book

Multivariate Calculus

Name: Multivariate Calculus
ISBN: 9781000906219

by Samiran Karmakar, Sibdas Karmakar

June 2023

Intermediate to advanced

330 pages

6h 58m

English

CRC Press

Read now

Unlock full access

Cover Page
Half Title page
Title Page
Copyright Page
Dedication
Preface
Contents
0 Preliminaries
0.1 Introduction0.2 Differential Calculus0.2.1 Function0.2.2 Hyperbolic functions0.2.3 Inverse hyperbolic functions0.2.4 Limit of a function0.2.5 Indeterminate forms0.2.6 Continuity of a function0.2.7 Uniform Continuity0.2.8 Differentiation0.2.9 Monotonicity0.2.10 Higher order derivatives0.2.11 The curve0.2.12 Properties of Derivative0.2.13 Mean Value Theorems and Expansion of Functions0.2.14 Maxima and Minima of a Function of One VariableLocal Maxima and Local MinimaGlobal Maxima and Global MinimaWorking rule for finding maxima and minima0.3 Integral Calculus0.3.1 Fundamental results0.3.2 Standard integrals0.3.3 Integration by parts0.3.4 Definite integral0.3.5 Properties of definite integrals0.3.6 Useful reduction formulae0.3.7 Beta and Gamma functions
1 Functions of Several Variables: Limits & Continuity
1.1 Introduction1.2 Some Concepts on Point Sets in ℝ2 and ℝ31 Neighbourhood of a point2 Interior point3 Open sets3 Limit point or Cluster point or Point of accumulation4 Closed sets5 Bounded sets1.3 Functions of Two Independent Variables1.4 Types of Functions1.4.1 Single and Multiple valued functions1.4.2 Explicit and Implicit functions1.5 Geometrical Representation of a Function of the form u = f(x, y)1.6 Examples of Functions1.7 Limit of a Function1.7.1 Simultaneous or Double limits1.8 Simultaneous Limit of a Function of Three Variables1.9 Repeated or Iterated Limits1.10 Algebra of Limits1.11 Continuity of Functions of Several VariablesDefinition 1Definition 2 (Analytical Definition)Definition 31.12 Some Properties of Continuous Functions of Several Variables1.13 Uniform Continuity1.13.1 Some important properties of continuous functions defined over a closed domain1.14 Miscellaneous Illustrative Examples1.15 ExercisesAnswers
2 Functions of Several Variables: Differentiation - I
2.1 Introduction2.2 Partial Derivatives2.2.1 Successive partial derivatives of higher order2.3 Continuity and Partial Derivatives2.4 Differentiability or Total Differentiability2.4.1 Differential2.5 Conditions for Total Differentiability2.6 Directional Derivatives2.7 Directional Derivatives: An Alternative Approach2.8 Higher Order Partial Derivatives2.9 Differential of Higher Order2.10 Expansion of Functions of Several Variables2.10.1 Mean Value theorem in total differential form2.10.2 Taylor's theorem for functions of two independent variables2.10.3 Generalised Taylor's theorem for functions of m independent variables2.10.4 Maclaurin's theorem for the function of two independent variables2.11 Perfect Differential (or Exact Differential)2.12 Calculation of Small Errors by Differentials2.13 Differentiation of Implicit Functions2.14 Miscellaneous Illustrative Examples2.15 ExercisesAnswers

3 Functions of Several Variables: Differentiation - II
3.1 Introduction3.2 RecapitulationPartial derivativesSuccessive partial derivatives3.3 Differentials and Exact (or Perfect) Differentials3.4 Composite Functions (Functions of Functions): Chain Rule3.5 Change of Variables3.6 Homogeneous Functions3.7 Miscellaneous Illustrative ExamplesAn Alternative Approach3.8 Exercises
4 Jacobians, Functional Dependence and Implicit Functions
4.1 Introduction4.2 Change of Variables by Jacobians4.3 Jacobian of Implicit Functions4.4 Some Properties of Jacobians4.5 Functional Dependence4.6 Implicit Functions4.7 Condition for the Existence of an Explicit Functionfrom an Implicit Function4.7.1 Existence Theorem (in case of two variables)Proof of the Theorem4.8 Generalized Form of Existence Theorem4.9 Derivatives of Implicit Functions4.10 Implicit Functions Defined by a System of Functional Equations (Problems of Solving Two Equations are Considered)4.11 Miscellaneous Illustrative Examples4.12 ExercisesAnswers
5 Extrema of Functions of Several Variables
5.1 Introduction5.2 Maxima and Minima of Functions of Two Variables5.3 The Necessary Conditions for Extreme Values of a Function of Two Variables5.4 Sufficient Conditions for Maximum or Minimum5.5 Stationary Value5.6 Working Rule to Find the Maximum and Minimum of a Function1 For Two Variables2 For Three VariablesAn Important Observation5.7 Constrained Optimization5.7.1 Constrained extrema of a function having two independent variables5.7.2 Constrained extrema of a function having three independent variables5.8 Lagrange’s Method of Undetermined Multipliers5.9 Miscellaneous Illustrative ExamplesAlternative Method:Alternative Method:5.10 ExercisesAnswer
6 Multiple Integrals
6.1 Introduction6.2 Double Integrals6.3 Properties of Double Integral6.4 Evaluation of Double Integrals6.5 Evaluation Procedure of Double Integrals6.6 Double Integrals in Polar Coordinates6.7 Evaluation of Double Integrals in Polar Coordinates6.8 Change of Order of Integration6.9 Change in the Variable in a Double Integral6.10 Triple Integral or Integral over a Volume6.11 Evaluation of Triple Integrals6.11.1 Triple integration over a parallelepiped6.11.2 Triple integral over any finite region6.11.3 Evaluation of triple integral over the region A6.12 Change of Variables in Triple Integration6.12.1 In General Transformation6.12.2 Triple Integral in Cylindrical Coordinates6.12.3 Triple Integral in Spherical Polar Coordinates6.13 Differentiation under the Sign of Integration6.14 Miscellaneous Illustrative Examples6.15 Exercises
7 Line, Surface and Volume Integrals
7.1 Introduction7.2 The Curve7.3 Line Integral7.4 Existence of a Line Integral7.5 Properties of a Line Integral7.6 Illustrative Examples7.7 Concept of Surfaces7.8 Equation of a Surface7.9 Surface Integrals7.10 Relationships among Line, Surface and Volume Integrals with Simple Integrals and other Multiple Integrals7.10.1 Simple integral vs Line integral7.10.2 Surface integral vs Double integral7.11 Calculating Area using Double Integrals7.12 Calculating Volume using Double Integral7.13 Illustrative Examples7.14 Green's Formula or Green's Theorem in a Plane7.15 Gauss's Theorem or Gauss's Divergence Theorem (Second Generalization of Green's Theorem)7.15.1 Application of Gauss's Theorem7.15.2 Alternative Statement of Gauss's Theorem in Vector form7.16 Stoke's Theorem7.17 Relationship between the Integral Theorems (Green's, Stoke's and Gauss's Divergence Theorems)7.18 Miscellaneous Illustrative Examples7.19 ExercisesAnswers
8 Dirichlet's Theorem and Liouville'sExtension
8.1 Introduction8.2 Dirichlet's Theorem8.3 Liouville's Theorem (Extension of Dirichlet's Theorem)8.4 Miscellaneous Illustrative Examples8.5 Exercises
Bibliography
Index

Content preview from Multivariate Calculus

Chapter 6 Multiple Integrals

6.1 Introduction

In this chapter we shall study integrals of functions of several variables involving multiple integrals. The motivations connected with these integrals are much the same as with simple integrals of functions of one variable which we have studied earlier. The chapter deals mainly with two types of integrals in ℝ² and ℝ³, viz., (i) Double integrals and (ii) Triple integrals. These integrals have important applications to geometry and mechanics.

6.2 Double Integrals

The definite integral $\int_{a}^{b} f (x) d x$ is defined as the limit of the sum $\sum_{i = 1}^{n} f (x_{i}) δ x_{i}$ where n→∞ and each of the lengthsδ_x1 δ_x2,...tends to zero[ref.§0.3.4]. A double integral is its counterpart in two dimensions.

Let f(x, y) be a function of ...

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.

Read now

Unlock full access

More than 5,000 organizations count on O’Reilly

O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.

Julian F.

Head of Cybersecurity

I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.

Addison B.

Field Engineer

I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.

Amir M.

Data Platform Tech Lead

I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.

Mark W.

Embedded Software Engineer

Publisher Resources

ISBN: 9781000906219

Cloud Computing

Data Engineering

Data Science

AI & ML

Programming Languages

Software Architecture

IT/Ops

Security

Design