Divergence and curl,

Q. For a position vector $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$ the norm of the vector can be defined as $|\overrightarrow{r}|=\sqrt{x^2+y^2+z^2}.$ Given a function $\varphi =ln|\overrightarrow{r}|,$ its gradient $▽\varphi$ is

1. $\overrightarrow{r}$
2. $\frac{\overrightarrow{r}}{|\overrightarrow{r}|}$
3. $\frac{\overrightarrow{r}}{\overrightarrow{r}.\overrightarrow{r}}$
4. $\frac{\overrightarrow{r}}{|\overrightarrow{r}{|}^3}$

Q. The divergence of the vector field $3xz\hat{i}+2xy\hat{j}-y{z}^2\hat{k}$ at a point $(1,1,1)$ is equal to

1. $7$
2. $4$
3. $3$
4. $0$

Q. $\text{Let}f=(x+y+1)\hat{i}+\hat{j}-(x+y)\hat{k}\text{then}f.\text{curl}f\text{is equal to\_\_\_\_\_}$.

1. 0

Q. Divergence of the vector field $x^{2}z\hat{i}+xy\hat{j}-y{z}^2\hat{k}$ at $(1,-1,1)$ is

1. 0
2. 3
3. 5
4. 6

Q. Divergence of the three dimensional radial vector field $\overrightarrow{r}$ is

1. $3$
2. $1/r$
3. $\hat{i}+\hat{j}+\hat{k}$
4. $3(\hat{i}+\hat{j}+\hat{k})$

Q. Given a vector $\overrightarrow{u}$ = $\frac{1}{3}$$(-y^3\hat{i}+x^3\hat{j}+z^3\hat{k})$ and $\hat{n}$ as the unit normal vector to the surface of the hemisphere ($x^2+y^2+z^2=1;z>0$), the value of integral $\int (▽\times \overrightarrow{u}).\hat{n}dS$ evaluated on the curved surface of the hemisphere S is

1. -$\frac{\pi }{2}$
2. $\pi$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{3}$

Q. The divergence of the vector field $\overrightarrow{u}=e^x(cosy\hat{i}+siny\hat{j})$ is

1. 0
2. $e^{x}cosy+e^{x}siny$
3. $2e^{x}cosy$
4. $2e^{x}siny$

Q. If the velocity vector in a two dimensional flow field is given by $\overrightarrow{v}=2xy\hat{i}+(2y^2-x^2)\hat{j}$, the vorticity vector, curl $\overrightarrow{v}$ will be

1. $2y^2\hat{j}$
2. $6y\hat{k}$
3. zero
4. $-4x\hat{k}$

Q. Divergence of the vector field $\overrightarrow{v}(x,y,z)=(-xcosxy+y)\hat{i}+\left(ycosxy\right)\hat{j}+[\left(sin{z}^2\right)+x^2+y^2]\hat{k}$

1. $2zcos{z}^2$
2. $sinxy+2zcos{z}^2$
3. $xsinxy-cosz$
4. none of these

Q. The vector field $\overrightarrow{F}=x\hat{i}-y\hat{j}$ (where $\hat{i}$ and $\hat{j}$ are unit vector) is

1. divergence free, but not irrotational
2. irrotational, but not divergence free
3. divergence free and irrorational
4. neither divergence free nor irrotational

Q. What is curl of the vector field $2x^{2}y\hat{i}+5z^2\hat{j}-4yz\hat{k}?$

1. $-14z\hat{i}-2x^2\hat{k}$
2. $6z\hat{i}+4x\hat{j}-2x^2\hat{k}$
3. $6z\hat{i}+8xy\hat{j}+2x^{2}y\hat{k}$
4. $-14z\hat{i}+6y\hat{j}+2x^2\hat{k}$

Q. The direction of vector $A$ is radially outward from the origin, with $|A|=K{r}^n$ where $r^2=x^2+y^2+z^2$ and $K$ is constant. The value of $n$ for which $▽.A=0$ is

1. $-2$
2. $2$
3. $1$
4. $0$

Q. For a vector $E$, which one of the following statement is NOT TRUE?

1. If $▽.E=0,E$ is called solenoidal
2. $▽\times E=0,E$ is called conservative
3. If $▽\times E=0,E$ is called irrotational
4. If $▽.E=0,E$ is called irrotational

Q. The figures show diagramatic representations of vector fields, $\overrightarrow{X},\overrightarrow{Y}$ and $\overrightarrow{Z}$, respectively. Which one of the following choices is true?

1. $▽.\overrightarrow{X}=0,▽\times \overrightarrow{Y}\ne 0,▽\times \overrightarrow{Z}=0$
2. $▽.\overrightarrow{X}\ne 0,▽\times \overrightarrow{Y}=0,▽\times \overrightarrow{Z}\ne 0$
3. $▽.\overrightarrow{X}\ne 0,▽\times \overrightarrow{Y}\ne 0,▽\times \overrightarrow{Z}\ne 0$
4. $▽.\overrightarrow{X}=0,▽\times \overrightarrow{Y}=0,▽\times \overrightarrow{Z}=0$

Q. Consider the two-dimensional velocity field given by $\overrightarrow{V}=(5+a_{1}x+b_{1}y)\hat{i}+(4+a_{2}x+b_{2}y)\hat{j}$. where $a_1,b_1,a_2$ and $b_2$ are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

1. $a_1+b_1=0$
2. $a_1+b_2=0$
3. $a_2+b_1=0$
4. $a_2+b_2=0$

Q. Find the divergence of the vector field.
$V=2x^{2}i+5y^{3}j+3z^{4}katx=2,y=3,z=4$

1. 1204
2. 602
3. 803
4. 911

Q. The value of $▽.A$ where $A=3xy{\overrightarrow{a}}_x+x{\overrightarrow{a}}_y+xyz{\overrightarrow{a}}_z$ at a point $(2,-2,2)$ is

1. $-10$
2. $-6$
3. $2$
4. $4$

Q. The divergence of the vector field $\overrightarrow{A}=x{\hat{a}}_x+y{\hat{a}}_y+z{\hat{a}}_z$ is

1. $0$
2. $1/3$
3. $1$
4. $3$