2000

1) In what direction from the point (-1,1,1) is the directional derivative of $f=x^{2}y{z}^3$ a maximum $?$ Compute its magnitude.

[10M]

2) Show that

(i) $(A+B)\cdot (B+C)\times (C+A)=2A\cdot B\times C$

(ii)$\mathrm{\nabla }\times (A\times B)=(B.\mathrm{\nabla })A-B(\mathrm{\nabla }.A)-(A.\mathrm{\nabla })B+A(\mathrm{\nabla }.B)$

[10M]

3) Evaluate ${\iint }_S\overrightarrow{F}\cdot \hat{n}dS$, where $\mathbf{F}=2xy\mathbf{i}+y{z}^2\mathbf{j}+xz\mathbf{k}$ and $\mathrm{S}$ is the surface of the parallelcpiped bounded by $\mathrm{x}=0$, $\mathrm{y}=0,\mathrm{z}=0$, $x=2$, $y=1$ and $z=3$.

[10M]

1999

1) If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are the position vectors of $A,B,C$ prove that $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}$ is a vector perpendicuar to the plane $ABC$.

[10M]

2) If $\overrightarrow{F}=\mathrm{\nabla }(x^3+y^3+z^3-3xyz),$ find $\mathrm{\nabla }\times \overrightarrow{F}.$

[10M]

3) Evaluate ${\int }_C(e^{-x}sinydx+e^{-x}cosydy)$ (Green’s theorem), where C is the rectangle whose vertices are $(0,0),(\pi ,0),(\pi ,{\displaystyle \frac{\pi }{2}})$ and $(0,{\displaystyle \frac{\pi }{2}})$.

[10M]

4) If $X,Y,Z$ are the components of contravariant vector in rectangular Cartesian coordinates $x,y,z$ in a three dimensional space, show that the components of the vectors in cylinderal coordinates $r,\theta ,z$ are $Xcos\theta +Ysin\theta ,{\displaystyle \frac{-X}{r}}sin\theta +{\displaystyle \frac{Y}{r}}cos\theta ,Z$

[10M]

5) Show that ${\mathrm{\nabla }}^2\varphi =g^{ij}({\displaystyle \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}}-{\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}^l}}\{{\displaystyle \frac{1}{ij}}\})$ where $\varphi$ is a scalar function of coordinates $x^i$.

[10M]

1998

1) If $r_1$ and $r_2$ are the vectors joining the fixed points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ respectively to a variable point $P(x,y,z)$, then find the value of $grad(r_1\cdot r_2)$ and $curl(r_1\times r_2)$.

[10M]

2) Show that $(a\times b)\times c=a\times (b\times c)$ if either $b=0$ (or any other vector is 0) or $c$ is collinear with $a$ or $b$ is orthogonal to $a$ and $c$(both).

[10M]

1997

1) Prove that if $\overrightarrow{A},\overrightarrow{B}$ and $\dot{C}$ are three given non-coplanar vectors, then any vector $\overrightarrow{F}$ can be put in the form $F=\alpha \overrightarrow{B}\times \overrightarrow{C}+\beta \overrightarrow{C}\times \overrightarrow{A}+\gamma \overrightarrow{A}\times \overrightarrow{B}$. For given determine $\alpha$, $\beta$, $\gamma$.

[10M]

2) Verify Gauss theorem for $\overrightarrow{F}=4xi-2y^2\hat{j}+z^2\hat{k}$ taken over the region bounded by $x^2+y^2=4$ $z=0$ and $z=3$.

[10M]

1996

1) If $x\overrightarrow{i}+y\hat{j}+z\hat{k}$ and $r=|\overrightarrow{r}|,$ show that:
(i) $\overrightarrow{r}\times$ grad f $(r)=0$
(ii) $\mathrm{div}\left(r^n\overrightarrow{r}\right)=(n+3)r^n$

[15M]

2) Verify Gauss’ divergence theorem for $\overrightarrow{F}=xy\hat{i}+z^2\hat{j}+2yz\hat{k}$ on the tetrahedron $x=y=z=0,x+y+z=1$

[15M]

1995

1) Let the region $\mathrm{V}$ be bounded by the smooth surface $\mathrm{S}$ and let $\mathrm{n}$ denote outward drawn unit normal vector at a point on $\mathrm{S}$. If $\varphi$ is harmonic in $\mathrm{V}$, show that ${\int }_S{\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }n}}dS=0$

[20M]

2) In the vector field $v(\mathrm{x})$, let there exist a surface on which $v=0\$.\$Showthat,atanarbitrarypointofthissurface,curl$v=0$$ is tangential to the surface or vanishes.

[20M]