Elementary Vector Calculus and Its Applications with MATLAB Programming (River Publishers Series in Mathematical, Statistical and Computational Modelling for Engineering)

✍ Scribed by Nita H. Shah, Jitendra Panchal

• Publisher: River Publishers
• Year: 2023
• Tongue: English
• Leaves: 228
• Edition: 1
• Category: Library

✦ Synopsis

Sir Isaac Newton, one of the greatest scientists and mathematicians of all time, introduced the notion of a vector to define the existence of gravitational forces, the motion of the planets around the sun, and the motion of the moon around the earth. Vector calculus is a fundamental scientific tool that allows us to investigate the origins and evolution of space and time, as well as the origins of gravity, electromagnetism, and nuclear forces. Vector calculus is an essential language of mathematical physics, and plays a vital role in differential geometry and studies related to partial differential equations widely used in physics, engineering, fluid flow, electromagnetic fields, and other disciplines. Vector calculus represents physical quantities in two or three-dimensional space, as well as the variations in these quantities.

The machinery of differential geometry, of which vector calculus is a subset, is used to understand most of the analytic results in a more general form. Many topics in the physical sciences can be mathematically studied using vector calculus techniques.

This book is designed under the assumption that the readers have no prior knowledge of vector calculus. It begins with an introduction to vectors and scalars, and also covers scalar and vector products, vector differentiation and integrals, Gauss's theorem, Stokes's theorem, and Green's theorem. The MATLAB programming is given in the last chapter.

This book includes many illustrations, solved examples, practice examples, and multiple-choice questions.

✦ Table of Contents

Front Cover
Elementary Vector Calculus and its Applications with MATLAB Programming
Contents
Preface
List of Figures
1 Basic Concept of Vectors and Scalars
1.1 Introduction and Importance
1.2 Representation of Vectors
1.3 Position Vector and Vector Components
1.4 Modulus or Absolute Value of a Vector
1.5 Zero Vector and Unit Vector
1.6 Unit Vectors in the Direction of Axes
1.7 Representation of a Vector in terms of Unit Vectors
1.8 Addition and Subtraction of Vectors
1.9 Product of a Vector with a Scalar
1.10 Direction of a Vector
1.11 Collinear and Coplanar Vectors
1.11.1 Collinear Vectors
1.11.2 Coplanar Vectors
1.12 Geometric Representation of a Vector Sum
1.12.1 Law of Parallelogram of Vectors
1.12.2 Law of Triangle of Vectors
1.12.3 Properties of Addition of Vectors
1.12.4 Properties of Scalar Product
1.12.5 Expression of Any Vector in Terms of the Vectors Associated with its Initial Point and Terminal Point
1.12.6 Expression of Any Vector in Terms of Position Vectors
1.13 Direction Cosines of a Vector
1.14 Exercise
2 Scalar and Vector Products
2.1 Scalar Product, or Dot Product, or Inner Product
2.2 The Measure of Angle Between two Vectors and Projections
2.2.1 Properties of a Dot Product
2.3 Vector Product or Cross Product or Outer Product of Two Vectors
2.4 Geometric Interpretation of a Vector Product
2.4.1 Properties of a Vector Product
2.5 Application of Scalar and Vector Products
2.5.1 Work Done by a Force
2.5.2 Moment of a Force About a Point
2.6 Exercise
3 Vector Differential Calculus
3.1 Introduction
3.2 Vector and Scalar Functions and Fields
3.2.1 Scalar Function and Field
3.2.2 Vector Function and Field
3.2.3 Level Surfaces
3.3 Curve and Arc Length
3.3.1 Parametric Representation of Curves
3.3.2 Curves with Tangent Vector
3.3.2.1 Tangent Vector
3.3.2.2 Important Concepts
3.3.3 Arc Length
3.3.3.1 Unit Tangent Vector
3.4 Curvature and Torsion
3.4.1 Formulas for Curvature and Torsion
3.5 Vector Differentiation
3.6 Gradient of a Scalar Field and Directional Derivative
3.6.1 Gradient of a Scalar Field
3.6.1.1 Properties of Gradient
3.6.2 Directional Derivative
3.6.2.1 Properties of Gradient
3.6.3 Equations of Tangent and Normal to the Level Curves
3.6.4 Equation of the Tangent Planes and Normal Lines to the Surfaces
3.7 Divergence and Curl of a Vector Field
3.7.1 Divergence of a Vector Field
3.7.1.1 Physical Interpretation of Divergence
3.7.2 Curl of a Vector Field
3.7.2.1 Physical Interpretation of Curl
3.7.3 Formulae for grad, div, curl Involving Operator
3.7.3.1 Formulae for grad, div, curl Involving Operator Once
3.7.3.2 Formulae for grad, div, curl Involving Operator Twice
3.8 Exercise
4 Vector Integral Calculus
4.1 Introduction
4.2 Line Integrals
4.2.1 Circulation
4.2.2 Work Done by a Force
4.3 Path Independence of Line Integrals
4.3.1 Theorem: Independent of Path
4.4 Surface Integrals
4.4.1 Flux
4.4.2 Evaluation of Surface Integral
4.4.2.1 Component form of Surface Integral
4.5 Volume Integrals
4.5.1 Component Form of Volume Integral
4.6 Exercise
5
Theorem, Stokes Theorem, and Gauss' Theorem
5.1 Green’s
Theorem (in the Plane)
5.1.1 Area of the Plane Region
5.2 Stokes' Theorem
5.3 Gauss'
Divergence Theorem
5.4 Exercise
6 MATLAB Programming
6.1 Basic of MATLAB Programming
6.1.1 Basic of MATLAB Programming
6.1.1.1 Introductory MATLAB programmes
6.1.1.2 Representation of a Vector in MATLAB
6.1.1.3 Representation of a Matrix in MATLAB
6.2 Some Miscellaneous Examples using MATLAB Programming
References
Index
About the Authors
Back Cover