UNIT – I
VECTOR CALCULAS
PART-B
⃗⃗ prove that (i) ∇𝑟 = 𝑟⃗⃗⃗⃗ , (ii) ∇𝑟 𝑛 = 𝑛𝑟 𝑛−2 ⃗⃗⃗
1. If 𝑟⃗ = 𝑥𝑖⃗ + 𝑦𝑗⃗ + 𝑧𝑘
𝑟 ,where 𝑟 = |𝑟⃗⃗⃗⃗⃗|.
𝑟
2.
Find the angle of intersection at the point (2 ,-1,2) of the surfaces 𝑥 2 + 𝑦 2 + 𝑧 2 = 9 and
𝑧 = 𝑥 2 + 𝑦 2 − 𝑧 − 3.
3. Find ‘a’ and ‘b’ such that the surfaces 𝑎𝑥 2 − 𝑏𝑦𝑧 = (𝑎 + 2)𝑥 and 4𝑥 2 𝑦 + 𝑧 3 = 4 cut
orthogonally at (1 ,-1,2).
⃗⃗⃗⃗, find the scalar potential 𝜑.
4. If ∇𝜑 = 2𝑥𝑦𝑧𝑖⃗⃗ + 𝑥 2 𝑧𝑗⃗⃗ + 𝑥 2 𝑦𝑘
⃗⃗⃗⃗⃗ where 𝐹
⃗⃗ and C is the straight line from A(0
⃗⃗⃗⃗ ∙ 𝑑𝑟
⃗⃗⃗⃗ = 3𝑥 2 𝑖⃗ + (2𝑥𝑧 − 𝑦)⃗⃗𝑗 + 𝑧𝑘
5. Evaluate ∫ 𝐹
𝐶
,0 ,0) to B(2 , 1, 3).
⃗⃗ , evaluate ∫ ⃗⃗⃗⃗
6. Given the vector field ⃗⃗⃗⃗
𝐹 = xz⃗⃗𝑖⃗ + 𝑦𝑧⃗⃗𝑗 − 𝑧 2 𝑘
𝐹 ∙ ⃗⃗⃗⃗⃗
𝑑𝑟 from the point (0,0,0) to
𝐶
(1,1,1) where C is the curve (i) x = t , y = 𝑡 2 , z = 𝑡 3 , (ii) the straight path from (0,0,0) to (1,1,1).
⃗⃗⃗⃗ = (2𝑥 − 𝑦 + 𝑧) 𝑖⃗⃗ +
7. Find the total work done in moving a particle in a force field given by 𝐹
⃗⃗⃗⃗ along a circle C in the XY plane 𝑥 2 + 𝑦 2 = 9, 𝑧 = 0.
(𝑥 + 𝑦 − 𝑧)𝑗⃗⃗ + (3𝑥 − 2𝑦 − 5𝑧)𝑘
8. Find the work done by the force ⃗⃗⃗⃗
𝐹 = (2𝑥𝑦 + 𝑧 3 ) 𝑖⃗⃗ + 𝑥 2 𝑗⃗⃗ + 3𝑥𝑧 2 ⃗⃗⃗⃗
𝑘 when it moves a particle
from (1,-2,1) to (3,1,4) along any path.
9. Evaluate
 F  nˆds where F  yzi  zx j  xyk. and S is that part of the surface of the sphere
S
x2+y2+z2
10. Evaluate
= 1 which lies in the first octant.
 F  nˆds where F  18zi  12 j  3 yk as S is the part of the plane 2x + 3y + 6z = 12
S
which is in the first octant.
11. Evaluate
 F  nˆds where F  ( x  y
2
)i  2 x j  2 yz k where S is the region bounded by 2x +
S
y + 2z = 6 in the first octant.
12. If F  (2 x 2  3z)i  2 xy j  4 xk , then evaluate (i)
   F dV
V
(ii)
   FdV ,where V is
V
the region bounded by x = 0 , y = 0 , z = 0 and 2x + 2y + z = 4.
⃗⃗⃗⃗ over the cube bounded by x
⃗⃗⃗⃗ = 4𝑥𝑧 𝑖⃗⃗ − 𝑦 2 𝑗⃗⃗ + 𝑦𝑧𝑘
13. Verify the Gauss divergence theorem for 𝐹
= 0 , x = 1, y = 0, y = 1, z = 0, z = 1.
⃗⃗⃗⃗ taken over
14. Verify the Divergence theorem for ⃗⃗⃗⃗
𝐹 = (𝑥 2 − 𝑦𝑧) 𝑖⃗⃗ + (𝑦 2 − 𝑧𝑥)𝑗⃗⃗ + (𝑧 2 − 𝑥𝑦)𝑘
the rectangular parallelepiped 0 ≤ x ≤ a ,0 ≤ y ≤ b , 0 ≤ z ≤ c .
15. Evaluate
 F  nˆds where F  4xzi  y
2
j  yz k. and S is the surface of the cube bounded by
S
x = 0 ,x = 1, y = 0, y = 1, z = 0, z = 1.
⃗⃗⃗⃗ and S is the surface bounding
⃗⃗⃗⃗ = 4𝑥 𝑖⃗⃗ − 2𝑦 2 𝑗⃗⃗ + 𝑧 2 𝑘
16. Use divergence theorem to evaluate 𝐹
2
2
the region x + y = 4 z = 0 and z = 3.
17. Verify Green’s theorem in a plane for the integral
 ( x  2 y)dx  xdy,taken around the circle x2
C
+
y2
= 1.
18. Verify Green’s theorem for
 ( x
2

 y 2 )dx  2 xydy , where C is the boundary of the rectangle in
C
the XOY – plane bounded by the lines x = 0,x = a, y = 0 and y = a.
19. Verify Green’s theorem for
 ( xy  y
2

)dx  x 2 dy , where C is the closed curve of the region
C
bounded by y = x and y = x2 .
20. Using Green’s theorem, evaluate
 ( y  sin x)dx  cos xdy,where C is the triangle bounded by
C
𝑦 = 0, 𝑥 =
𝜋
2
,𝑦 =
2𝑥
𝜋
.
21. By applying Green’s theorem prove that the area bounded by a simple closed curve C is =
1
2
 ( xdy  ydx) and hence find the area of the ellipse.
C
⃗⃗⃗⃗ = (𝑥 2 − 𝑦 2 ) 𝑖⃗ + 2𝑥𝑦⃗⃗𝑗 in the rectangular
22. Verify Stoke’s theorem for a vector defined by 𝐹
region in the XOY plane bounded by the lines x = 0, x = a, y = 0 and y = b.
⃗⃗⃗⃗ ,where S is the upper half of
23. Verify Stoke’s theorem for a vector defined by ⃗⃗⃗⃗
𝐹 = y 𝑖⃗ + 𝑧⃗⃗𝑗 + 𝑥𝑘
the surface of the sphere x2 + y2 + z2 = 1 and C is its boundary.
24. Evaluate the integral
 ( x  y)dx  (2 x  z)dy  ( y  z)dz where C is the boundary of the
C
triangle with vertices (2,0,0), (0,3,0) and (0,0,6) using Stoke’s theorem.
25. Evaluate  ( xydx  xy 2 dy) by the Stoke’s theorem where C is the square in the XY plane with
C
vertices (1,0),
(-1,0), (0,1) and (0,-1).
26. Prove that  r  dr  2 nˆds where S is the surface enclosing a circuit C.
S