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466 Vectors<br />
6.<br />
<br />
Verify Divergence Theorem for 2<br />
F ( x y ) iˆ<br />
– 2 xj ˆ 2 yzkˆ<br />
and the volume of a tetrahedron bounded<br />
by co-ordinate planes and the plane 2 x + y + 2 z = 6.<br />
(Nagpur, Winter 2000, A.M.I.E.T.E.. Winter 2000)<br />
7. Verify Divergence Theorem for the function<br />
x 2 + y 2 = 9, z = 0 and z = 2.<br />
8. Use the Divergence Theorem to evaluate<br />
<br />
s<br />
ˆ ˆ 2 ˆ over the region bounded by<br />
F yi xj z k<br />
3 2 2<br />
x dydz x ydzdx x zdxdy,<br />
where S is the surface of the region bounded by the closed cylinder<br />
x 2 + y 2 = a 2 , (0 z b) and z = 0, z = b.<br />
<br />
Ans.<br />
4<br />
5 ab<br />
4<br />
2<br />
9. Evaluate the integral ( z x) dy dz xy dx dz 3 zdxdy,<br />
where S is the surface of closed region<br />
s<br />
bounded by z = 4 – y 2 and planes x = 0, x = 3, z = 0 by transforming it with the help of Divergence<br />
Theorem to a triple integral. Ans. 16<br />
10. Evaluate<br />
<br />
s<br />
ds<br />
2 2 2 2 2 2<br />
a x b y c z<br />
over the closed surface of the ellipsoid ax 2 + by 2 + cz 2 = 1 by<br />
applying Divergence Theorem. Ans.<br />
11. Apply Divergence Theorem to evaluate 2 2 2<br />
( lx my nz ) ds<br />
<br />
4<br />
( abc)<br />
taken over the sphere (x – a) 2 + (y – b) 2 + (z – c) 2 = r 2 , l, m, n being the direction cosines of the external<br />
normal to the sphere. (AMIETE June 2010, 2009) Ans.<br />
8 ( )<br />
3<br />
3 a b c r<br />
12. Show that ( uV u<br />
V ) dv<br />
V<br />
<br />
<br />
= .<br />
s<br />
<br />
uV ds<br />
13. If E = grad and 2<br />
= 4 , prove that E <br />
n ds = 4 dv<br />
S<br />
V<br />
where n is the outward unit normal <strong>vector</strong>, while dS and dV are respectively surface and volume<br />
elements.<br />
Pick up the correct option from the following:<br />
14. If F <br />
is the velocity of a fluid particle then F.<br />
dr represents.<br />
(a) Work done (b) Circulation<br />
C<br />
(c) Flux (d) Conservative field.<br />
(U.P. Ist Semester, Dec 2009) Ans. (b)<br />
15. If f = ax i by j cz k , a, b, c, constants, then f.<br />
dS where S is the surface of a unit sphere is<br />
<br />
(a) ( )<br />
3 a b c (b) 4 ( a b c)<br />
(c) 2 ( a b c)<br />
(d) (a + b + c)<br />
3<br />
(U.P., Ist Semester, 2009) Ans. (b)<br />
16. A force field F is said to be conservative if<br />
<br />
(a) Curl F 0 (b) grad F 0 (c) Div F 0 (d) Curl (grad F ) = 0<br />
(AMIETE, Dec. 2006) Ans. (a)<br />
17. The line integral<br />
<br />
<br />
2 2<br />
x dx y dy, where C is the boundary of the region x 2 + y 2 < a 2 equals<br />
c<br />
(a) 0, (b) a (c) a 2 1 2<br />
(d) a<br />
2<br />
(AMIETE, Dec. 2006) Ans. (b)
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