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.
The material is available free to all
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or without permission from the author. However, the documents
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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
- Gradients and Directional Derivatives
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
Chapter 10: PARAMETRIZED
SURFACES
- 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