Multivariable and Vector Analysis – Lectures
by WWL Chen
A — MULTIVARIABLE ANALYSIS
Chapt 1: FUNCTIONS OF SEVERAL VARIABLES
* Basic Definitions
* Open Sets
* Limits and Continuity
* Limits and Continuity: Proofs
Chapt 2: DIFFERENTIATION
* Partial Derivatives
* Total Derivatives
* Consequences of Differentiability
* Conditions for Differentiability
* Properties of the Derivative
* Gradients and Directional Derivatives
Chapt 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
Chapt 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
B — VECTOR ANALYSIS
Chapt 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
Chapt 9: INTEGRALS OVER PATHS
* Integrals of Scalar Functions over Paths
* Line Integrals
* Equivalent Paths
* Simple Curves
Chapt 10: PARAMETRIZED SURFACES
* Introduction
* Surface Area
Chapt 11: INTEGRALS OVER SURFACES
* Integrals of Scalar Functions over Parametrized Surfaces
* Surface Integrals
* Equivalent Parametrized Surfaces
* Parametrization of Surfaces
Chapt 12: INTEGRATION THEOREMS
* Green’s Theorem
* Stokes’s Theorem
* Gauss’s Theorem
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