# Vector Calculus Questions and Answers – Divergence and Curl of a Vector Field

This set of Vector Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Divergence and Curl of a Vector Field”.

1. What is the divergence of the vector field $$\vec{f} = 3x^2 \hat{i}+5xy^2\hat{j}+xyz^3\hat{k}$$ at the point (1, 2, 3).
a) 89
b) 80
c) 124
d) 100

Explanation: $$\vec{f} = 3x^2 \hat{i}+5xy^2\hat{j}+xyz^3\hat{k}$$
∴ div $$\vec{f}= ∇.\vec{f} = (\frac{∂}{∂x}\hat{i}+ \frac{∂}{∂y}\hat{j} +\frac{∂}{∂z}\hat{k}).(3x^2\hat{i} + 5xy^2\hat{j} + xyz^3\hat{k})$$
$$= 6x +10xy+ 3xyz^2$$
At the point (1, 2, 3)
div $$\vec{f} = 6(1)+10(1)(2)+3(1)(2)(3)^2 = 80.$$

2. Divergence of $$\vec{f}(x,y,z) = \frac{(x\hat{i}+y\hat{j}+z\hat{k})}{(x^2+y^2+z^2)^{3/2}}, (x, y, z) ≠ (0, 0, 0).$$
a) 0
b) 1
c) 2
d) 3

Explanation: Here $$\vec{f}(x,y,z) = \frac{x}{r^3}\hat{i} + \frac{y}{r^3}\hat{j} + \frac{z}{r^3}\hat{k},$$ where $$r=(x^2+y^2+z^2)$$
Also here we can see that
$$\frac{∂r}{∂x} = \frac{x}{r} , \frac{∂r}{∂y} = \frac{y}{r} , \frac{∂r}{∂z} = \frac{z}{r}$$
∴ div $$\vec{f} = \frac{∂}{∂x} (\frac{x}{r}) + \frac{∂}{∂y} (\frac{y}{r}) + \frac{∂}{∂z} (\frac{z}{r})$$
$$\frac{∂}{∂x} (\frac{x}{r^3}) = \frac{r^3-x 3r^2 (x/r)}{(r^3)^2} = \frac{1}{r^3} -\frac{3x^2}{r^5}$$
$$\frac{∂}{∂y} (\frac{y}{r^3}) = \frac{1}{r^3} – \frac{3y^2}{r^5} , \frac{∂}{∂y} (\frac{y}{r}) = \frac{1}{r^3} – \frac{3z^2}{r^5}$$
∴ we get div $$\vec{f} = \frac{3}{r^3} – \frac{3x^2+3y^2+3z^2}{r^5} = 0.$$

3. Divergence of $$\vec{f} (x, y, z) = e^{xy} \hat{i} -cos⁡y \hat{j}+(sinz)^2 \hat{k}.$$
a) yexy+ cos⁡y + 2 sinz.cosz
b) yexy– sin⁡y + 2 sinz.cosz
c) 0
d) yexy+ sin⁡y + 2 sinz.cosz

Explanation: $$div \vec{f} = (\frac{∂}{∂x}\hat{i} + \frac{∂}{∂y}\hat{j}+ \frac{∂}{∂z}\hat{k}) .( e^{xy} \hat{i} – cos⁡y \hat{j} + (sinz)^2 \hat{k})$$
$$= \frac{∂}{∂x}(e^{xy}) + \frac{∂}{∂y}(-cos⁡y) + \frac{∂}{∂z}((sinz)^2)$$
$$= ye^{xy} + sin⁡y + 2 sinz.cosz .$$

4. Curl of $$\vec{f} (x, y, z) = 2xy \hat{i}+ (x^2+z^2)\hat{j} + 2zy\hat{k}$$ is ________
a) $$xy^2\hat{i} – 2xyz\hat{k}$$ & irrotational
b) 0 & irrotational
c) $$xy^2\hat{i} – 2xyz \hat{k}$$ & rotational
d) 0 & rotational

Explanation: Curl $$vec{f} = ∇ ⤫ \vec{f} = \begin{vmatrix} i & j & k\\ \frac{∂}{∂x} & \frac{∂}{∂y} & {∂}{∂z}\\ 2xy & x^2 + z^2 & 2zy\\ \end{vmatrix}$$
$$= (2z – 2z) \hat{i} – (0-0)\hat{j} + (2x-2x)\hat{k}$$
$$= 0$$
Hence F is irrotational field as Curl $$\vec{f} = 0.$$

5. Chose the curl of $$\vec{f} (x ,y ,z) = x^2 \hat{i} + xyz \hat{j} – z \hat{k}$$ at the point (2, 1, -2).
a) $$2\hat{i} + 2\hat{k}$$
b) $$-2\hat{i} – 2\hat{j}$$
c) $$4\hat{i} – 4\hat{j} + 2\hat{k}$$
d) $$-2\hat{i} – 2\hat{k}$$

Explanation:$$\vec{f} (x ,y ,z) = x^2 \hat{i} + xyz \hat{j} – z \hat{k}$$
⇨ Curl $$\vec{f} = ∇ ⤫ \vec{f} = \begin{vmatrix} i & j & k\\ \frac{∂}{∂x} & \frac{∂}{∂y} & \frac{∂}{∂z}\\ x^2 & xyz & -z\\ \end{vmatrix}$$
$$= \bigg\{\frac{∂}{∂y} (-z)-\frac{∂}{∂z}(xyz)) \hat{i} + (\frac{∂}{∂z} (x^2 )- \frac{∂}{∂x}(-z)) \hat{j} + (\frac{∂}{∂x} (xyz)- \frac{∂}{∂y} (x^2 ) \hat{k}\bigg\}$$
$$= (0-xy) \hat{i} + (0-0) \hat{j} + (yz-0) \hat{k}$$
$$= – xy \hat{i} + yz \hat{k}$$
$$∇ ⤫ \vec{f} |_{(1, -1, 2)} = -2(1) \hat{i} + (1) (-2) \hat{k} = -2\hat{i} – 2\hat{k}.$$
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6. A vector field which has a vanishing divergence is called as ____________
a) Solenoidal field
b) Rotational field
c) Hemispheroidal field
d) Irrotational field

Explanation: By the definition: A vector field whose divergence comes out to be zero or
Vanishes is called as a Solenoidal Vector Field.
i.e.
If $$∇. \vec{f} = 0 ↔ \vec{f}$$ is a Solenoidal Vector field.

7. Divergence and Curl of a vector field are ___________
a) Scalar & Scalar
b) Scalar & Vector
c) Vector & Vector
d) Vector & Scalar

Explanation: Let $$\vec{f} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$
div$$\vec{f} = (\frac{∂}{∂x} \hat{i} + \frac{∂}{∂y} \hat{j}+ \frac{∂}{∂z} \hat{k}).(a_1\hat{i} + a_2\hat{j} + a_3\hat{k})$$
$$= \frac{∂a_1}{∂x} + \frac{∂a_2}{∂y} + \frac{∂a_3}{∂z}$$ which is a scalar quantity.
Also curl $$\vec{f} = \begin{vmatrix} i & j & k\\ \frac{∂}{∂x} & \frac{∂}{∂y} & \frac{∂}{∂z}\\ a1 & a2 & a3\\ \end{vmatrix}$$ = $$b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$$
Which is going to be a vector quantity as cross product of two vectors is again a vector, where dot product gives a scalar outcome.

8. A vector field with a vanishing curl is called as __________
a) Irrotational
b) Solenoidal
c) Rotational
d) Cycloidal

Explanation: By the definition: Mathematically,
If $$∇ ⤫ \vec{f} = 0 ↔ \vec{f}$$ is an Irrotational Vector field.

9. The curl of vector field $$\vec{f} (x,y,z) = x^2\hat{i} + 2z \hat{j} – y \hat{k}$$ is _________
a) $$-3\hat{i}$$
b) $$-3\hat{j}$$
c) $$-3\hat{k}$$
d) 0

Explanation:Curl,
$$\vec{f} = ∇ ⤫ \vec{f} = \begin{vmatrix} i & j & k\\ \frac{∂}{∂x} & \frac{∂}{∂y} & \frac{∂}{∂z}\\ x^2 & 2z & -y\\ \end{vmatrix}$$ $$= \bigg\{\frac{∂}{∂y} (-y)-\frac{∂}{∂z}(2z)) \hat{i} + (\frac{∂}{∂z} (x^2 )-\frac{∂}{∂x}(-y)) \hat{j} + (\frac{∂}{∂x} (2z)-\frac{∂}{∂y} (x^2 ) \hat{k}\bigg\}$$
$$= (-1-2) \hat{i} + (0-0) \hat{j} + (0-0) \hat{k} = -3\hat{i}.$$

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