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E-raamat: Elementary Vector Calculus and Its Applications with MATLAB Programming

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Elementary Vector Calculus and Its Applications with MATLAB ProgrammingSuurem pilt
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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.
Preface ix
List of Figures
xi
1 Basic Concept of Vectors and Scalars
1(28)
1.1 Introduction and Importance
1(1)
1.2 Representation of Vectors
1(1)
1.3 Position Vector and Vector Components
2(1)
1.4 Modulus or Absolute Value of a Vector
3(1)
1.5 Zero Vector and Unit Vector
4(1)
1.6 Unit Vectors in the Direction of Axes
4(1)
1.7 Representation of a Vector in terms of Unit Vectors
5(1)
1.8 Addition and Subtraction of Vectors
6(1)
1.9 Product of a Vector with a Scalar
6(1)
1.10 Direction of a Vector
7(1)
1.11 Collinear and Coplanar Vectors
8(1)
1.11.1 Collinear Vectors
8(1)
1.11.2 Coplanar Vectors
8(1)
1.12 Geometric Representation of a Vector Sum
8(4)
1.12.1 Law of Parallelogram of Vectors
8(1)
1.12.2 Law of Triangle of Vectors
9(1)
1.12.3 Properties of Addition of Vectors
9(1)
1.12.4 Properties of Scalar Product
10(1)
1.12.5 Expression of Any Vector in Terms of the Vectors Associated with its Initial Point and Terminal Point
10(1)
1.12.6 Expression of Any Vector in Terms of Position Vectors
11(1)
1.13 Direction Cosines of a Vector
12(14)
1.14 Exercise
26(3)
2 Scalar and Vector Products
29(26)
2.1 Scalar Product, or Dot Product, or Inner Product
29(1)
2.2 The Measure of Angle Between two Vectors and Projections
30(7)
2.2.1 Properties of a Dot Product
30(7)
2.3 Vector Product or Cross Product or Outer Product of Two Vectors
37(1)
2.4 Geometric Interpretation of a Vector Product
38(7)
2.4.1 Properties of a Vector Product
39(6)
2.5 Application of Scalar and Vector Products
45(7)
2.5.1 Work Done by a Force
46(1)
2.5.2 Moment of a Force About a Point
46(6)
2.6 Exercise
52(3)
3 Vector Differential Calculus
55(56)
3.1 Introduction
55(1)
3.2 Vector and Scalar Functions and Fields
55(2)
3.2.1 Scalar Function and Field
56(1)
3.2.2 Vector Function and Field
56(1)
3.2.3 Level Surfaces
56(1)
3.3 Curve and Arc Length
57(7)
3.3.1 Parametric Representation of Curves
57(1)
3.3.2 Curves with Tangent Vector
58(1)
3.3.2.1 Tangent Vector
59(1)
3.3.2.2 Important Concepts
60(1)
3.3.3 Arc Length
61(1)
3.3.3.1 Unit Tangent Vector
61(3)
3.4 Curvature and Torsion
64(6)
3.4.1 Formulas for Curvature and Torsion
67(3)
3.5 Vector Differentiation
70(3)
3.6 Gradient of a Scalar Field and Directional Derivative
73(13)
3.6.1 Gradient of a Scalar Field
73(1)
3.6.1.1 Properties of Gradient
73(1)
3.6.2 Directional Derivative
74(1)
3.6.2.1 Properties of Gradient
75(9)
3.6.3 Equations of Tangent and Normal to the Level Curves
84(1)
3.6.4 Equation of the Tangent Planes and Normal Lines to the Surfaces
85(1)
3.7 Divergence and Curl of a Vector Field
86(18)
3.7.1 Divergence of a Vector Field
86(1)
3.7.1.1 Physical Interpretation of Divergence
86(3)
3.7.2 Curl of a Vector Field
89(1)
3.7.2.1 Physical Interpretation of Curl
89(7)
3.7.3 Formulae for grad, div, curl Involving Operator V
96(1)
3.7.3.1 Formulae for grad, div, curl Involving Operator V Once
96(4)
3.7.3.2 Formulae for grad, div, curl Involving Operator V Twice
100(4)
3.8 Exercise
104(7)
4 Vector Integral Calculus
111(24)
4.1 Introduction
111(1)
4.2 Line Integrals
111(2)
4.2.1 Circulation
112(1)
4.2.2 Work Done by a Force
112(1)
4.3 Path Independence of Line Integrals
113(9)
4.3.1 Theorem: Independent of Path
113(9)
4.4 Surface Integrals
122(7)
4.4.1 Flux
123(1)
4.4.2 Evaluation of Surface Integral
123(1)
4.4.2.1 Component form of Surface Integral
124(5)
4.5 Volume Integrals
129(2)
4.5.1 Component Form of Volume Integral
129(2)
4.6 Exercise
131(4)
5 Green's Theorem, Stokes' Theorem, and Gauss' Theorem
135(32)
5.1 Green's Theorem (in the Plane)
135(11)
5.1.1 Area of the Plane Region
137(9)
5.2 Stokes' Theorem
146(8)
5.3 Gauss' Divergence Theorem
154(9)
5.4 Exercise
163(4)
6 MATLAB Programming
167(40)
6.1 Basic of MATLAB Programming
167(21)
6.1.1 Basic of MATLAB Programming
167(1)
6.1.1.1 Introductory MATLAB programmes
168(15)
6.1.1.2 Representation of a Vector in MATLAB
183(3)
6.1.1.3 Representation of a Matrix in MATLAB
186(2)
6.2 Some Miscellaneous Examples using MATLAB Programming
188(19)
Index 207(6)
About the Authors 213
Professor Shah received her PhD in Statistics from Gujarat University in 1994. From February 1990 until now Professor Shah has been Head of the Department of Mathematics in Gujarat University, India. She is a post-doctoral visiting research fellow of University of New Brunswick, Canada. Professor Shahs research interests include inventory modeling in supply chains, robotic modeling, mathematical modeling of infectious diseases, image processing, dynamical systems and their applications, etc. She has published 13 monograph, 5 textbooks, and 475+ peer-reviewed research papers. Four edited books have been prepared for IGI-global and Springer with coeditor Dr. Mandeep Mittal. Her papers are published in high-impact journals such as those published by Elsevier, Interscience, and Taylor and Francis. According to Google scholar, the total number of citations is over 3334 and the maximum number of citations for a single paper is over 177. She has guided 28 PhD Students and 15 MPhil students. She has given talks in USA, Singapore, Canada, South Africa, Malaysia, and Indonesia. She was Vice-President of the Operational Research Society of India. She is Vice-President of the Association of Inventory Academia and Practitioner and a council member of the Indian Mathematical Society.

Dr Panchal is an Assistant Professor in the Department of Applied Sciences and Humanities, Parul University, India. He has 7+ years of teaching experience and 5+ years of research experience. His research interests are in the fields of mathematical control theory applied to various types of impulsive and/or fractional differential inclusions/systems with non-local conditions, and integer/fractional-order mathematical modeling of dynamical systems based on real-life phenomena. His 10 articles have been published in international journals indexed in SCIE, Scopus and Web of Science.
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