Paper 1: Mathematical Physics I

Syllabus

Differential equations and their solutions, calculus, Vector calculus and its differentiation, vector integration, Dirac delta function, Orthogonal curvilinear coordinates, functional algebra and error analysis.

Unit I

Calculus -I: Plotting of functions, Intuitive ideas of continuous, differentiable functions and plotting of curves, Approximation: Taylor and binomial series (statements only), First Order Differential Equations and Integrating Factor, Second Order Differential equations: Homogeneous Equations with constant coefficients, Wronskian and general solution, Statement of existence and Uniqueness Theorem for Initial Value Problems, Particular Integral.
Calculus-II: Calculus of functions of more than one variable: Partial derivatives, exact and inexact differentials. Integrating factor with simple illustration, Constrained Maximization using Lagrange Multipliers.

Unit II

Vector algebra: 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.
Vector Differentiation: Directional derivatives and normal derivative, Gradient of a scalarfield and its geometrical interpretation, Divergence and curl of a vector field, Del and Laplacian operators, Vector identities.

Unit III

Vector Integration: Ordinary Integrals of Vectors, Multiple integrals, Jacobian, Notion of infitesimal 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)
Dirac Delta function and its properties: Definit ion of Dirac delta function. Representation as limit of a Gaussian function and rectangular function, Properties of Dirac delta function.

Unit IV

Orthogonal Curvilinear Coordinates: Orthogonal Curvilinear Coordinates, Derivation of Gradient, Divergence, Curl and Laplacian in Cartesian, Spherical and Cylindrical Coordinate Systems, Comparison of velocity and acceleration in cylindrical and spherical coordinate system.

Text Books:

Mathematical Methods for Physicists, G. B. Arfken, H. J. Weber, F. E. Harris (2013, 7th Edn., Elsevier)
Advanced Engineering Mathematics, Erwin Kreyszig (Wiley India)

Reference books:

Mathematical Physics C. Harper (Prentice Hall India)
Complex Variable: Schaum’s Outlines Series M. Spiegel (2nd Edition, Mc- Graw Hill Education)
Complex variables and applications, J. W. Brown and R.V. Churchill
Mathematical Physics, Satya Prakash (SultanChand)
Mathematical Physics, B. D. Gupta (4th edition, Vikas Publication)
Mathematical Physics and Special Relativity, M. Das, P.K. Jena and B.K.Dash (Srikrishna Prakashan)
Mathematical Physics –H.K.Das, Dr. Rama Verma (S. Chand Publishing)
Mathematical Physics, B.S. Rajput,(Pragati Prakashana)

Questions

2024 | +3-I-S-NEP-Major-I-P-1-Sc-Phy | Full Marks : 100

PART-I | 1. Answer the following Questions. | 1x10

The trace of $y=3^x$ is symmetrical about ____ axis.
The second term of Taylor series expansion for the function ${(1-x)}^{-2}$ is ____.
Write the unit vector perpendicular to ($\hat{i} + \hat{j}$) lies in same plane.
For any scaler function $\phi$, the value of curl grad $\phi$ is ____.
The Dirac delta function $\delta(-x)=$ _____.
The function $\tan x$ is not continuous at $x=$_____.
Does the function $f(x)$ which is continuous at $x_0$ will be necessarily differentiable at xo ? (yes/no/can not say)
The value of $\oint_c \vec{r}.\vec{dr}$ is _____.
In spherical polar co-ordinate system $\hat{r}\times\hat{\theta}=$____.
Does the vector triple product is associative ? (yes/no/ can not say)

PART-II | Answer all | 2x9

Solve $dy/dx = 2x-7$ where $y(2)= 0$.
Trace the curve $y = x \sin x$.
If $\vec{a}$ is a constant vector then show that $\hat{\nabla}. (\vec{a}\times \vec{r})=0$.
If $x=r \cos \theta$, and $y=r \sin \theta$ then evaluate $\partial (x, y) / \partial (r, \theta)$.
Prove that, $\hat{i}\times (\vec{a}\times \hat{i})+\hat{j}\times (\vec{a}\times \hat{j})+\hat{k}\times (\vec{a}\times \hat{k})=2\vec{a}$.
Show that $\vec{a} \ne \vec{c}$ although $\vec{a}.\vec{b}-\vec{b}.\vec{c}$.
Evaluate the directional derivative of the function $\phi =x^2-y^2+2z^2$ from P(1,2,3) to Q(2,3,4).
Prove that $\vec{\nabla}\times (\phi \vec{A})=\phi\vec{\nabla}\times \vec{A}+\vec{A}\times \vec{\nabla}\phi$.
For position vector $\vec{r}$ show that $\vec{\nabla}\vec{r}^n = n\vec{r}^{n-1}\hat{r}$.

PART-III | Answer any eight questions | 5x8

Soslve the differential equation $(1+x^2) dy - (1+y^2)dx = 0$.
Using complementary function and particular Integral solve $y'' +4y= 2\sin 2x$.
Verify the linear independence of the functions $e^{ax} \sin bx$ and $e^{ax}\cos bx$.
State and prove Green's theorem.
Prove that $\vec{\nabla}\times (\vec{\nabla}\times \vec{F}) =\vec{\nabla}.(\vec{\nabla}.\vec{F})-\nabla^2 \vec{F}$.
Explain the existance and Uniqueness theorem.
Obtain the acceleration expression in circular cylindrical co-ordinate system.
Evaluate $\int\int_R (x^2 + y^2 ) dxdy$ where R is the region in 1st quadrant of XY plane with side 2 and one vertex at origin as a squire.
Prove that $\int_{-\infty}^\infty \delta (x-a) \delta (x-b) dx = \delta (a-b)$.
Show that scalar product of two vectors remains invariant under rotations.

PART-IV | Answer any four of the following questions. | 8x4

What are Lagrange's multipliers ? Using them show that "The Maximum volume of solid inscribed in a sphere is a cube".
Establish the physical significance of scalar triple product. Obtain the total surface area and volume of a parallelepiped whose edges ${(\hat{i}+2\hat{j}+3\hat{k})}$, ${(3\hat{i}+4\hat{j}-\hat{k})}$ and ${(\hat{i}+2\hat{j}+\hat{k})}$.
State and prove Gauss divergence theorem. Using it obtain the volume of a sphere having radius r.
Derive the expression for Laplacian ${(\nabla^2)}$ in spherical polar co-ordinate system.
Solve the differential equation ${(x^2+y^2)} dx - 2xy dy = 0$ and obtain Dirac delta function as the limitation of Gaussian function.

2023 | +3-I-S-CBCS(MS)-Sc(H)-Core-I-Phy-R&B | Full Marks : 60

PART-I | Answer the following questions. 1×8

The second term in binomial expansion of $(1+x)^{-1}$ is _____.
If $|\vec A+\vec B | = | \vec A - \vec B | $, then angle between $\vec A$ and $\vec B$ is ____.
Curl of an irrotational vector field is ______.
Value of $x\delta (x-2)$ is _____.
Given $f(x,y)=xy$, value of $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}$ is _____.
If $|\vec A \times \vec B| =\vec A.\vec B$, then angle between $\vec A$ and $\vec B$ is _____.
Scalar triple product is expressed as _____.
Laplacian operator in Cartesian co-ordinate system is given by _____.

PART-II | Answer any eight of the following. | 1.5x8

Expand sinx about x=0.
Find the angle between $\hat i + 2\hat j -\hat k$ and $-\hat i +\hat j+\hat k$.
Find the volume of the cube with sides $\vec a = 2\hat i$, $\vec b = \hat j$ and $\vec c = 2\hat k$.
Write the condition for exactness of a differential equation.
If $\vec A = -y\hat i + x\hat j$, find $\vec\nabla \times \vec A$.
State Gauss' divergence theorem.
Find first order partial derivative of $\sqrt {x^2+y^2}$.
Find integrating factor of differential equation. $$(1+x)\frac{dy}{dx}-y=e^x$$
Show that $(\vec a - \vec b)\times (\vec a + \vec b) = 2(\vec a\times \vec b)$.
State uniqueness theorem.

PART-III | Answer any eight of the following | 2×8

Expand $e^x \cos y$ at (0, 0) up to three terms.
Evaluate $\frac{\partial^2}{\partial y^2}[\exp (5\ln (xy))]$.
Give geometrical interpretation of gradient of a scalar field.
$(D^2+4)y=\cos 2x$.
Discuss steps of Lagrange undetermined methods to obtain extremum value of a function.
Find area of a parallelogram whose adjacent sides are $\hat i - 2\hat j +3\hat k$ and $2\hat i + \hat j -4\hat k$.
Using Green's theorem evaluate $\int (x^2+xy) dx + (x^2+y^2) dy$ for a square formed by the lines $y=\pm 1$, $x=\pm 1$.
Prove that $x\delta' (x) = - \delta (x)$.
Test the equation $y dx - x dy = xy^3 dy$ is exact or not.
Show that curl of gradient of a scalar function is zero.

PART-IV | Answer the following question. | 6x4

Solve the differential equations
a. $(e^y +1)\cos x dx + e^y \sin x dy = 0$
b. $(1+x^2) dy + 2xy dx = 0$
OR
Trace the following curves:
a. $y^2(a^2+x^2)=x^2(a^2-x^2)$
b. $y^2x^2= x^2-a^2$
Solve the following differential equations :
a. $y''-3y'-4y=0$ subject to $y(0)=1$ and $y'(0)=0$.
b. $x\ln x \frac{dy}{dx} +y=2\ln x $
OR
a. Prove that : $\vec a \times (\vec b \times \vec c )= (\vec a.\vec c)\vec b - (\vec a .\vec b)\vec c$.
b. Prove that: $(\vec a \times \vec b).(\vec c \times \vec d) = (\vec a.\vec c)(\vec b . \vec d)-(\vec a.\vec d)(\vec b.\vec c)$.
Derive expression for divergence of a vector in Orthogonal curvilinear coordinates and hence in Cartesian co-ordinates.
OR
a. Given $\vec A = 3y^2 \hat i +4z \hat j+6y \hat k$. Find $\oint \vec A . \vec{dr}$ for the circle of radius 2 centered at (0, 0, 0) in the yz plane.
b. Prove that $\delta (x^2-a^2)= \frac{1}{2a} [\delta (x+a) + \delta (x-a)]$.
Prove the vector identity: $$ \vec \nabla \times (\vec A \times \vec B)=\vec A (\vec \nabla. \vec B) - \vec B (\vec \nabla. \vec A) + (\vec B .\vec \nabla) \vec A- (\vec A. \vec \nabla) \vec B $$
OR
a. Applying Stoke's theorem prove that $\oint \vec r . \vec{dr}=0$.
b. Show that $r^{-3}\vec r$ is a solenoidal vector, where $\vec=$ is the position vector.

2022 | +3-I-S-CBCS(MS)-Sc(H)-Core-I-Phy-R&B | Full Marks : 60

PART-I | Fill in the blank with appropriate words | 1x8

What is the average value of the function $f(x) = \sin x$ in the interval $[0, \pi]$?
$x^2\frac{\partial^2f}{\partial x^2}+y^2\frac{\partial ^2f}{\partial y^2}$ is a _____ differential equation.
The integrating factor of homogeneous equation $M.dx+N.dy=0$ is _____.
The value of $\vec{A}.(\vec{A}\times \vec{B})$ is ____ .
The scale factors of cylindrical coordinates are _____.
The value of $\int_{-\infty}^\infty \delta (x).dx =$_____.
The value of $\vec{\nabla}. r^n$ is _____.
Represent Jacobian of (u,v) with respect to (x,y).

PART-II | Answer any eight within two to three sentences | 1.5x8

Solve $\frac{dy}{dx} =\cot x . \tan y$.
What is Wronskian ?
Write condition for exactness.
What is integrating factor ?
If $\vec{A}+ \vec{B}+\vec{C}=0$, then prove that $\vec{A}.( \vec{B}\times\vec{C})=0$.
Prove that $x.\delta (x)= 0$.
Express $\vec{\nabla}\times \vec{A}$ in terms of spherical polar coordinates.
State Green's Theorem in a plane.
Show that $\vec{F} = (2xy+z^3)\hat{i}+x^2\hat{j}+3xz^2\hat{k}$ is a conservative force field.
Write volume integrals of vector field.

PART-III | Answer any eight of the following (in maximum 75 words.) | 2x8

Give physical interpretation of Gauss Divergence Theorem.
Evaluate $\oint_s d\sigma$, where 's' is a closed surface.
Find the directional derivative of f= xy + yz + zx in the direction of vector $\vec{A}=\hat{i}+2\hat{j}+2\hat{k}$ at the point p(1,2,0).
Explain Laplacian ($\vec{\nabla}$) in terms of orthogonal curvilinear coordinates.
Define Dirac Delta function and how it is related with step function.
Write relation between cylindrical and spherical polar coordinates.
Explain scalar and vector field with examples.
If the implicit function is $f(x,y) = x^2- xy -y^2= 0$ then find dy/dx.
Solve the equation $dy/dx = (y+1)/(x+1)$.
Write Taylor series for a real function of Real variables.

PART-IV | Answer within 500 words each | 6x4

4) Derive an expression for linear Differential Equation of First order and find its solution by the Method of Integrating Factor.
OR
a. Find the approximate value of $\sqrt{10}$ using Binomial series.
b. Solve the differential equation $dy/dx=(y+1)\tan x$.

5) Describe and derive an expression for Exact differential.
OR
Explain scalar Triple product with its physical significance and important features.

6) Derive an expression for velocity and accleration in cylindrical and spherical polar coordinates in 3D.
OR
a. Describe Dirac Delta function as a limit of rectangular function.
b. Prove that $x.\delta'(x)=-\delta(x)$.

7) Explain gradient of a scalar field, Also explain its physical significance and geometrical interpretation.
OR
a. If $x=r\cos \theta$ and $y= r\sin \theta$, evaluate $\partial(x,y)/\partial (r,\theta)$.
b. Prove that $\int_c \phi.d\vec{r} = \int_s \hat{n} \times \vec{\nabla} \phi.ds$ using stoke's Theorem.

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