Mathematical Physics-I
Teacher
Gyanendra Kumar Mishra
Category
Core Courses
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Course Name : Mathematical Physics-I
Code(Credit) : CUTM1483(3-2-1)
Course Objectives
1. To introduce the students to understand the physical meaning of different mathematical methods
2. The emphasis of course is on applications in solving problems of interest to physicists.
3. Highlights the use of computational methods to solve physical problems
Learning Outcomes
| COs | Course outcomes | Mapping COs with POs (High-3, Medium-2, Low-1) |
| CO1 | Understand the mathematical and physical interpretation of vector. | PO1-3, PO2-3, PO5-3 |
| CO2 | Handle different coordinate systems. | PO1-3, PO2-3, PO5-3, PO9-1 |
| CO5 | Introduced to the concept of probability. | PO1-3, PO9-1 |
Course Syllabus
Module-I Calculus:
Special types of matrices (conjugate and its transpose, periodic, Idempotent, Nilopotent, symmetric and antisymmetric, Hermitian and skew-hermitian, orthogonal and unitary matrix). Rank of a matrix. Eigen value and Eigen value. Caley -Hamilton theorem . Solution of linear equation by using Gauss-Elimination and Gauss-Jordan Method.
Approximation, Taylor and binomial series (statements only).Solution of 1st order differential equation(linear, homogeneous, non-homogeneous and exact). Wronskian and general solution. Complimentary function and Particular Integral.
practice-1 Introduction and Overview of 'C'
practice-2 Basics of scientific computing
Module-II
Vector Calculus:
Recapitulation of vectors, Properties of vectors under rotations. Scalar product and its invariance under rotations. Vector product, Scalar triple product and their interpretation in terms of area and volume respectively. Scalar and Vector fields.
practice-3 Errors and error Analysis
practice-4 Review of C & C++ Programming fundamentals
Module-III
Vector Differentiation:
Directional derivatives and normal derivative. Gradient of ascalar field and its geometrical interpretation. Divergence and curl of a vector field. Del and Laplacian operators. Vector identities.
practice-5 Programs:(Sum & average of a list of numbers, largest of a given list of numbers)
practice-6 Description with Applications
Module-IV
Vector Integration:
Ordinary Integrals of Vectors. Multiple integrals, Jacobian. Notion of infinitesimal line, surface and volume elements. Line, surface and volume integrals of Vector fields. Flux of a vector field. Gauss' divergence theorem, Green's and Stokes Theorems and their applications (no rigorous proofs).
practice-7 Area of circle, area of square
practice-8 volume of sphere, value of pi (π)
Module-V
Orthogonal Curvilinear Coordinates:
Orthogonal Curvilinear Coordinates. Derivation of Gradient, Divergence, Curl andLaplacian in Cartesian, Spherical and Cylindrical Coordinate Systems.
practice-9 Solution of linear and quadratic equation
Module-VI
Introduction to probability:
Independent random variables: Probability distribution functions; binomial, Gaussian and Poisson, with examples. Mean and variance. Dependent events: Conditional Probability. Bayes' Theorem and the idea of hypothesis testing.
practice-10 solving 𝛼=𝑡𝑎𝑛𝛼; 𝐼=𝐼0(𝑠𝑖𝑛𝛼𝛼)2in optics
practice- 11 Evaluation of trigonometric functions sin and cosine of an angle
Module-VII
Dirac Delta function and its properties:
Definition of Dirac delta function. Representation as limit of a Gaussian function and rectangular function. Properties of Dirac delta function.
practice-12 Evaluation of trigonometric functions tangent of a function
Text Books:
- Mathematical Physics by Satyaprakash, Chand&Co.
- Mathematical Methods for Physicists, G.B. Arfken, H.J. Weber, F.E. Harris, 2013, 7th Edn., Elsevier.
Reference Books:
- An introduction to ordinary differential equations, E.A. Coddington, 2009, PHI learning
- Differential Equations, George F. Simmons, 2007, McGraw
- Mathematical Tools for Physics, James Nearing, 2010, Dover
- Mathematical methods for Scientists and Engineers, D.A. McQuarrie, 2003, Viva Book
- Advanced Engineering Mathematics, D.G. Zill and W.S. Wright, 5 Ed., 2012, Jones and Bartlett Learning
- Mathematical Physics, Goswami, 1st edition, Cengage Learning
- Engineering Mathematics, S.Pal and S.C. Bhunia, 2015, Oxford University Press
- Advanced Engineering Mathematics, Erwin Kreyszig, 2008, Wiley
- Essential Mathematical Methods, K.F.Riley&M.P.Hobson, 2011, Cambridge Univ. Press
Session Plan
Session 1
Special types of matrices (conjugate and its transpose, periodic, Idempotent, Nilopotent, symmetric and antisymmetric, Hermitian and skew-hermitian, orthogonal and unitary matrix).Rank of a matrix.
Session 2
Rank of a matrix. Eigen value and Eigen vector of a matrix . Caley -Hamilton theorem .
https://www.youtube.com/watch?v=6agQBHWP8IE
https://www.slideshare.net/christopherchibangu/presentation-binomial-theorem
Session 3
First Order and Second Order Differential equations
https://www.youtube.com/watch?v=vmFz0P9Dvyc
https://www.slideshare.net/alibhalli43/differential-equations-48242801
Session 4
practice-1(2 hrs.)
Computer architecture and organization, memory andInput/output devices
Using 'C' Lab
https://www.youtube.com/watch?v=HDv_2Wv_a00
https://www.slideshare.net/abdulugc/input-output-organization-45974656
Session 5
First Order Differential Equations and Integrating Factor
https://www.youtube.com/watch?v=v5C0CtRAK10
https://www.slideshare.net/pokkarnmahajan/ode-powerpoint-presentation1
Session 6
Homogeneous Equations with constant coefficients
https://www.youtube.com/watch?v=StJ1LriDtq8
https://www.slideshare.net/SeyidKadher1/differential-equations-68070117
Session 7
Assignment-1/Flipped class-1-:
solutions of linear differential equation
https://www.youtube.com/watch?v=Et4Y41ZNyao
https://www.slideshare.net/pratiksudra10/linear-differential-equation
Session 8
Statement of existence and Uniqueness Theorem for Initial Value Problems
Session 9
Practice-2(2hrs.):-
Binary and decimal arithmetic, Floating point numbers, algorithms, Sequence, Selection and Repetition, single and double precision arithmetic, underflow &overflowemphasizethe importance of making equations in termsof dimensionless variables, Iterative methods
https://www.youtube.com/watch?v=TWm9phoTILw
https://www.youtube.com/watch?v=bNAP9ejTqM0
https://www.slideshare.net/mumthasshaikh/data-representation-and-arithmetic-algorithms
Session 11
Assignment-2 /Flipped class-2:-
Solutions of differential equation by using Complimentary solution and particular integral Method
Session 12
Recapitulation of vectors: Properties of vectors under rotations
https://www.youtube.com/watch?v=ml4NSzCQobk
https://www.slideshare.net/UrmilaBhardwaj/vector-algebra-58263932
Session 13
Scalar product and its invariance under rotations
https://www.youtube.com/watch?v=tLuykrw8GR4
https://www.slideshare.net/beddhakal9/scalar-product-of-vectors
Session 14
Vector product and its representations in terms of area
https://www.youtube.com/watch?v=eYF_QVdS3yY
https://www.slideshare.net/AshrafulTauhid/dot-cross-product-of-vectors
Session 16
Scalar triple product and their interpretation in terms of volume
Session 17
Session 18
Assignment-3/flipped class-3:-
Geometrical meaning of vector and scalar product with examples
https://www.slideshare.net/beddhakal9/cross-product-of-vectors
Session 19
Assignment-4, flipped class-4:-
Concept of work and force in terms of scalar and vectors respectively
Session 20
Directional derivatives and normal derivative
Session 21
Gradient of a scalar field and its geometrical interpretation
Session 22
Assignment-5, flipped class-5:-
Use of Gradient of a scalar field in different fields of physics.
Session 23
Divergence and curl of a vector field
https://www.youtube.com/watch?v=ZtQyuN7DdKE
https://www.slideshare.net/KunjPatel4/vector-calculus-and-linear-algebra
Session 24
Practice-4(2hrs.)
Review of C & C++ Programming fundamentals
using c lab
Session 25
Session 27
practice-5(2hrs.)
Programs:(Sum & average of a list of numbers, largest of a given list of numbers)
using c lab
Session 30
practice-6(2hrs.):-
Area of circle, area of square
(using c lab)
Session 31
Notion of infinitesimal line, surface and volume elements
https://www.slideserve.com/penda/chapter-11-line-surface-and-volume-integrals
Session 32
Assignment -6 , Flipped class-6:-
Electric field and application of Gauss law
https://www.slideshare.net/GyanraoPhysics/gauss-law-and-application-arun-kumar
Session 33
Line, surface and volume integrals of Vector fields
https://www.slideshare.net/GyanraoPhysics/liner-surface-and-volum-intergrals-karishma-mansuri
Session 34
Assignment-7, Flipped class-7:-
Application and problems of Line, surface and volume integrals of Vector fields
Session 35
Flux of a vector field. Gauss' divergence theorem
Session 36
Green's and Stokes Theorems and their applications
Session 37
practice-7(2hrs.):-
Area of a square
using c lab
Session 38
Assignment-8, Flipped class-8:-
Applications Green's and Stokes Theorems
Session 39
Orthogonal Curvilinear Coordinates. Derivation of Gradient
Session 40
Practice-8(2hrs.):-
volume of sphere, value of pi (π)
using c lab
Session 41
Divergence, Curl and Laplacian in Cartesian
Session 42
Assignment-9 , Flipped class-9:-
Divergence, Curl and Laplacian in Cartesian and Spherical
Session 43
Divergence, Curl and Laplacian in cylindrical system
https://www.youtube.com/watch?v=AOT719oYmt0
https://www.slideserve.com/norah/divergence-curl-and-laplacian
Session 44
Practice-9(2hrs.):-
Solution of linear and quadratic equation
(using c lab)
Session 45
Assignment-10, Flipped class-10:-
Application and example of Divergence, Curl and Laplacian in cyllindrical system
https://www.youtube.com/watch?v=AOT719oYmt0
https://www.slideserve.com/norah/divergence-curl-and-laplacian
Session 46
Independent random variables: Probability distribution functions
Session 47
Practice-10(2hrs.):-
solving
𝛼=𝑡𝑎𝑛𝛼; 𝐼=𝐼0(𝑠𝑖𝑛𝛼𝛼)2in optics
using c lab
Session 48
Binomial, Gaussian integrals
https://www.slideshare.net/mrrahimi2012/gaussian-integration
Session 49
Assignment-11 ,Flipped class-11:-
Mean and variance of binomial, Gaussian and Poisson
https://www.slideshare.net/numanmunir01/binomial-distribution-59502855
Session 50
Practice-11(2hrs.):-
Evaluation of trigonometric functions sin and cosine of an angle
using c lab
Session 51
Bayes' Theorem and the idea of hypothesis testing.
Session 52
Assignment-12, Flipped class-12:-
Problems from conditional probability,Bayes' Theorem.
Session 53
Dependent events: Conditional Probability
https://www.slideshare.net/MariaRominaAngustia/conditional-probability-66117183
Session 54
Practice-12(2hrs.):-
Evaluation of trigonometric functions tangent of a function
(using c lab)
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