Multivariable and Vector Analysis

by WWL Chen

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.

To read the notes, click the chapters below for connection to the appropriate PDF files. You will need Adobe Acrobat Reader Version 4.0 or later.

The material is available free to all individuals, on the understanding that it is not to be used for financial gains, 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 (last uploaded on 2 December 2002)

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

Chapter 2: DIFFERENTIATION (last uploaded on 2 December 2002)

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

Chapter 3: IMPLICIT AND INVERSE FUNCTION THEOREMS (last uploaded on 29 August 2007)

• Implicit Function Theorem
• Inverse Function Theorem

Chapter 4: HIGHER ORDER DERIVATIVES (last uploaded on 2 December 2002)

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

Chapter 5: DOUBLE AND TRIPLE INTEGRALS (last uploaded on 3 April 2006)

• 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 (last uploaded on 2 December 2002)

• Introduction
• Planar Transformations
• The Jacobian
• Triple Integrals

SECTION B --- VECTOR ANALYSIS

Chapter 7: PATHS (last uploaded on 2 December 2002)

• Introduction
• Differentiable Paths
• Arc Length

Chapter 8: VECTOR FIELDS (last uploaded on 2 December 2002)

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

Chapter 9: INTEGRALS OVER PATHS (last uploaded on 2 December 2002)

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

Chapter 10: PARAMETRIZED SURFACES (last uploaded on 25 September 2003)

• Introduction
• Surface Area

Chapter 11: INTEGRALS OVER SURFACES (last uploaded on 2 December 2002)

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

Chapter 12: INTEGRATION THEOREMS (last uploaded on 2 December 2002)

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