**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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