2004

1) Evaluate ${\int }_e\overrightarrow{F}\cdot \overrightarrow{d}r$ for the field $\overrightarrow{F}=$ grad $\left(x{y}^2{z}^3\right)$ where $e$ is the ellipse in which the plane $\mathrm{z}=2\mathrm{x}+3\mathrm{y}$ cuts the cylinder ${\mathrm{x}}^2+{\mathrm{y}}^2=12$ counterclockwise as viewed from the positive end of the z-axis looking towards the origin.

[10M]

2) Show that $\mathrm{div}(\overrightarrow{A}\times \overrightarrow{B})=\overrightarrow{B}\cdot \mathrm{curl}\overrightarrow{A}-\overrightarrow{A}\cdot \mathrm{curl}\overrightarrow{B}.$

[10M]

3) Evaluate curl $\left[{\displaystyle \frac{(2\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k})\times \overrightarrow{r}}{r^3}}\right]$, where $\overrightarrow{r}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$ and $r^3=x^3+y^3+z^3$.

[10M]

4) Evaluate ${\iint }_S(x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k})\cdot \overrightarrow{n}ds$, where $S$ is the surface $x+y+z=1$ lying in the first octant.

[10M]

5) Evaluate ${\mathrm{\nabla }}^{2}u$ in spherical polar coordinates.

[10M]

2003

1) Find expressions for curvature and torsion at a point on the curve $x=a\cos \theta$, $y=a\sin \theta$, $z=a\theta \cot \beta$

[10M]

2) If $\overline{r}$ is the position vector of the point $(\mathrm{x},\mathrm{y},2)$ with respect to the origin, prove that ${\mathrm{\nabla }}^{2}f(r)=f^{\mathrm{\prime }\mathrm{\prime }}(r)+{\displaystyle \frac{2}{r}}{f}^{\mathrm{\prime }}(r)$
Find $f(r)$ such that ${\mathrm{\Delta }}^{2}f(r)=0$.

[10M]

3) If $\overrightarrow{F}$ is solenoidal, prove that Curl Curl Curl Curl $\overrightarrow{F}={\mathrm{\nabla }}^4\overrightarrow{F}$.

[10M]

4) Verify Stoke’s Theorem when
$\overline{\mathbf{F}}=(2xy-x^2)\overline{i}-(x^2-y^2)\overline{d}$
and $C$ is the boundary of the region enclosed by the parabolas $y^2=x$ and $x2=y$

[10M]

TBC

5) Express $\mathrm{\nabla }\times \overrightarrow{F}$ and ${\mathrm{\nabla }}^2\mathrm{\Phi }$ in cylindrical co-ordinates,

[10M]

2002

1) Find the eurvature and torsion of the curve $x={\displaystyle \frac{(2t+1)}{t-1}}$, $y={\displaystyle \frac{t^2}{t-1}}$, $z=t+2$. Interpret your answer.

[10M]

2) State Stoke’s theorem and then verify it for $\overline{A}=(x^2+1)\overline{i}+xy\overline{j}$ integrated round the square in the plane

$z=0$ whose sides are along the lines

[10M]

1. Prove that: (i) $\overrightarrow{\mathrm{\nabla }}\times (\overrightarrow{\mathbf{A}}\times \overrightarrow{\mathbf{B}})=\overrightarrow{\mathbf{A}}(\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{\mathbf{B}})-\overrightarrow{\mathbf{B}}(\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{\mathbf{A}})+(\overrightarrow{\mathbf{B}}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{\mathbf{A}}-(\overrightarrow{\mathbf{A}}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{\mathbf{B}}$

[10M]

(ii) $\mathrm{curl}{\displaystyle \frac{\overrightarrow{a}\times \overrightarrow{r}}{r^3}}$= $-{\displaystyle \frac{\overrightarrow{a}}{r^3}}+{\displaystyle \frac{3\overrightarrow{r}}{r^3}}(\overrightarrow{a}\cdot \overrightarrow{r})$, $\overrightarrow{a}$ = constant vector.

4) Show that if $\overrightarrow{A}\ne \overrightarrow{0}$ and both of the conditions $\overrightarrow{A}\cdot \overrightarrow{B}=\overrightarrow{A}\overrightarrow{C}$ and $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{A}\times \overrightarrow{C}$ hold simultancously then $\overrightarrow{B}=\overrightarrow{C}$ but if only one of these conditions holds then $\overline{B}\ne \overline{C}$ necessarily.

[10M]

5) Prove the following
(i) If $u_1,u_2,u_3$ are general coordinates, then
${\displaystyle \frac{\mathrm{\partial }\overrightarrow{r}}{\mathrm{\partial }{u}_1}}\times {\displaystyle \frac{\mathrm{\partial }\overrightarrow{r}}{\mathrm{\partial }{u}_2}}\times {\displaystyle \frac{\mathrm{\partial }vecr}{\mathrm{\partial }{u}_3}}$ and $\overrightarrow{\mathrm{\nabla }}{u}_1,\overrightarrow{\mathrm{\nabla }}{u}_2,\overrightarrow{\mathrm{\nabla }}{u}_3$ are reciprocal system of vectors.

[5M]

(ii) $({\displaystyle \frac{\mathrm{\partial }\overrightarrow{r}}{u_1}}\cdot {\displaystyle \frac{\mathrm{\partial }\overrightarrow{r}}{u_2}}\times {\displaystyle \frac{\mathrm{\partial }\overrightarrow{r}}{u_3}})(\overrightarrow{\mathrm{\nabla }}{u}_1\cdot \overrightarrow{\mathrm{\nabla }}{u}_2\times \overrightarrow{\mathrm{\nabla }}{u}_3)=1$.

[5M]

2001

1) Find an equation for the plane passing through the points $P_1(3,1,-2)$, $P_2(-1,2,4)$, $P_3(2,-1,1)$ by using vector method.

[10M]

2) Prove that $\mathrm{\nabla }\times (\mathrm{\nabla }\times \overline{A})=-{\mathrm{\nabla }}^2\overline{A}+\mathrm{\nabla }(\mathrm{\nabla }\cdot \overline{A})$.

[10M]

3) If $\mathrm{\nabla }\cdot \overline{E},\mathrm{\nabla }\cdot \overline{H},\mathrm{\nabla }\times \overline{E}=-{\displaystyle \frac{\mathrm{\partial }\overline{H}}{\mathrm{\partial }t}}$, $\mathrm{\nabla }\times \overline{H}={\displaystyle \frac{\mathrm{\partial }\overline{E}}{\mathrm{\partial }t}}$ Show that $\overline{E}$ & $\overline{H}$ satisfy ${\mathrm{\nabla }}^{2}u=-{\displaystyle \frac{{\mathrm{\partial }}^2\overline{u}}{\mathrm{\partial }{t}^2}}$

[10M]

4) Given the space curve $x=t$, $y=t^2$, $z={\displaystyle \frac{2}{3}}{t}^3$. Find
(i) the curvature $\rho$ (ii) the torsion $\tau$.

[10M]

5) If $F=(y^2+z^2-x^2)i+(z^2+x^2-y^2)j+(x^2+y^2-z^2)k$ evaluate ${\iint }_S\mathrm{curl}\overline{F}\cdot \hat{n}ds$, taken over the portion of the surface $x^2+y^2+z^2-2ax+az=0$ above the plane $z=0$ and verify Stokes’ theorem.

[10M]

2000

1) Solve for x the vector equation $px+x\times a=b,p\ne 0$

[10M]

2) Prove the identities:
(i) Curl grad $\varphi =0,(ii)$ div curl $f=0$.

If $\overrightarrow{OA}=ai,\overrightarrow{OB}=aj,\overrightarrow{OC}=ak$ form three coterminous edges of a cube and $\mathrm{s}$ denotes the surface of the cube, evaluate ${\int }_1^{\mathrm{\prime }}(x^3-yz)i-2x^{2}yj+2k\}$ nds by expressing it as volume integral, Where $n$ is the unit outward normal to $ds$.

[20M]