Differential Calculus in Several Variables A Learning By Doing Approach 1st Edition

Differential Calculus in Several Variables A Learning By Doing Approach 1st Edition

1  Vectors and Sets in Rm  

1.1   Vectors in Rm

1.2 Lines and Planes in R3

1.3 Points or Vectors?

1.4 Convergent Sequences in Rm

1.5 Sets in Rm

2  Functions of Several Variables

2.1 Functions, Domains and Codomains

2.2 The Graph of a Function

2.3 Level Sets

2.4 Quadric Surfaces

2.5 Curves

3  Limits and Continuity

3.1 Limit of a Function

3.2 Two-Path Test for the Nonexistence of a Limit

3.3 Functions with Limit at a Point

3.4 Continuous Functions

3.5 Continuous Extensions

4  Differentiable Functions 

4.1 Partial Derivatives

4.2 The Tangent Plane to a Surface

4.3 Differentiable Functions

4.4 A Criterion for Differentiability

4.5 Differentiability of Vector Valued Functions

5  Chain Rule

5.1 Chain Rule for Several Variable Functions

5.2 Implicit Differentiation

5.3 Mean Value Theorem in Several Variables

6  Directional Derivative

6.1 Directional Derivative

6.2 A Geometric Insight

6.3 The Gradient and the Level Sets

7  Second Order Derivatives

7.1 Second Order Derivatives

7.2 Chain Rule for Second Order Derivatives

7.3 The Laplace Operator

8  Taylor’s Theorem

8.1 Higher Order Derivatives

8.3   Taylor’s Theorem

8.3 Linear and Quadratic Approximation

9  Implicit Function Theorem

9.1 Preliminaries

9.2 Two Variable Case

9.3 Three Variable Case

9.4 The General Case

10 Local and Global Extrema

10.1 Extreme Values and Critical Points

10.2 Second Order Derivative Test

10.3 The Inconclusive Case

11 Constrained Optimization 

11.1 Motivation

11.2 Lagrange Multipliers Method, Part I

11.3 Lagrange Multipliers Method, Part II

11.4 Extrema of Functions on General Compact Sets

11.5 Some Applications to Business and Economics

12 Solutions

Appendix: Useful Facts in Linear Algebra

A.1 Basics

A.2 Determinants

A.3 Inverse Matrices

A.4 The Rank of a Matrix

A.5 Positive and Negative Definite Matrices