Tutorial 3+(vc)

EQT 203 : ENGINEERING MATHEMATICS III Chapter 2 : Vector Calculus (Part II)

Tutorial 3

1. If V = x 3 y + 2 xy 2 + yz , evaluate

∫ Vdr C

~

from A(0,0,0) to B(2,1,-3) along the

curve defined by x = 2t , y = t dan z = −3t 3 . 2

Ans : 3 i + ~

2 3 2 2 2. If F~ = x y i~ + yz ~j + zx k~ , evaluate

∫

18 81 j− k 7 ~ 8 ~

F . d r along the curve x = 3u 2 , y = u ~

C ~

and z = 2u 3 from A(3,-1,-2) to B(3,1,2).

3. Evaluate

∫

Ans :12

F dV where F~ = 3 i~ + z j + 2 y k~ and V is the closed region bounded ~

V ~

by planes z = 0, z = 3 and surface x 2 + y 2 = 4. Ans :18 (2 i  j ) ~

4. If V is a scalar field where V  xyz 2 , evaluate

∫VdS S

~

~

on surface S defined by

x 2 + y 2 = 9 between z = 0 and z = 2 in the first octant. Ans : 24( i + j ) ~

5. Evaluate

∫ F .d S S ~

~

on surface

S

defined by

x2 + y2 + z2 = 4

~

where

F = x i + 2 z j + y k and bounded by x = 0, y = 0 and z = 0 in the first octant. ~ ~ ~ ~     1  6 

Ans : 8 

6. Evaluate

∫ F .d S S ~

~

where F~ = x i~ + xy ~j + 2 k~ and S is the surface of the region

bounded by planes z = 0, z = 4, x = 0, y = 0 and x 2 + y 2 = 9 in the first octant. Ans : 9(  4)

7. Evaluate

∫ (( xy + y C 2

2

)dx + x 2 dy ) where C is a closed curve bounded by y = x

dan y = x . Ans : −

8. Evaluate

∫

V

1 20

2 2 2 div F dV where F~ = 2 x y i~ − y j + 4 xz k~ taken over the region in ~ ~

the first octant bounded by the cylinder y 2 + z 2 = 9 and the plane x = 2. Ans :180

9. A surface is bounded by planes x = 0, x = 2, y = 0, y = 2 and z = 3 − y at z ≥ 0. Evaluate ∫S curl F~ . d S~ where F~ = 2 x i~ + xz ~j + yz k~ . Ans : 0 oooooooooooooo