1st Edition
Differential Calculus in Several Variables A Learning-by-Doing Approach
The aim of this book is to lead the reader out from the ordinary routine of computing and calculating by engaging in a more dynamic process of learning. This Learning-by-Doing Approach can be traced back to Aristotle, who wrote in his Nicomachean Ethics that “For the things we have to learn before we can do them, we learn by doing them”.
The theory is illustrated through many relevant examples, followed by a large number of exercises whose requirements are rendered by action verbs: find, show, verify, check and construct. Readers are compelled to analyze and organize analytical skills.
Rather than placing the exercises in bulk at the end of each chapter, sets of practice questions after each theoretical concept are included. The reader has the possibility to check their understanding, work on the new topics and gain confidence during the learning activity. As the theory unfolds, the exercises become more complex – sometimes they span over several topics. Hints have been added in order to guide the reader in the process.
This book stems from the Differential Calculus course which the author taught for many years. The goal of this book is to immerse the reader in the subtleties of Differential Calculus through an active perspective. Particular attention was paid to continuity and differentiability topics, presented in a new course of action.
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.
