1st Edition
Differential Calculus in Several Variables A Learning-by-Doing Approach
Preface
Foreword to the Student
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
Bibliography
Index
Biography
Marius Ghergu is an Associate Professor at University College Dublin. He holds a Ph.D. from Universite de Savoie, France. His interests lie at the interface of Calculus and Partial Differential Equations. He is the author and co-author of four research monographs. He has published over 60 research articles in major journals in the field and has been invited to give talks at various international meetings such as conferences and summer schools for graduate students.
