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.
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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