Skip to main content
Mathematical Methods of Physics:
For Undergraduate
Sunil Karna
Contents
Index
Search Book
close
Search Results:
No results.
Prev
Up
Next
\(\newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \newcommand{\T}{\mathcal T} \newcommand\comb[2]{^{#1}C{_{#2}}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Dedication
Acknowledgements
Preface
1
Vector Analysis
1.1
Vector Algebra
1.1.1
Addition of Vectors
1.1.1.1
Subtraction of Vectors
1.1.1.2
Scalar Multiplication of a Vector
1.1.1.3
Components of a Vector
1.1.2
Distance between two points
1.1.3
Dividing the line segments
1.1.4
Linear Combination of Vectors
1.1.5
Product of Two Vectors
1.1.5.1
Scalar or Dot Product
1.1.5.2
Vector or Cross Product
1.1.6
Scalar Triple Product
1.1.7
Vector Triple Product
1.1.8
Reciprocal Vectors
1.1.9
Differentiation of a Vector with respect to Scalars
1.2
Field
1.2.1
Gradient of a Scalar Function
1.2.1.1
Physical Significance of
\(\vec{ \nabla \phi}\)
1.2.1.2
Directional Derivative
1.2.2
Divergence of a Vector Function
1.2.2.1
Physical Significance of
\(\vec{\bigtriangledown}\cdot\vec{F}\)
1.2.3
Curl of a Vector Function
1.2.3.1
Physical Significance of
\(\vec{\nabla}\times \vec{F}\)
1.3
Examples A
1.4
Vector Integration
1.4.1
The Line Integral
1.4.2
The Surface Integral
1.4.3
The Volume Integral
1.4.4
Gauss’s Divergence Theorem
1.4.5
Stoke’s Theorem
1.5
Examples B
1.6
Curvilinear Coordinates
1.6.1
The Gradient in Curvilinear Coordinates
1.6.2
The Divergence in Curvilinear Coordinates
1.6.3
The Curl in Curvilinear Coordinates
1.6.4
Laplacian in Curvilinear Coordinate Systems
1.6.5
Cylindrical Orthogonal Curvilinear Coordinates
1.6.6
Spherical Orthogonal Curvilinear Coordinates
1.7
Examples C
1.8
Exercise
2
Linear Vector Space
2.1
Vector Space
2.1.1
Addition of Vectors
2.1.1
Multiplication of Vectors by Scalars
2.1.2
Linear Dependence and Independence
2.1.3
Dimensionality of a Vector Space
2.1.3.1
Inner Product
2.1.3.2
Gram Schmidt’s Orthogonalization
2.1.4
Linear Transformations
2.1.5
Linear Operators
2.2
ExamplesA
2.3
Matrix
2.3.1
Matrix Algebra
2.3.1.1
Addition
2.3.1.2
Multiplication
2.3.2
The Transpose of a Matrix,
\(A^{t}\)
2.3.3
Complex - Conjugate of a Matrix,
\(A^{*}\)
2.3.4
Hermitian Conjugate of a Matrix,
\(A^{\dagger}\)
2.3.5
Special Square Matrices
2.3.5.1
Unit or Identity Matrix:
2.3.5.2
Diagonal Matrix
2.3.5.3
The Inverse, Singular, and Non - Singular Matrices
2.3.5.4
Cofactor Matrix,
\(A^{c}\)
2.3.5.5
Adjoint of a Matrix,
\(\hat{A}\)
2.3.5.6
Self - Adjoint Matrix
2.3.5.7
Symmetric Matrix
2.3.5.8
Antisymmetric (Skew) Matrix
2.3.5.9
Hermitian Matrix
2.3.5.10
Unitary Matrix
2.3.5.11
Orthogonal Matrix
2.3.6
The Trace of a Matrix
2.4
ExamplesB
2.5
Eigen Value Problem
2.6
ExamplesC
2.7
Python Code for Matrix
2.8
Exercise
3
Tensor Analysis
3.1
Rank of a Tensor
3.1.1
Contravarient and Covarient Tensors
3.1.1.1
Discussion
3.1.2
Symmetric and Antisymmetric Tensors
3.1.3
Transformation of Coordinates
3.2
Tensor Algebra
3.2.1
Dummy Suffix
3.2.2
The line element and a metric tensor
3.3
ExamplesA
3.4
Moment of Inertia Tensor
3.4.1
The Stress Tensor
3.5
Exercise
4
Differential Equations
4.1
Solution of a Differential Equation
4.1.1
First Order Linear Differential Equation
4.1.2
Second Order Linear Differential Equations
4.1.2.1
Linear Equations with Constant Coefficients
4.1.2.2
General Method of Finding the Particular Integral of Any Function
\(\phi(x)\)
4.1.2.3
Linear Equations with Variable Coefficients
4.2
ExamplesA
4.3
Power Series Method
4.3.1
When
\(x=0\)
is a Regular Singular Point of the Equation
4.4
Examples B
4.5
Special Functions
4.5.1
Legendre’s Differential Equation
4.5.1.1
Rodrigue’s Formula
4.5.1.2
Legendre Polynomials
4.5.1.3
Generating Function for
\(P_{n}(x)\)
4.5.1.4
Some Important Results
4.5.1.5
Recurrence Relations for
\(P_{n}(x)\)
4.5.1.6
Orthogonality of Legendre’s Polynomials
4.5.1.7
The Associated Legendre’s Polynomials
4.6
Examples C
4.7
Bessel’s Differential Equation
4.7.1
Bessel’s Functions,
\(J_{n}(x)\)
4.7.1.1
Generating Function for
\(J_{n}(x)\)
4.7.1.2
Integral Represntation of
\(J_{n}(x)\)
4.7.1.3
Recurrence Relations for
\(J_{n}(x)\)
4.7.1.4
Orthogonality of Bessel’s Functions
4.8
Examples D
4.9
Hermite Differential Equation
4.9.1
Hermite Polynomials
\(H_{n}(x)\)
4.9.1.1
Generating Function for
\(H_{n}(x)\)
4.9.1.2
Rodrigue’s Formula for Hermite Polynomials
4.9.1.3
Recurrence Relation for Hermite Polynomials
4.9.1.4
Orthogonality of Hermite Polynomials
4.10
Examples E
4.11
Laguerre Differential Equation
4.11.1
Laguerre’s Polynomial
4.11.1.1
Generating Function for
\(L_{n}(x)\)
4.11.1.2
Rodrigue’s Differential Formula for
\(L_{n}(x)\)
4.11.1.3
Recurrence Relation for
\(L_{n}(x)\)
4.11.1.4
Orthogonal Property of Laguerre Polynomials
4.12
Examples F
4.13
Exercise
5
Fourier Series
5.1
Direchlet’s Conditions
5.1.1
Determination of Fourier Coefficients
5.2
Examples A
5.3
The Fourier Cosine and Sine Series
5.3.1
Summing of Fourier Series
5.3.1.1
Half - Range Series
5.4
Examples B
5.5
Change of Interval
5.5.1
Complex Form of Fourier Series
5.5.2
Fourier Integral
5.5.3
Gibb’s Phenomenon
5.6
Examples C
5.7
Exercise
6
Integral Transform
6.1
Dirac Delta Function
6.1.1
Properties of Delta functions
6.1.2
Fourier Transforms
6.1.2.1
Fourier Integral Theorem
6.1.2.2
Different Forms of Fourier Integral.
6.1.3
Parseval’s Indentity for Fourier Transform. (Rayleigh’s Theorem)
6.1.4
The (Infinite) Fourier Sine Transform of
\(\mathcal{F}(k)\)
6.1.5
Some important Integrals:
6.1.6
Properties of Fourier Transforms
6.1.7
The Fourier Tansform of Derivative of
\(f(x)\)
6.1.8
Convolution Theorem
6.1.9
Choice of infinite Sine or Cosine Transforms to Boundary Value Problems
6.2
ExamplesA
6.3
Laplace Transforms
6.3.1
Properties of Laplace Transforms
6.3.1.1
Linear Property
6.3.1.2
First-Shifting Property
6.3.1.3
Inverse Laplace Transforms
6.3.2
Laplace Transform of the Derivative of
\(f(t)\)
6.3.2.1
Laplace Transform of Derivative of order
\(n\)
6.3.2.2
Laplace Transform of Integral of
\(f(t)\)
6.3.3
Laplace Transform of
\(t^{n}f(t)\)
6.3.4
The Change of Scale Property
6.3.5
Laplace Transform of
\(\frac{1}{t}f(t)\)
6.4
ExamplesB
6.5
Solution of Differential Equations by Laplace Transforms
6.5.1
Unit Step Function
6.5.1.1
Laplace Transform of Unit Step Function
6.5.2
Second-Shifting Property
6.5.3
Priodic Functions
6.5.4
Convolution Theorem
6.5.4.1
Differential Equation with Variable Coefficients
6.6
Examples C
6.7
Exercise
7
Partial Differential Equation (PDE)
7.1
One Dimensional Wave Equation in Stretched String
7.1.1
Solution of Wave Equation for Vibrating String
7.1.2
D’Alembert’s Solution of Vibrating String
7.1.3
Two Dimensional Wave Equation for Vibrating Membrane
7.1.4
Solution of Two Dimensional Wave Equation
7.1.4.1
Rectangular Membrane
7.1.4.2
Circular Membrane
7.1.4.3
Three Dimensional Wave Equation
7.2
Examples A
7.3
Heat Equation
7.3.1
One Dimensional Equation of Heat Flow
7.3.1.1
If ends of a bar are at temperature zero
7.3.1.2
If ends of the bar are insulated
7.3.1.3
If one end of a bar at temperature
\(u_{o}\)
and the other at temperature zero
7.3.1.4
Temperature in an infinite bar
7.3.2
Two Dimensional Equation of Heat Flow
7.3.3
Laplace’s Equation
7.3.4
Two Dimensional Laplace’s Equation
7.3.4.1
In Cartesian Coordinates
7.3.4.2
In Cylindrical Coordinates
7.3.5
Three Dimensional Equation of Heat Flow
7.3.6
Three Dimensional Laplace’s Equation
7.3.6.1
In Rectangular Coordinates
7.3.6.2
In Cylindrical Coordinates
7.3.6.3
In Spherical Coordinates
7.4
ExamplesB
7.5
Exercise
Backmatter
A
List of Symbols
B
The partial differential equation
C
Laplace’s equation in cylindrical coordinates
D
Laplace’s equation in spherical coordinates
Index
References
Colophon
Dedication
Dedication
To my spouse, who supported me through countless late nights and early mornings, and never once complained.
