MCQs of Vector Calculus (Mathematics - II - 3110015)

Mathematics - II (3110015) MCQs

MCQs of Vector Calculus

Showing 21 to 30 out of 79 Questions

21.

For the scalar function

ϕ

, curl (grad

ϕ

)

=

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

22.

ϕ = 3 xy + 2 {yzx}^{2} - 12 {xz}^{3}

, then

\nabla \times (\nabla ϕ) =

(a)	$5$
(b)	$0$
(c)	$3 y \hat{i} + 2 {zx}^{2} \hat{j} - 36 {xz}^{2} \hat{k}$
(d)	$3 y + 2 z - 36 {xz}^{2}$

23.

A Vector field

F

is irrotational/conservative if

(a)	$\nabla \times F = 0$
(b)	$\nabla \cdot F = 0$
(c)	$\nabla^{2} F = 0$
(d)	None of above

24.

If the vector field

F = (x + 2 y + mz) \hat{i} + (2 x - 3 y - z) \hat{j} + (4 x - y + 2 z) \hat{k}

is irrotational, then the value of

m

(a)	$- 3$
(b)	$3$
(c)	$4$
(d)	$- 4$

25.

Let

F = M (x, y, x) \hat{i} + N (x, y, x) \hat{j} + P (x, y, x) \hat{k}

be a field on an open simply connected domain. Also, first partial derivatives of

M

N

and

P

are continuous. Then

F

is conservative if and only if

(a)	$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, \frac{\partial M}{\partial z} = \frac{\partial P}{\partial y}, \frac{\partial N}{\partial z} = \frac{\partial P}{\partial x}$
(b)	$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, \frac{\partial P}{\partial x} = \frac{\partial M}{\partial z}, \frac{\partial N}{\partial z} = \frac{\partial P}{\partial y}$
(c)	$\frac{\partial N}{\partial y} = \frac{\partial M}{\partial x}, \frac{\partial M}{\partial z} = \frac{\partial P}{\partial y}, \frac{\partial N}{\partial z} = \frac{\partial P}{\partial x}$
(d)	$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, \frac{\partial P}{\partial z} = \frac{\partial M}{\partial y}, \frac{\partial N}{\partial z} = \frac{\partial P}{\partial x}$

26.

For the vector

A = (a_{1}, a_{2}, a_{3})

\nabla \cdot (\nabla \times A) =

(a)	$\frac{\partial^{2} a_{1}}{\partial x^{2}} \hat{i} + \frac{\partial^{2} a_{2}}{\partial y^{2}} \hat{j} + \frac{\partial^{2} a_{3}}{\partial z^{2}} \hat{k}$
(b)	$\frac{\partial a_{1}}{\partial x} \hat{i} + \frac{\partial a_{2}}{\partial y} \hat{j} + \frac{\partial a_{3}}{\partial z} \hat{k}$
(c)	$\frac{\partial^{2} a_{1}}{\partial x^{2}} + \frac{\partial^{2} a_{2}}{\partial y^{2}} + \frac{\partial^{2} a_{3}}{\partial z^{2}}$
(d)	$0$

27.

For a vector field

F

, if

F = \nabla ϕ

, then

ϕ

is called

(a)	Curl of $F$
(b)	Scalar potential function of $F$
(c)	div( $F$ )
(d)	Solenoidal

28.

Let the vector field

F = (6 xy + z^{3}) \hat{i} + (3 x^{2} - z) \hat{j} + (3 {xz}^{2} - y) \hat{k}

be irrotational. Then the corresponding scalar potential function

f

satisfying

F = \nabla f

(a)	$3 x^{2} y + z^{3} x - yz + c$
(b)	$3 x^{2} y + z^{3} x + c$
(c)	$6 x^{2} y + x^{3} + xy - yz + c$
(d)	$x^{2} y + z^{3} x - y^{3} + c$

29.

For an irrotational vector field

F = (y^{2} \cos (x) + z^{2}) \hat{i} + (2 ysin (x) - 4) \hat{j} + (2 zx + 2) \hat{k}

such that

F = \nabla ϕ

, then the scalar potential function

ϕ

(a)	$- y^{2} \sin (x) + z^{2} x + y^{2} \sin (x) + {xz}^{2} + c$
(b)	$y^{2} \sin (x) + z^{2} x - 4 y + 2 z + c$
(c)	$y^{2} \cos (x) + z^{2} x + y^{2} \sin (x) - 4 y + {xz}^{3} + c$
(d)	$\frac{y^{2}}{3} \sin (x) + z^{2} y + 2 ycos (x) - 4 x + c$

30.

F = yz \hat{i} + zx \hat{j} + xy \hat{k}

and

F = \nabla ϕ

, then

ϕ

(a)	$x + y + z + c$
(b)	$x^{2} + y^{2} + z^{2} + c$
(c)	$xyz + c$
(d)	$x^{2} + y + z + c$

Showing 21 to 30 out of 79 Questions