Multivariable and Vector Analysis – Lectures


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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