Mathematical Physics I

Description

Mathematical Physics 1 by Dr. Rajesh Kumar Verma is designed for students following the syllabi of 1st Semester GU, AU, MU, NU, 2nd Semester Bodoland University, and 3rd Semester DU, RTU, Bhattadev, and other Indian universities as per FYUGP (NEP). This comprehensive text covers essential topics such as vector calculus, curvilinear coordinates, the Dirac delta function, and matrices, offering clear explanations, solved examples, and practice questions. It is an essential resource for mastering the mathematical foundations necessary for advanced studies in physics.

CONTENT :

PAGE CONTENTS
VECTOR CALCULUS ………………………………………………………………….. PAGE 1-40
1.1. Scalars and Vectors Quantities ………………………………………………………………. 1
1.2. Laws of Scalar Addition and Multiplication …………………………………………… 1
Associativity: ………………………………………………………………………………………… 2
1.3. Subtraction of Vectors …………………………………………………………………………… 3
1.4. Components of a Vector ………………………………………………………………………… 3
1.5 Rectangular Components of a Vector in a Plane ……………………………………… 3
1.5. Right- Handed and Left-Handed Coordinate Systems ……………………………. 4
1.6. Rectangular Components of a Three-Dimensional Vector ……………………….. 5
1.7. Position Vector ……………………………………………………………………………………… 7
Collinear or Parallel Vectors ………………………………………………………………….. 8
1.8. Coplanar Vectors …………………………………………………………………………………… 9
Solved Examples …………………………………………………………………………………. 10
1.9. Product of Two Vectors ……………………………………………………………………….. 12
1.10. Scalar or Dot Product of Two Vectors …………………………………………………… 13
1.11. Applications of Scalar Product …………………………………………………………….. 14
Solved Examples …………………………………………………………………………………. 15
1.12 Vectors or Cross Product……………………………………………………………………… 21
1.13 Applications of Cross Product ……………………………………………………………… 22
1.14 Null or Zero Vector ……………………………………………………………………………… 25
Solved Examples …………………………………………………………………………………. 25
1.15 Triple Products ……………………………………………………………………………………. 30
Solved Examples …………………………………………………………………………………. 33
1.16 Fields …………………………………………………………………………………………………….. 36
1.17 Scalar Field ……………………………………………………………………………………………. 37
1.18 Vector Field ……………………………………………………………………………………………. 37
Multiple Choice Questions …………………………………………………………………… 38
Short Answer Type Questions ………………………………………………………………. 40
CALCULUS………………………………………………………..………Page 43-73
2.1 Vector Calculus ……………………………………………………………………………………… 43
2.2 Derivative of a Vector Field ……………………………………………………………………. 44
Solved Examples ……………………………………………………………………………………. 45
2.3 Derivative Operators of Vectors ……………………………………………………………… 45
2.3.1. Gradient of a scalar field is normal to the surface …………………………………… 45
2.4 Partial Derivatives and Gradient ……………………………………………………………. 46
2.5 Physical Significance of Gradient …………………………………………………………… 48
2.6 Divergence of a Vector Field …………………………………………………………………… 49
2.7 Physical Interpretation and Significance of Div ………………………………………. 49
2.8 Physical significance of divergent …………………………………………………………… 51
2.9 Vector Integration………………………………………………………………………………….. 51
2.10 Curl ………………………………………………………………………………………………………. 52
2.11 Physical Interpretation and Significance of curl ……………………………………… 53
2.12 Laplacian Operator ……………………………………………………………………………….. 54
2.13 Gradient Theorem …………………………………………………………………………………. 54
2.14 Gauss’ Theorem …………………………………………………………………………………….. 56
2.15 Application of Gauss’ Theorem ………………………………………………………………. 58
2.16 Green’s Theorem …………………………………………………………………………………… 58
2.17 Stroke’s Theorem …………………………………………………………………………………… 59
Solved Examples ……………………………………………………………………………………. 61
Multiple Choice Questions ……………………………………………………………………… 71
Long Answer Questions …………………………………………………………………………. 73
Short Answers Questions ……………………………………………………………………….. 73
CURVILINEAR COORDINATES…………………………………..……Page 75-110
3.1 Curvilinear Coordinates ……………………………………………………………………….. 75
3.2 Curvilinear Coordinates System ……………………………………………………………. 75
3.3 Area and Volume in Curvilinear Coordinates ………………………………………… 78
3.4 Differential Operators in Curvilinear Coordinates …………………………………. 79
3.4.1 Gradient ……………………………………………………………………………………… 79
Solved Examples …………………………………………………………………………………… 80
3.4.2 Divergence ……………………………………………………………………………………. 80
3.4.3 Curl ……………………………………………………………………………………………… 81
3.4.4 Laplacian Operators …………………………………………………………………….. 83
Solved Examples …………………………………………………………………………………… 83
3.5 Cartesian Coordinate System ………………………………………………………………… 85
3.5.1. Area and Volume …………………………………………………………………………. 86
3.5.2 Gradient ………………………………………………………………………………………. 86
3.5.3 Divergence ……………………………………………………………………………………. 86
3.5.4 Curl ……………………………………………………………………………………………… 86
3.5.5 Laplacian Operator ………………………………………………………………………. 87
3.6 Circular Cylindrical Coordinates System ………………………………………………. 87
3.6.1 Area and Volume ………………………………………………………………………….. 88
3.6.2 Gradient ………………………………………………………………………………………. 89
3.6.3 Divergence ……………………………………………………………………………………. 89
3.6.4. Curl …………………………………………………………………………………………….. 89
3.6.5. Laplacian Operator ……………………………………………………………………… 89
3.6.6. Resolution of Circular Cylindrical Unit Vectors Into Their Cartesian
Components …………………………………………………………………………………………. 90
3.6.7. Resolution of Cartesian Components Unit Vectors into their Circular
Cylindrical Components ………………………………………………………………………. 91
Solved Examples …………………………………………………………………………………… 91
3.6.8 Velocity and Acceleration of a Particle * ………………………………………… 93
3.7 Spherical Coordinates System ……………………………………………………………….. 95
3.7.1. Area and Volume …………………………………………………………………………. 97
3.7.2. Gradient ……………………………………………………………………………………… 97
3.7.3. Divergence …………………………………………………………………………………… 97
3.7.4. Curl …………………………………………………………………………………………….. 98
3.7.5. Laplacian Operator ……………………………………………………………………… 98
3.7.6. Resolution of Spherical Unit Vectors into their Cartesian Components
…………………………………………………………………………………………………………….. 98
3.7.7. Resolution of Cartesian Components Unit Vector into their Spherical
Components ……………………………………………………………………………………….. 100
Solved Example ………………………………………………………………………………….. 100
3.7.8 Velocity and Acceleration of a Particle …………………………………………. 103
Multiple Choice Questions …………………………………………………………………… 106
Short Answer Type Questions ……………………………………………………………… 107
Long Answer Type Questions ………………………………………………………………. 110
DIRAC DELTA FUNCTION…………………………………….………Page 111-124
4.1 Introduction ………………………………………………………………………………………… 111
4.2 Definition …………………………………………………………………………………………….. 111
4.3 Representation of Dirac Delta Function ………………………………………………… 111
4.4 Properties of Dirac Delta Function ……………………………………………………….. 114
Solved Examples ………………………………………………………………………………….. 117
4.5  Function in three Dimensions(3D) …………………………………………………….. 118
4.6 Laplace Transformation of  Function …………………………………………………. 118
4.7 Fourier Transformation of  Function………………………………………………….. 119
Solved Examples ………………………………………………………………………………….. 119
Multiple Choice Questions ……………………………………………………………………. 121
Short Answer Type Questions ………………………………………………………………. 122
Long Answer Type Questions ……………………………………………………………….. 124
MATRIX………………………………………………………………..Page 126-187
5.1 Introduction …………………………………………………………………………………….. 126
5.2 Matrix and its Elements……………………………………………………………………. 126
5.3 Different Types of Matrices ………………………………………………………………. 127
5.4 Matrix Algebra ………………………………………………………………………………… 129
5.4.1 Addition of Matrices ………………………………………………………………………… 129
Solved Examples………………………………………………………………………………. 130
5.4.2 Properties of Matrix Addition …………………………………………………………… 130
5.4.3 Properties of Matrix Multiplication ………………………………………………….. 134
5.5 Some Special Matrices ……………………………………………………………………… 139
5.5.1 Nilpotent Matrix ……………………………………………………………………………… 139
5.5.2 Idempotent Matrix …………………………………………………………………………… 140
5.6 Transpose Matrix …………………………………………………………………………….. 142
5.6.1 Properties of Transpose of a Matrix ………………………………………………….. 143
5.6.2 Symmetric and Skew Symmetric Matrix …………………………………………… 145
5.7 Determinant of a Matrix…………………………………………………………………… 150
5.7.1 Properties of a Determinant ……………………………………………………………… 151
5.8 Inverse of a Matrix…………………………………………………………………………… 153
5.8.1 Finding Inverse of a Matrix………………………………………………………………. 154
5.9 Solution of Linear Equation by Matrix Method…………………………………. 163
5.10 Complex Conjugate or Adjoint of a Matrix ……………………………………….. 172
5.10.1 Properties of Conjugate Transpose……………………………………………………. 173
5.11 Hermitian Matrix …………………………………………………………………………….. 174
5.12 Anti Hermitian Matrix……………………………………………………………………… 179
Multiple Choice Questions ……………………………………………………………….. 183
Short Answer Questions……………………………………………………………………. 186
Long Answer Questions ……………………………………………………………………. 187