Elementary vector calculus and its applications with MATLAB programming / Nita H. Shah, Jitendra Panchal.
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...
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| Language: | English |
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RIVER PUBLISHERS,
2022.
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| Series: | River Publishers series in mathematical, statistical and computational modelling for engineering.
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| 245 | 1 | 0 | |a Elementary vector calculus and its applications with MATLAB programming / |c Nita H. Shah, Jitendra Panchal. |
| 264 | 1 | |a [Place of publication not identified] : |b RIVER PUBLISHERS, |c 2022. | |
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| 490 | 1 | |a River Publishers series in mathematical, statistical and computational modelling for engineering | |
| 520 | |a 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. | ||
| 505 | 0 | |a Preface ix List of Figures xi 1 Basic Concept of Vectors and Scalars 1 1.1 Introduction and Importance 1 1.2 Representation of Vectors 1 1.3 Position Vector and Vector Components 2 1.4 Modulus or Absolute Value of a Vector 3 1.5 Zero Vector and Unit Vector 4 1.6 Unit Vectors in the Direction of Axes 4 1.7 Representation of a Vector in terms of Unit Vectors 5 1.8 Addition and Subtraction of Vectors 6 1.9 Product of a Vector with a Scalar 6 1.10 Direction of a Vector 7 1.11 Collinear and Coplanar Vectors 8 1.11.1 Collinear Vectors 8 1.11.2 Coplanar Vectors 8 1.12 Geometric Representation of a Vector Sum 8 1.12.1 Law of Parallelogram of Vectors 8 1.12.2 Law of Triangle of Vectors 9 1.12.3 Properties of Addition of Vectors 9 1.12.4 Properties of Scalar Product 10 1.12.5 Expression of Any Vector in Terms of the Vectors Associated with its Initial Point and Terminal Point 10 1.12.6 Expression of Any Vector in Terms of Position Vectors 11 1.13 Direction Cosines of a Vector 12 1.14 Exercise 26 2 Scalar and Vector Products 29 2.1 Scalar Product, or Dot Product, or Inner Product 29 2.2 The Measure of Angle Between two Vectors and Projections 30 2.2.1 Properties of a Dot Product 30 2.3 Vector Product or Cross Product or Outer Product of Two Vectors 37 2.4 Geometric Interpretation of a Vector Product 38 2.4.1 Properties of a Vector Product 39 2.5 Application of Scalar and Vector Products 45 2.5.1 Work Done by a Force 46 2.5.2 Moment of a Force About a Point 46 2.6 Exercise 52 3 Vector Differential Calculus 55 3.1 Introduction 55 3.2 Vector and Scalar Functions and Fields 55 3.2.1 Scalar Function and Field 56 3.2.2 Vector Function and Field 56 3.2.3 Level Surfaces 56 3.3 Curve and Arc Length 57 3.3.1 Parametric Representation of Curves 57 3.3.2 Curves with Tangent Vector 58 3.3.2.1 Tangent Vector 59 3.3.2.2 Important Concepts 60 3.3.3 Arc Length 61 3.3.3.1 Unit Tangent Vector 61 3.4 Curvature and Torsion 64 3.4.1 Formulas for Curvature and Torsion 67 3.5 Vector Differentiation 70 3.6 Gradient of a Scalar Field and Directional Derivative 73 3.6.1 Gradient of a Scalar Field 73 3.6.1.1 Properties of Gradient 73 3.6.2 Directional Derivative 74 3.6.2.1 Properties of Gradient 75 3.6.3 Equations of Tangent and Normal to the Level Curves 84 3.6.4 Equation of the Tangent Planes and Normal Lines to the Surfaces 85 3.7 Divergence and Curl of a Vector Field 86 3.7.1 Divergence of a Vector Field 86 3.7.1.1 Physical Interpretation of Divergence 86 3.7.2 Curl of a Vector Field 89 3.7.2.1 Physical Interpretation of Curl 89 3.7.3 Formulae for grad, div, curl Involving Operator ∇ 96 3.7.3.1 Formulae for grad, div, curl Involving Operator ∇ Once 96 3.7.3.2 Formulae for grad, div, curl Involving Operator ∇ Twice 100 3.8 Exercise 104 4 Vector Integral Calculus 111 4.1 Introduction 111 4.2 Line Integrals 111 4.2.1 Circulation 112 4.2.2 Work Done by a Force 112 4.3 Path Independence of Line Integrals 113 4.3.1 Theorem: Independent of Path 113 4.4 Surface Integrals 122 4.4.1 Flux 123 4.4.2 Evaluation of Surface Integral 123 4.4.2.1 Component form of Surface Integral 124 4.5 Volume Integrals 129 4.5.1 Component Form of Volume Integral 129 4.6 Exercise 131 5 Green's Theorem, Stokes' Theorem, and Gauss' Theorem 135 5.1 Green's Theorem (in the Plane) 135 5.1.1 Area of the Plane Region 137 5.2 Stokes' Theorem 146 5.3 Gauss' Divergence Theorem 154 5.4 Exercise 163 6 MATLAB Programming 167 6.1 Basic of MATLAB Programming 167 6.1.1 Basic of MATLAB Programming 167 6.1.1.1 Introductory MATLAB programmes 168 6.1.1.2 Representation of a Vector in MATLAB 183 6.1.1.3 Representation of a Matrix in MATLAB 186 6.2 Some Miscellaneous Examples using MATLAB Programming 188 Index 207 About the Authors 213. | |
| 630 | 0 | 0 | |a MATLAB. |
| 630 | 0 | 7 | |a MATLAB |2 fast |
| 650 | 0 | |a Vector analysis. | |
| 650 | 7 | |a Vector analysis |2 fast | |
| 700 | 1 | |a Panchal, Jitendra, |e author. | |
| 776 | 0 | 8 | |i Print version: |z 8770223874 |z 9788770223874 |w (OCoLC)1342621288 |
| 830 | 0 | |a River Publishers series in mathematical, statistical and computational modelling for engineering. | |
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