over Parametrized Surfaces
• Surface Integrals
• Equivalent Parametrized Surfaces
• Parametrization of Surfaces

### Chapter 12 : INTEGRATION THEOREMS >>

• Green's Theorem
• Stokes's Theorem
• Gauss's Theorem