Vector Calculus Multiple Choice Questions on “Divergence and Curl of a Vector Field”.
1. What is the divergence of the vector field ( vec{f} = 3x^2 hat{i}+5xy^2hat{j}+xyz^3hat{k} ) at the point (1, 2, 3).
a) 89
b) 80
c) 124
d) 100
View Answer
Answer: b
Explanation: ( vec{f} = 3x^2 hat{i}+5xy^2hat{j}+xyz^3hat{k} )
∴ div (vec{f}= ∇.vec{f} = (frac{∂}{∂x}hat{i}+ frac{∂}{∂y}hat{j} +frac{∂}{∂z}hat{k}).(3x^2hat{i} + 5xy^2hat{j} + xyz^3hat{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{(xhat{i}+yhat{j}+zhat{k})}{(x^2+y^2+z^2)^{3/2}}, (x, y, z) ≠ (0, 0, 0).)
a) 0
b) 1
c) 2
d) 3
View Answer
Answer: a
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} -cosy hat{j}+(sinz)^2 hat{k}.)
a) yexy+ cosy + 2 sinz.cosz
b) yexy– siny + 2 sinz.cosz
c) 0
d) yexy+ siny + 2 sinz.cosz
View Answer
Answer: d
Explanation: (div vec{f} = (frac{∂}{∂x}hat{i} + frac{∂}{∂y}hat{j}+ frac{∂}{∂z}hat{k}) .( e^{xy} hat{i} – cosy hat{j} + (sinz)^2 hat{k}) )
( = frac{∂}{∂x}(e^{xy}) + frac{∂}{∂y}(-cosy) + frac{∂}{∂z}((sinz)^2) )
( = ye^{xy} + siny + 2 sinz.cosz .)
4. Curl of (vec{f} (x, y, z) = 2xy hat{i}+ (x^2+z^2)hat{j} + 2zyhat{k} ) is ________
a) (xy^2hat{i} – 2xyzhat{k}) & irrotational
b) 0 & irrotational
c) (xy^2hat{i} – 2xyz hat{k} ) & rotational
d) 0 & rotational
View Answer
Answer: b
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) (2hat{i} + 2hat{k} )
b) (-2hat{i} – 2hat{j} )
c) (4hat{i} – 4hat{j} + 2hat{k} )
d) (-2hat{i} – 2hat{k} )
View Answer
Answer: d
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} = -2hat{i} – 2hat{k}. )
6. A vector field which has a vanishing divergence is called as ____________
a) Solenoidal field
b) Rotational field
c) Hemispheroidal field
d) Irrotational field
View Answer
Answer: a
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
View Answer
Answer: b
Explanation: Let (vec{f} = a_1hat{i} + a_2hat{j} + a_3hat{k} )
div(vec{f} = (frac{∂}{∂x} hat{i} + frac{∂}{∂y} hat{j}+ frac{∂}{∂z} hat{k}).(a_1hat{i} + a_2hat{j} + a_3hat{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_1hat{i} + b_2hat{j} + b_3hat{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
View Answer
Answer: a
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^2hat{i} + 2z hat{j} – y hat{k} ) is _________
a) (-3hat{i} )
b) (-3hat{j} )
c) (-3hat{k} )
d) 0
View Answer
Answer: a
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} = -3hat{i}. )
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