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

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

• 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