2008

1) Show that
${\oint }_{\mathrm{S}}{\displaystyle \frac{\mathrm{d}\mathrm{s}}{\sqrt{{\mathrm{a}}^2{\mathrm{x}}^2+{\mathrm{b}}^2{\mathrm{y}}^2+{\mathrm{c}}^2{\mathrm{z}}^2}}}={\displaystyle \frac{4\pi }{\sqrt{\mathrm{a}\mathrm{b}\mathrm{c}}}}$, where $\mathrm{s}$ is the surface of the ellipsoid $a{x}^2+b{y}^2+c{z}^2=1$.

2) Find the unit vector along the normal to the surface $z=x^2+y^2$ at the point $(-1,-2,5)$.

[10M]

3) Prove that the necessary and sufficient condition for the vector function $\overrightarrow{\mathrm{V}}$ of the scalar variable $t$ to have constant magnitude is $\overrightarrow{\mathrm{V}}\cdot {\displaystyle \frac{\mathrm{d}\overrightarrow{\mathrm{v}}}{\mathrm{d}\mathrm{t}}}=0$.

[10M]

4) Prove that the shortest distances between a diagonal of a rectangular parallelopiped whose sides are of lengths a, $\mathrm{b}$ and $\mathrm{c}$ and the edges not meeting it are

[10M]

5) If $\overrightarrow{F}=2x^2\hat{i}-4yz\hat{j}+zx\hat{k}$, evaluate

${\iint }_5\overrightarrow{F}\cdot \hat{n}ds$,

where $s$ is the surface of the cube bounded by the planes $x=0$, $x=1$, $y=0$, $y=1$, $z=0$, $z=1$.

[10M]

2007

1) Evaluate

where $\overline{F}=c[-3a{\sin }^2\theta \cos \theta \overline{i}+a(2\sin \theta -3{\sin }^2\theta )\hat{j}+b\sin 2\theta \overrightarrow{k}]$ and the curve $\mathrm{C}$ is given by $\overrightarrow{r}=a\cos \theta \overrightarrow{i}+a\sin \theta j+b\theta \overline{k}$ $\theta$ varying from ${\displaystyle \frac{\pi }{4}}$ to ${\displaystyle \frac{\pi }{2}}$.

2) Show that

Where $\overrightarrow{a}$ is a constant vector and

[10M]

3) Find the curvature and torsion at any point of the curve
$x=a\cos 2t,y=a\sin 2t,z=2a\sin t$

[10M]

4) Evaluate the surface integral ${\int }_S(yz\overline{i}+zx\overline{j}+xy\overline{k}).d\overrightarrow{a}$ Where $\mathrm{S}$ is the surface of the sphere $x^2+y^2+z^2=1$ in the first octant.

[10M]

5) Apply Stokes’ theorem to prove that

Where $C$ is curve given by

[10M]

2006

1) If $\overrightarrow{f}=3xyi-y^{2}j$, determine the value of ${\int }_C\overrightarrow{f}.d\overrightarrow{r}$, where $C$ is the curve $y=2x^2$ in the $xy$-plane from $(0,0)$ to $(1,2)$.

[10M]

2) If $u\overrightarrow{f}=\overrightarrow{\mathrm{\nabla }}v$, where $u$, $y$ are scalar fields and $\overrightarrow{f}$ is a vector field, find the value of $\overrightarrow{f}\cdot$ curt $\overrightarrow{f}$.

[10M]

3) If $O$ be the origin, $A,B,$ two fixed points and $P(x,y,z)$ a vaniable point, show that $\mathrm{curl}(\overrightarrow{P}A\times \overrightarrow{PB})=2\overrightarrow{AB}$

[10M]

4) Using Stokes’ theorem, determine the value of the integral

where $\mathrm{\Gamma }$ is the curve defined by

[10M]

5) Prove that the cylindrical co-ordinate system is orthogonal.

[10M]

2005

1) For the curve $$ \vec{r}=a\left(3 t-t^{3}\right) \vec{i}+3 a t^{2}\vec{j}+a\left(3 t+b^{3}\right) \vec{k} $$ being $a$ constant. Show that the radius of curvature is equal to its radius of torsion.

2) Find $f(r)$ if $f(x)=6$ both solenoidal and irrotational.

[10M]

3) Evaluate ${\iint }_S\overline{F}\cdot d\overline{s}$, where $\overline{F}=yz\overline{i}+zx\hat{j}+xy\overline{k}$ and $S$ is the part of the sphere $x^2+y^2+z^2=1$ that lies in the first octant.

[10M]

4) Verify the divergence theorem for $\overline{F}=4x\overline{i}-2y^2\overline{j}+z^2\overline{k}$ taken over the region bounded by $x^2+y^2=4$, $z=0$ and $z=3$.

[10M]

5) By using vector methods, find an equation for the tangent plane to the surface $z=x^2+y^2$ at the point $(1,-1,2)$.

[10M]