Mathematical Physics I Vectors, Differential Equations…

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

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

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

PART-II | Answer all | 2x9

1. Solve $dy/dx = 2x-7$ where $y(2)= 0$.
2. Trace the curve $y = x \sin x$.
3. If $\vec{a}$ is a constant vector then show that $\hat{\nabla}. (\vec{a}\times \vec{r})=0$.
4. If $x=r \cos \theta$, and $y=r \sin \theta$ then evaluate $\partial (x, y) / \partial (r, \theta)$.
5. Prove that, $\hat{i}\times (\vec{a}\times \hat{i})+\hat{j}\times (\vec{a}\times \hat{j})+\hat{k}\times (\vec{a}\times \hat{k})=2\vec{a}$.
6. Show that $\vec{a} \ne \vec{c}$ although $\vec{a}.\vec{b}-\vec{b}.\vec{c}$.
7. Evaluate the directional derivative of the function $\phi =x^2-y^2+2z^2$ from P(1,2,3) to Q(2,3,4).
8. Prove that $\vec{\nabla}\times (\phi \vec{A})=\phi\vec{\nabla}\times \vec{A}+\vec{A}\times \vec{\nabla}\phi$.
9. 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

1. Soslve the differential equation $(1+x^2) dy - (1+y^2)dx = 0$.
2. Using complementary function and particular Integral solve $y’’ +4y= 2\sin 2x$.
3. Verify the linear independence of the functions $e^{ax} \sin bx$ and $e^{ax}\cos bx$.
4. State and prove Green’s theorem.
5. Prove that $\vec{\nabla}\times (\vec{\nabla}\times \vec{F}) =\vec{\nabla}.(\vec{\nabla}.\vec{F})-\nabla^2 \vec{F}$.
6. Explain the existance and Uniqueness theorem.
7. Obtain the acceleration expression in circular cylindrical co-ordinate system.
8. 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.
9. Prove that $\int_{-\infty}^\infty \delta (x-a) \delta (x-b) dx = \delta (a-b)$.
10. Show that scalar product of two vectors remains invariant under rotations.

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

1. What are Lagrange’s multipliers ? Using them show that “The Maximum volume of solid inscribed in a sphere is a cube”.
2. 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 ${(\hat{i}+2\hat{j}+\hat{k})}$.
3. State and prove Gauss divergence theorem. Using it obtain the volume of a sphere having radius r.
4. Derive the expression for Laplacian ${(\nabla^2)}$ in spherical polar co-ordinate system.
5. Solve the differential equation ${(x^2+y^2)} dx - 2xy dy = 0$ and obtain Dirac delta function as the limitation of Gaussian function.

+3-I-S-CBCS(MS)-Sc(H)-Core-I-Phy-R&B

2022 | 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.