Department of Mathematics

Practice Set-3
1. Evaluate the following double integrals by converting rectangular coordinate to polar coordinates:
R 1 R √1−x2
(a) −1 0 dy dx, Ans: π2
√
R 1 R 1−x2
(b) −1 −√1−x2 dy dx, Ans: π
√
R 1 R 1−y2 2
(c) 0 0 (x + y 2 ) dx dy, Ans: π8
√
R 1 R 1−y2
(d) −1 √ 2 (x2 + y 2 ) dx dy, Ans: π2
− 1−y
R a R √a2 −x2
(e) √
−a − a2 −x2
dy dx, Ans: πa2
R 2 R √4−y2
(f) 0 0
(x2 + y 2 ) dx dy, Ans: 2π

2. Evaluate the cylindrical coordinate integrals:

R 2π R 1 R √2−r2 √
4π( 2−1)
(a) 0 0 r dz r dr dθ, Ans: 3
R 2π R 3 R √18−r2 √
9π(8 2−7)
(b) 0 0 r2 /3 dz r dr dθ, Ans: 2
R 2π R θ/2π R 3+24r2 17π
(c) 0 0 0
dz r dr dθ, Ans: 5
R 2π R θ/π R 3√4−r2 37π
(d) 0 0 √ z dz r dr dθ, Ans:
− 4−r 2 15

3. Evaluate the spherical coordinate integrals:

R π R π R 2 sin ϕ 2
(a) 0 0 0 ρ sin ϕ dρ dϕ dθ, Ans: π 2
R 2π R π/4 R 2
(b) 0 0 0
(ρ cos ϕ) ρ2 sin ϕ dρ dϕ dθ, Ans: 2π
R 2π R π R (1−cos ϕ)/2 2 π
(c) 0 0 0 ρ sin ϕ dρ dϕ dθ, Ans: 3
R 3π/2 R π R 1 3 3 5π
(d) 0 0 0
5ρ sin ϕ dρ dϕ dθ, Ans: 2
R 2π R π/3 R 2
(e) 0 0 sec ϕ
3ρ2 sin ϕ dρ dϕ dθ, Ans: 5π
R 2π R π/4 R sec ϕ π
(f) 0 0 0
(ρ cos ϕ) ρ2 sin ϕ dρ dϕ dθ, Ans: 4.

1
 4. Find the gradient fields of the functions :
⃗ = −xî−y ĵ−z k̂
(i) f (x, y, z) = (x2 + y 2 + z 2 )−1/2 , Ans:∇f (x2 +y 2 +z 2 )3/2

⃗ = xî+y ĵ+z k̂
p
(ii) f (x, y, z) = ln x2 + y 2 + z 2 , Ans: ∇f x2 +y 2 +z 2
   
(iii) g(x, y, z) = ez − ln(x2 + y 2 ), ⃗ =
Ans: ∇g −2x
î − 2y
ĵ + ez k̂
x2 +y 2 x2 +y 2

(iv) g(x, y, z) = xy + yz + xz, ⃗ = (y + z)î + (x + z)ĵ + (x + y)k̂.

Ans: ∇g

69
xy dx + (x + y) dy, along the curve y = x2 from (−1, 1) to (2, 4).
R
5. Evaluate C
Ans: 4

F⃗ · d⃗r, for the vector field F⃗ = x2 ı̂ − yȷ̂ along the curve x = y 2 from (4, 2) to (1, −1).
R
6. Evaluate C
Ans: − 39
2

7. Find the work done by F⃗ over the curve in the direction of increasing t.
(i) F⃗ = xy î + y ĵ − yz k̂, ⃗r(t) = tî + t2 ĵ + tk̂, 0 ≤ t ≤ 1, Ans: 1
2

(ii) F⃗ = 2y î + 3xĵ + (x + y)k̂, ⃗r(t) = (cos t)î + (sin t)ĵ + t

6 k̂, 0 ≤ t ≤ 2π, Ans: π

(iii) F⃗ = z î + xĵ + y k̂, ⃗r(t) = (sin t)î + (cos t)ĵ + tk̂, 0 ≤ t ≤ 2π, Ans: −π

(iv) F⃗ = 6z î + y 2 ĵ + 12xk̂, ⃗r(t) = (sin t)î + (cos t)ĵ + t

6 k̂, 0 ≤ t ≤ 2π, Ans: 0

8. Find the work done by the force F⃗ = xy î + (y − x)ĵ over the straight line from (1, 1) to (2, 3).
Ans: 25
6

9. Find the work done by the force F⃗ = y 2 z 3 î + 2xyz 3 ĵ + 3xy 2 z 2 k̂ along the line segment from (1, 1, 1)
to (2, 1, −1).
Ans: −3
10. F⃗ is the velocity field of a fluid flowing through a region in space. Find the flow along the given curve
in the direction of increasing t.
(i) F⃗ = −4xy î + 8y ĵ + 2k̂, ⃗r(t) = tî + 2t2 ĵ + k̂, 0 ≤ t ≤ 2, Ans: 48
(ii) F⃗ = x2 ı̂ + yzȷ̂ + y 2 k̂, ⃗r(t) = 3tȷ̂ + 4tk̂, 0 ≤ t ≤ 1, Ans: 24
⃗
(iii) F = (x − z)ı̂ + xk̂, ⃗r(t) = (cos t)ı̂ + (sin t)k̂, 0 ≤ t ≤ π, Ans: π
(iv) F⃗ = −yı̂ + xȷ̂ + 2k̂, ⃗r(t) = (−2 cos t)ı̂ + (2 sin t)ȷ̂ + 2tk̂, 0 ≤ t ≤ 2π, Ans: 0

11. Find the circulation and flux of the fields

F⃗1 = xı̂ + yȷ̂ and F⃗2 = −yı̂ + xȷ̂

around and across each of the following curves.

(a) The circle ⃗r(t) = (cos t)ı̂ + (sin t)ȷ̂, 0 ≤ t ≤ 2π, Ans: Circulation1 = 0, Circulation2 =
2π, Flux1 = 2π, Flux2 = 0

2
 (b) The ellipse ⃗r(t) = (cos t)ı̂ + (4 sin t)ȷ̂, 0 ≤ t ≤ 2π Ans: Circulation1 = 0, Circulation2 =
8π, Flux1 = 8π, Flux2 = 0

12. Which fields are conservative, and which are not?

(i) F⃗ = yz î + xz ĵ + xy k̂, Ans: Conservative
(ii) F⃗ = (y sin z)î + (x sin z)ĵ + (xy cos z)k̂, Ans: Conservative
(iii)F⃗ = y î + (x + z)ĵ − y k̂, Ans: Not Conservative
(iv) F⃗ = −y î + xĵ, Ans: Not Conservative

13. Check C F⃗ · d⃗r is path independent or not if

R

(i) F⃗ = (z + y)î + z ĵ + (y + x)k̂, Ans: Not path independent

(ii) F⃗ = (ex cos y)î − (ex sin y)ĵ + z k̂, Ans: path independent

14. Check the field is conservative or not, if it is conservative, then find the scalar potential function f for
the field F⃗ .
2
(a) F⃗ = 2xî + 3y ĵ + 4z k̂, Ans: f (x, y, z) = x2 + 3y2 + 2z 2 + c
(b) F⃗ = (y + z)î + (x + z)ĵ + (x + y)k̂, Ans: f (x, y, z) = x(y + z) + yz + c
(c) F⃗ = ey+2z (î + xĵ + 2xk̂), Ans: f (x, y, z) = xey+2z + c
(d) F⃗ = (y sin z)î + (x sin z)ĵ + (xy cos z)k̂, Ans: f (x, y, z) = xy sin z + c

15. Verify the conclusion of Green’s Theorem over R : x2 + y 2 ≤ a2 and its bounding circle C : ⃗r =
(a cos t)ı̂ + (a sin t)ȷ̂, 0 ≤ t ≤ 2π.
(i) F⃗ = −y î + xĵ
(ii) F⃗ = y î
(iii) F⃗ = 2xî − 3y ĵ
(iv) F⃗ = −x2 y î + xy 2 ĵ

16. Apply Green’sI Theorem to evaluate the following integrals:

(i) Evaluate (y 2 dx + x2 dy),
C
where C is the triangle bounded by x = 0, x + y = 1, and y = 0.
Ans: 0
I
(ii) Evaluate (3y dx + 2x dy),
C
where C is the boundary of 0 ≤ x ≤ π, 0 ≤ y ≤ sin x. Ans: −2
I
(iii) Evaluate (6y + x) dx + (y + 2x) dy,
C
where C is the circle (x − 2)2 + (y − 3)2 = 4.
Ans: −16π
I
(iv) Evaluate (2x + y 2 ) dx + (2xy + 3y) dy,
C
where C is any simple closed curve in the plane for which Green’s Theorem holds.
Ans: 0

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17. Evaluate S F⃗ · n̂ ds, using Gauss divergence theorem for
RR

(i) F⃗ = x2 î + y 2 ĵ + z 2 k̂, over the cube bounded by x = ±1, y = ±1, z = ±1.

Ans: 0
(ii) F⃗ = xî + y ĵ + 2k̂, over the sphere x2 + y 2 + z 2 = a2 .
3
Ans: 8πa 3

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