1988

1) Define the divergence of a vector point function, prove that $\mathrm{div}(\overrightarrow{u}\times \overrightarrow{v})=\overrightarrow{v}\cdot \mathrm{curl}\overrightarrow{u}-\overrightarrow{u}\mathrm{curl}\overrightarrow{v}$.

2) Using Gauss divergence theorem, evaluate ${\iint }_s(x\hat{i}+y\hat{i}+z^2\hat{k})\cdot \hat{n}ds$ where $\mathrm{S}$ is the closed surface bounded by the cone $x^2+y^2=2$ and the plane $\mathrm{Z}=1$ and $\hat{n}$ is the outward unit normal to $\mathrm{S}$.

1987

1) Show that for a vector field $\overrightarrow{f},$ curl (curl $\overrightarrow{f})$=$\mathrm{grad}(\mathrm{div}\overrightarrow{f})-{\mathrm{\nabla }}^2\overrightarrow{f}$.

2) If $\overrightarrow{r}$ is the position vector to a point whose distance from the origin is $\mathrm{r},$ prove that $\mathrm{div}\overrightarrow{f}=0$ if $\overrightarrow{f}={\displaystyle \frac{\overrightarrow{r}}{r^3}}$.

3) Prove that for a three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{b}(\overrightarrow{a}\cdot \overrightarrow{c})-\overrightarrow{c}(\overrightarrow{a}\cdot \overrightarrow{b})$ and explain its geometric meaning.

1986

1) Let $\overrightarrow{a},\overrightarrow{b}$ be given vectors in the three dimensional Euclidean space $E_3$ and let $\varphi (\overrightarrow{x})$ be a scalar field of the vectors $\overrightarrow{x}$ also of $E_3$. If $\varphi (\overrightarrow{x})=(\overrightarrow{x}\times \overrightarrow{a})\cdot (\overrightarrow{x}\times \overrightarrow{b}),$ show that
grad$\varphi (i.e,\mathrm{\nabla }\varphi (\overrightarrow{x}))$=$\overrightarrow{b}\times (\overrightarrow{x}\times \overrightarrow{a})+\overrightarrow{a}\times (\overrightarrow{x}\times \overrightarrow{b})$

2) If $\overrightarrow{f},\overrightarrow{g}$ are two vector fields in $E_3$ and if ‘div’, ‘curl’ are defined on an open set $S\subset E_3$ show that $\mathrm{div}(\overrightarrow{f}\times \overrightarrow{g})=\overrightarrow{g}.\mathrm{curl}\overrightarrow{f}-\overrightarrow{f}\cdot \mathrm{curl}\overrightarrow{g}$.

1985

1) If $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ are points $(3,-2,-1)$, $(1,3,4)$, $(2,1,-2)$ respectively. Find the distance from $\mathrm{P}$ to the plane $\mathrm{O}\mathrm{Q}\mathrm{R}$, where ${}^{\prime }{\mathrm{O}}^{\prime }$ is the origin.

2) Find the angle between the tangents to the curve $\overrightarrow{r}=t^2\hat{i}-2t\hat{j}+t^3\hat{k}$ at the points $\mathrm{t}=1$ and $\mathrm{t}=2$.

3) Find div $\mathrm{F}$ and curl $\mathrm{F}$, where $F=\mathrm{\nabla }(x^3+y^3+z^3-3xyz)$.

1983

1) Prove that curl (curl $\mathrm{F}$ ) $=$ grad div $\mathrm{F}-{\mathrm{\nabla }}^2\mathrm{F}$.