# Mathematical Physics-I

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# 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

Upon successful completion of this course, students will be :

1. Able to understand the mathematical and physical interpretation of vector

2. Able to handle different coordinate systems

3. Introduced to the concept of probability

## 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:
1. Mathematical Physics by Satyaprakash, Chand&Co.
2. Mathematical Methods for Physicists, G.B. Arfken, H.J. Weber, F.E. Harris, 2013, 7th Edn., Elsevier.
##### Reference Books:
1. An introduction to ordinary differential equations, E.A. Coddington, 2009, PHI learning
2. Differential Equations, George F. Simmons, 2007, McGraw
3. Mathematical Tools for Physics, James Nearing, 2010, Dover
4. Mathematical methods for Scientists and Engineers, D.A. McQuarrie, 2003, Viva Book
5. Advanced Engineering Mathematics, D.G. Zill and W.S. Wright, 5 Ed., 2012, Jones and Bartlett Learning
6. Mathematical Physics, Goswami, 1st edition, Cengage Learning
7. Engineering Mathematics, S.Pal and S.C. Bhunia, 2015, Oxford University Press
8. Advanced Engineering Mathematics, Erwin Kreyszig, 2008, Wiley
9. Essential Mathematical Methods, K.F.Riley&M.P.Hobson, 2011, Cambridge Univ. Press

## 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.slideshare.net/christopherchibangu/presentation-binomial-theorem

## Session 4

practice-1(2 hrs.)

Computer architecture and organization, memory andInput/output devices

Using 'C' Lab

https://www.slideshare.net/abdulugc/input-output-organization-45974656

## Session 7

Assignment-1/Flipped class-1-:

solutions of linear differential equation

https://www.slideshare.net/pratiksudra10/linear-differential-equation

## 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.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

https://youtu.be/2-2_TZLsalQ

https://www.slideshare.net/vrajes/higherorder-non-homogeneous-partial-differrential-equations-maths-3-power-point-representation

## Session 15

practice-3(2hrs.):-

Errors and error Analysis

https://youtu.be/UPZfpY6PFm0

using c lab

## Session 16

Scalar triple product and their interpretation in terms of volume

https://youtu.be/ApY-mtOpot0

https://youtu.be/mbeELeZP9_Y

## Session 18

Assignment-3/flipped class-3:-

Geometrical meaning of vector and scalar product with examples

https://youtu.be/eu6i7WJeinw

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

https://youtu.be/fNk_zzaMoSs

https://youtu.be/fNk_zzaMoSs

## Session 22

Assignment-5, flipped class-5:-

Use of Gradient of a scalar field in different fields of physics.

https://youtu.be/fZ231k3zsAA

https://youtu.be/18wS1TICPUA

## Session 24

Practice-4(2hrs.)

Review of C & C++ Programming fundamentals

using c lab

## 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 32

Assignment -6 , Flipped class-6:-

Electric field and application of Gauss law

https://youtu.be/GWBXW1vpZQI

https://www.slideshare.net/GyanraoPhysics/gauss-law-and-application-arun-kumar

## Session 34

Assignment-7, Flipped class-7:-

Application and problems of Line, surface and volume integrals of Vector fields

https://youtu.be/9M5koXl2mmY

https://youtu.be/lNdZE-bfxn8

## 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

https://youtu.be/TPov_v9mLWc

https://www.slideshare.net/samiul11/application-of-gsg

## Session 40

Practice-8(2hrs.):-

volume of sphere, value of pi (π)

using c lab

## Session 42

Assignment-9 , Flipped class-9:-

Divergence, Curl and Laplacian in Cartesian and Spherical

https://youtu.be/_DFTpRxZVl8

https://youtu.be/AOT719oYmt0

## 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.slideserve.com/norah/divergence-curl-and-laplacian

## Session 46

Independent random variables: Probability distribution functions

https://youtu.be/NF0lrkqXIkQ

## Session 47

Practice-10(2hrs.):-

solving
𝛼=𝑡𝑎𝑛𝛼; 𝐼=𝐼0(𝑠𝑖𝑛𝛼𝛼)2in optics

using c lab

## Session 49

Assignment-11 ,Flipped class-11:-

Mean and variance of binomial, Gaussian and Poisson

https://youtu.be/65n_v92JZeE

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.

https://youtu.be/Zxm4Xxvzohk

https://youtu.be/OqmJhPQYRc8

## Session 52

Assignment-12, Flipped class-12:-

Problems from conditional probability,Bayes' Theorem.

https://youtu.be/OqmJhPQYRc8

https://www.slideshare.net/Tish997/bayes-rule-bays-law

## Session 54

Practice-12(2hrs.):-

Evaluation of trigonometric functions tangent of a function

(using c lab)