Multivariate Calculus

Preface This book entitled "Multivariate Calculus" is a compilation of all basic topics of Calculus and provides us with the tools to do so by extending the concepts that we find in Calculus, such as the computation of the rate of change to multiple variables, determination of the gradient of a multivariate function by finding its derivatives in different directions, extensive use in Neural Networks to update the model parameters etc. It is intended to serve as an introductory text aimed towards undergraduate and postgraduate students in Science and Technology covering the newly introduced B.A./B.Sc. Honours syllabus for Mathematics of Indian universities, recommended by the University Grants Commission (UGC) in Choice Based Credit System (CBCS).The treatments of theories are a special feature of this book, which are presented in a systematic and interesting manner. Care has been taken to explain the subject matter in a clear and perspicuous style, so that even an average student can understand it independently. The purpose in writing this book has been to provide a development of the subject matter which is well-motivated, rigorous and up-to-date as best as possible.This book contains nine chapters including the chapter of Preliminaries (Chapter 0), in which we have tried to make a thorough systematic discussion on this subject. The topics of this chapter have frequently been used in the subsequent chapters of this book.Sincere attempts have been made to present the topics in a simple and lucid manner to create interest into the subject along with various types of worked out examples. A large number of problems have been suitably framed, properly graded and supplied with answers. Exercises contain motivated problems and are given in right places. Majority of them are straight forward; hints are occasionally given for harder once.Authors hope that the book will prove useful not only to the Mathematics Honours students for whom it is intended but also to the Engineering students and professionals and to the candidates of different competitive examinations as well.Criticisms and constructive suggestions for improvement, modification and correction of this book, if any, from the teachers, students and readers will be gratefully acknowledged. Cover Half Title Title Page Copyright Page Dedication Preface Table of Contents 0 Preliminaries 0.1 Introduction 0.2 Differential Calculus 0.2.1 Function 0.2.2 Hyperbolic functions 0.2.3 Inverse hyperbolic functions 0.2.4 Limit of a function 0.2.5 Indeterminate forms 0.2.6 Continuity of a function 0.2.7 Uniform Continuity 0.2.8 Differentiation 0.2.9 Monotonicity 0.2.10 Higher order derivatives 0.2.11 The curve 0.2.12 Properties of Derivative 0.2.13 Mean Value Theorems and Expansion of Functions 0.2.14 Maxima and Minima of a Function of One Variable 0.3 Integral Calculus 0.3.1 Fundamental results 0.3.2 Standard integrals 0.3.3 Integration by parts 0.3.4 Definite integral 0.3.5 Properties of definite integrals 0.3.6 Useful reduction formulae 0.3.7 Beta and Gamma functions 1 Functions of Several Variables: Limits & Continuity 1.1 Introduction 1.2 Some Concepts on Point Sets in R2 and R3 1.3 Functions of Two Independent Variables 1.4 Types of Functions 1.4.1 Single and Multiple valued functions 1.4.2 Explicit and Implicit functions 1.5 Geometrical Representation of a Function of the form 1.6 Examples of Functions 1.7 Limit of a Function 1.7.1 Simultaneous or Double limits 1.8 Simultaneous Limit of a Function of Three Variables 1.9 Repeated or Iterated Limits 1.10 Algebra of Limits 1.11 Continuity of Functions of Several Variables 1.12 Some Properties of Continuous Functions of Several Variables 1.13 Uniform Continuity 1.13.1 Some important properties of continuous functions defined over a closed domain 1.14 Miscellaneous Illustrative Examples 1.15 Exercises 2 Functions of Several Variables: Differentiation - I 2.1 Introduction 2.2 Partial Derivatives 2.2.1 Successive partial derivatives of higher order 2.3 Continuity and Partial Derivatives 2.4 Differentiability or Total Differentiability 2.4.1 Differential 2.5 Conditions for Total Differentiability 2.6 Directional Derivatives 2.7 Directional Derivatives: An Alternative Approach 2.8 Higher Order Partial Derivatives 2.9 Differential of Higher Order 2.10 Expansion of Functions of Several Variables 2.10.1 Mean Value theorem in total differential form 2.10.2 Taylor’s theorem for functions of two independent variables 2.10.3 Generalised Taylor’s theorem for functions of m independent variables 2.10.4 Maclaurin’s theorem for the function of two independent variables 2.11 Perfect Differential (or Exact Differential) 2.12 Calculation of Small Errors by Differentials 2.13 Differentiation of Implicit Functions 2.14 Miscellaneous Illustrative Examples 2.15 Exercises 3 Functions of Several Variables: Differentiation - II 3.1 Introduction 3.2 Recapitulation 3.3 Differentials and Exact (or Perfect) Differentials 3.4 Composite Functions (Functions of Functions): Chain Rule 3.5 Change of Variables 3.6 Homogeneous Functions 3.7 Miscellaneous Illustrative Examples 3.8 Exercises 4 Jacobians, Functional Dependence and Implicit Functions 4.1 Introduction 4.2 Change of Variables by Jacobians 4.3 Jacobian of Implicit Functions 4.4 Some Properties of Jacobians 4.5 Functional Dependence 4.6 Implicit Functions 4.7 Condition for the Existence of an Explicit Function from an Implicit Function 4.7.1 Existence Theorem (in case of two variables) 4.8 Generalized Form of Existence Theorem 4.9 Derivatives of Implicit Functions 4.10 Implicit Functions Defined by a System of Functional Equations (Problems of Solving Two Equations are Considered) 4.11 Miscellaneous Illustrative Examples 4.12 Exercises 5 Extrema of Functions of Several Variables 5.1 Introduction 5.2 Maxima and Minima of Functions of Two Variables 5.3 The Necessary Conditions for Extreme Values of a Function of Two Variables 5.4 Sufficient Conditions for Maximum or Minimum 5.5 Stationary Value 5.6 Working Rule to Find the Maximum and Minimum of a Function 5.7 Constrained Optimization 5.7.1 Constrained extrema of a function having two independent variables 5.7.2 Constrained extrema of a function having three independent variables 5.8 Lagrange’s Method of Undetermined Multipliers 5.9 Miscellaneous Illustrative Examples 5.10 Exercises 6 Multiple Integrals 6.1 Introduction 6.2 Double Integrals 6.3 Properties of Double Integral 6.4 Evaluation of Double Integrals 6.5 Evaluation Procedure of Double Integrals 6.6 Double Integrals in Polar Coordinates 6.7 Evaluation of Double Integrals in Polar Coordinates 6.8 Change of Order of Integration 6.9 Change in the Variable in a Double Integral 6.10 Triple Integral or Integral over a Volume 6.11 Evaluation of Triple Integrals 6.11.1 Triple integration over a parallelepiped 6.11.2 Triple integral over any finite region 6.11.3 Evaluation of triple integral over the region A 6.12 Change of Variables in Triple Integration 6.12.1 In General Transformation 6.12.2 Triple Integral in Cylindrical Coordinates 6.12.3 Triple Integral in Spherical Polar Coordinates 6.13 Differentiation under the Sign of Integration 6.14 Miscellaneous Illustrative Examples 6.15 Exercises 7 Line, Surface and Volume Integrals 7.1 Introduction 7.2 The Curve 7.3 Line Integral 7.4 Existence of a Line Integral 7.5 Properties of a Line Integral 7.6 Illustrative Examples 7.7 Concept of Surfaces 7.8 Equation of a Surface 7.9 Surface Integrals 7.10 Relationships among Line, Surface and Volume Integrals with Simple Integrals and other Multiple Integrals 7.10.1 Simple integral vs Line integral 7.10.2 Surface integral vs Double integral 7.11 Calculating Area using Double Integrals 7.12 Calculating Volume using Double Integral 7.13 Illustrative Examples 7.14 Green’s Formula or Green’s Theorem in a Plane 7.15 Gauss’s Theorem or Gauss’s Divergence Theorem (Second Generalization of Green’s Theorem) 7.15.1 Application of Gauss’s Theorem 7.15.2 Alternative Statement of Gauss’s Theorem in Vector form 7.16 Stoke’s Theorem 7.17 Relationship between the Integral Theorems (Green’s, Stoke’s and Gauss’s Divergence Theorems) 7.18 Miscellaneous Illustrative Examples 7.19 Exercises 8 Dirichlet’s Theorem and Liouville’s Extension 8.1 Introduction 8.2 Dirichlet’s Theorem 8.3 Liouville’s Theorem (Extension of Dirichlet’s Theorem) 8.4 Miscellaneous Illustrative Examples 8.5 Exercises Bibliography Index This book is a compilation of all basic topics on functions of Several Variables and is primarily meant for undergraduate and post graduate students. Topics covered are: Limits, continuities and differentiabilities of functions of several variables. Properties of Implicit functions and Jacobians. Extreme values of multivariate functions. Various types of integrals in planes and surfaces and their related theorems including Dirichlet and Liouville's extension to Dirichlet. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan)