1994

1) Show that $r^n\overrightarrow{r}$ is an irrotational vector for any value of $n$, but is solenoidal only if $n=-3$.

[10M]

2) If $\overrightarrow{F}=y\overrightarrow{i}+(x-2xz)\overrightarrow{j}-xy\overrightarrow{k},$ evaluate ${\iint }_S(\mathrm{\nabla }\times \overrightarrow{F})\cdot \overrightarrow{n}dS$,
where $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ above the xy plane.

[10M]

1993

1) Prove that the angular velocity of rotation at any point is equal to one half of the curl of the velocity vector $V$.

[10M]

–

2) Evaluate ${\iint }_s\mathrm{\nabla }\times FhdS$ where $S$ is the upper half surface of the unit sphere $x^2+y^2+z^2=1$ and $\overrightarrow{F}=z\hat{i}+x\hat{j}+y\hat{k}$.

[10M]

1992

1) If $\overrightarrow{f}(x,y,z)=(y^2+z^2)\overrightarrow{i}+(z^2+x^2)\overrightarrow{j}+(x^2+y^2)\overrightarrow{k}$, then calculate ${\int }_c\overrightarrow{f}\overrightarrow{dx},$ where $\mathrm{C}$ consists of (i) the line segment from $(0,0,0)$ to $(1,1,1)$ (ii) the three line segments $\mathrm{A}\mathrm{B},\mathrm{B}\mathrm{C}$ and $\mathrm{C}\mathrm{D},$ where $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$ are respectively the points $(0,0,0)$, $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$ (iii) the curve $\overline{x}+\overrightarrow{u}+\overline{u^2}j+\overline{u^{2}k}u$, from 0 to 1

[10M]

2) If $\overrightarrow{a}$ and $\overrightarrow{b}$ are constant vectors, show that

(i) $\mathrm{div}\{x\times (\overrightarrow{a}\times \overrightarrow{x})\}=-2\overrightarrow{x}\overrightarrow{a}$
(ii) $\mathrm{div}\{(\overrightarrow{a}\times \overrightarrow{x})\times (\overrightarrow{b}\times \overrightarrow{x})\}=2\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{x})-2b\cdot (\overrightarrow{a}\times x)$

[10M]

3) Obtain the formula

[10M]

1991

1) If $\varphi$ be a scalar point function and $\mathrm{F}$ be a vector point function, show that the components of $\mathrm{F}$ normal and tangential to surface $\varphi =0$ at any point there of are ${\displaystyle \frac{(F.\mathrm{\nabla }\varphi )\mathrm{\nabla }\varphi }{(\mathrm{\nabla }\varphi {)}^2}}$ and ${\displaystyle \frac{\mathrm{\nabla }\varphi \times (F\times \mathrm{\nabla }\varphi )}{(\mathrm{\nabla }\varphi {)}^2}}$.

2) Find the value of $\int$ curl F. dS taken over the portion of the surface $x^2+y^2-2ax+az=0,$ for which $z\ge 0$ when $F=(y^2+z^2-x^2)\hat{i}+(z^2+x^2-y^2)\hat{j}+(x^2+y^2-z^2)\hat{k}$

1989

1) Define the curl of a vector point function.

2) Prove that $\mathrm{\nabla }\times \left({\displaystyle \frac{\overrightarrow{r}}{r^2}}\right)=0$ where $\overrightarrow{r}=(x,y,z)$ and $r=|\overrightarrow{r}|$.