PYQs

1) Let $\overrightarrow{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$. Show that $curl(curl\overrightarrow{v})=grad(div\overrightarrow{v})-{\mathrm{\nabla }}^2\overrightarrow{v}$

[2018, 12M]

2) Calculate ${\mathrm{\nabla }}^2\left(r^n\right)$ and find its expression in terms of $r$ and $n$, $r$ being the distance of any point $(x,y,z)$ from the origin, $n$ being a constant and ${\mathrm{\nabla }}^2$ being the Laplace operator.

[2013, 10M]

3) If $u$ and $v$ are two scalar fields and $\overrightarrow{f}$ is a vector field, such that $u\overrightarrow{f}=gradv$, find the value of $\overrightarrow{f}\cdot curl\overrightarrow{f}$.

[2011, 10M]

4) Prove that $\mathrm{div}(f\overrightarrow{V})=f(div\overline{V})+(\mathrm{grad}.f)\overrightarrow{V}$, where $f$ is a scalar function.

[2010, 20M]

5) Prove that ${\mathrm{\nabla }}^{2}f(r)={\displaystyle \frac{d^{2}f}{d{r}^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{df}{dr}}$ where $r={(x^2+y^2+z^2)}^{1/2}$. Hence find $f(x)$ such that ${\mathrm{\nabla }}^{2}f(r)=0$.

[2008, 15M]

6) Show that $\mathrm{div}(\mathrm{grad}{r}^n)=n(n+1)r^{n-2}$, where $r=\sqrt{x^2+y^2+z^2}$.

[2009, 12M]

7) Show that curl $(k\times \mathrm{grad}{\displaystyle \frac{1}{r}})+\mathrm{grad}(k\mathrm{grad}{\displaystyle \frac{1}{r}})=0$ where $r$ is the distance from the origin and $k$ is the unit vector in the direction $OZ$.

[2005, 15M]

8) Prove the identity $\mathrm{\nabla }(\overline{A}\cdot \overline{B})=(\overline{B}.\overline{A})\overline{A}+(\overline{A}\cdot \mathrm{\nabla })\overline{B}+\overline{B}\times (\mathrm{\nabla }\times \overline{A})+\overline{A}\times (\mathrm{\nabla }\times \overline{B})$.

[2004, 15M]

9) Derive the identity ${\iiint }_V(\varphi {\mathrm{\nabla }}^2\psi -\psi {\mathrm{\nabla }}^2\varphi )dV={\iint }_S(\varphi \mathrm{\nabla }\psi -\psi \mathrm{\nabla }\varphi )\hat{n}dS$ where $V$ is the volume bounded by the closed surface $S$.

[2004, 15M]

10) Show that if $a^{\mathrm{\prime }}$, $b^{\mathrm{\prime }}$ and $c^{\mathrm{\prime }}$ are the reciprocals of the non-coplanar vectors $a$, $b$ and $c$, then any vector $r$ may be expressed as $r=(r.a^{\mathrm{\prime }})a+(r.c^{\mathrm{\prime }})c$.

[2003, 12M]

11) Prove the identity $\mathrm{\nabla }{A}^2=2(A.\mathrm{\nabla })A+2A\times (\mathrm{\nabla }\times A)$ where $\mathrm{\nabla }=\hat{i}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}}+\hat{j}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }y}}+\hat{k}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}$.

[2003, 15M]

12) Let $\overline{R}$ be the unit vector along the vector $\overline{r}(t)$. Show that $\overline{R}\times {\displaystyle \frac{\overline{dR}}{dt}}={\displaystyle \frac{\overline{r}}{r^2}}\times {\displaystyle \frac{\overline{dr}}{dt}}$ where $r=|r|$.

[2002, 15M]

13) Show that $(curl\overline{v})=\mathrm{grad}(div\overline{v})-{\mathrm{\nabla }}^{2}v$.

[2002, 15M]

14) Show that $curl{\displaystyle \frac{a\times r}{r^3}}=-{\displaystyle \frac{a}{r^3}}+{\displaystyle \frac{3r}{r^5}}(a\cdot r)$, where $a$ is constant vector.

[2001, 12M]

1) Prove that the vector $\overrightarrow{a}=3\overrightarrow{i}+\hat{j}-2\hat{k}$, $\overrightarrow{b}=-\hat{i}+3\hat{j}+4\hat{k}$, $\overrightarrow{c}=4\hat{i}-2\hat{j}-6\hat{k}$ can from the sides of a triangle. Find the length of the medians of the triangle.

[2016, 10M]

2) If $\overrightarrow{A}=x^{2}yz\overrightarrow{i}-2x{z}^3\overrightarrow{j}+x{z}^2\overrightarrow{k}$, $\overrightarrow{B}=2z\overrightarrow{i}+y\overrightarrow{j}-x^2\overrightarrow{k}$, find the value of ${\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }x\mathrm{\partial }y}}(\overrightarrow{A}+\overrightarrow{B})$ at $(1,0,-2)$.

[2012, 12M]

3) If $\overline{\mathrm{A}}=2\hat{i}+\hat{k}$, $\overline{\mathrm{B}}=\hat{i}+\hat{j}+\hat{k}$, $\overline{C}=4\hat{i}-3\hat{j}-7\hat{k}$, determine a vector $\overline{R}$ satisfying the vector equation $\overline{R}\times \overline{B}=\overline{C}\times \overline{B}$ and $\overline{R}\cdot \overline{\mathrm{A}}=0$.

[2006, 15M]

4) Let $D$ be a closed and bounded region having boundary $S$. Further, let $f$ is a scalar function having second partial derivatives defined on it. Show that ${\iint }_s(\text{ fgradf })\hat{n}ds={\iiint }_v[|gradf{|}^2+f{\mathrm{\nabla }}^{2}f]dv$. Hence ${\iint }_{s}f(\text{ grad }f)\cdot \hat{n}ds$ or otherwise evaluate for $f=2x+y+2z$ over $s:x^2+y^2+z^2=4$.

[2002, 15M]