MCQs of Vector Calculus (Mathematics - II - 3110015)

Showing 1 to 10 out of 79 Questions

f (x, y, z) = 3 x^{2} y + 2 y - 3 z, then \nabla f =

The gradient of a scalar function

ϕ (x, y, z)

(a)	$\frac{\partial^{2} ϕ}{\partial x^{2}} \hat{i} + \frac{\partial^{2} ϕ}{\partial y^{2}} \hat{j} + \frac{\partial^{2} ϕ}{\partial z^{2}} \hat{k}$
(b)	$\frac{\partial ϕ}{\partial x} + \frac{\partial ϕ}{\partial y} + \frac{\partial ϕ}{\partial z}$
(c)	$\frac{\partial ϕ}{\partial x} \hat{i} + \frac{\partial ϕ}{\partial y} \hat{j} + \frac{\partial ϕ}{\partial z} \hat{k}$
(d)	$\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}} + \frac{\partial^{2} ϕ}{\partial z^{2}}$

Let

\bar{r} = x \hat{i} + y \hat{j} + z \hat{k}

and

r = |\bar{r}|

. Then

\nabla e^{r} =

Let

\bar{r} = x \hat{i} + y \hat{j} + z \hat{k}

and

r = |\bar{r}|

. Then

\nabla \log (r) =

F = f_{1} (x, y, z) \hat{i} + f_{2} (x, y, z) \hat{j} + f_{3} (x, y, z) \hat{k}

is a vector function, then the divergence of

F

(a)	$\frac{\partial^{2} f_{1}}{\partial x^{2}} \hat{i} + \frac{\partial^{2} f_{2}}{\partial y^{2}} \hat{j} + \frac{\partial^{2} f_{3}}{\partial z^{2}} \hat{k}$
(b)	$\frac{\partial f_{1}}{\partial x} + \frac{\partial f_{2}}{\partial y} + \frac{\partial f_{3}}{\partial z}$
(c)	$\frac{\partial f_{1}}{\partial x} \hat{i} + \frac{\partial f_{2}}{\partial y} \hat{j} + \frac{\partial f_{3}}{\partial z} \hat{k}$
(d)	$\frac{\partial^{2} f_{1}}{\partial x^{2}} + \frac{\partial^{2} f_{2}}{\partial y^{2}} + \frac{\partial^{2} f_{3}}{\partial z^{2}}$

The divergence of a vector function

A = 3 x^{2} \hat{i} + 5 {xy}^{2} \hat{j} + {xyz}^{3} \hat{k}

The divergence of a vector field

F = (x^{2} y, y^{2}, z^{2} x)

at a point

(- 1, 2, 3)

F = e^{xy} \hat{i} - \cos (y) \hat{j} + {(\sin (z))}^{2} \hat{k}

, then

\nabla \cdot F =

For the scalar function

f

, div(grad

f

) is equal to

(a)	$\frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k}$
(b)	$0$
(c)	$\frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}}$
(d)	$\frac{\partial^{2} f}{\partial x^{2}} \hat{i} + \frac{\partial^{2} f}{\partial y^{2}} \hat{j} + \frac{\partial^{2} f}{\partial z^{2}} \hat{k}$

10.

ϕ (x, y, z) = xyz - 2 y^{2} z + x^{2} z^{2}

, then

\nabla \cdot (\nabla ϕ)

at the point

(2, 4, 1)

Showing 1 to 10 out of 79 Questions