Begin2.DVI

8-31. Let
r denote a position vector to a general point on a closed surface S,

which encloses a volume V. Evaluate the surface integral

∫∫

S

r ·dS
=

∫∫

S

r ·ˆendS

8-32. The Gauss Theorem Let
r denote the position vector from the origin to a

general point on a closed surface S. Show that

∫∫

S

ˆen·
r

r^3 dS =

{ 0 , if the origin is outside the closed surface S

4 π, if the origin is inside the closed surface S

Hint: Use the divergence theorem and when the origin is inside S, construct a small

sphere of radius about the origin.

8-33. For F
=x^2 zˆe 1 +xyz ˆe 2 +yz ˆe 3 and φ=xyz^2 , calculate ∇(φF
)

8-34. For A,
B
vector fields and f a scalar field, verify each of the following:

(i) curl (fA
) = (grad f)×A
+fcurl A

(ii) curl (A
×B
) = A
(div B
)−B
(div A
) + (B
·∇ )A
−(A
·∇ )B

(iii) div (fA
) = (grad f)·A
+fdiv A

(iv) grad (A
×B
) = B
·curl A
−A
·curl B

(v) grad (fg ) = fgrad g+ggrad f

(vi) grad (A
·B
) = (A
·∇ )B
+ (B
·∇ )A
+A
×curl B
+B
×curl A

8-35. Evaluate the line integral

A=^1

2

∫

C

©x dy −y dx

around the triangle having the vertices (0 ,0), (b,0) and (c, h )where b, c, h are positive

constants. Evaluate this integral using Green’s theorem in the plane.

8-36. Evaluate the integral

I=

∫∫

S

(∇× F
)·dS ,

where F
= (y− 2 x)ˆe 1 + (3 x+ 2y)ˆe 2 and Sis the surface of the cone

x=ucos v, y =usin v, z =ufor 0 ≤u≤ 9 and 0 ≤v≤ 2 π.

Hint: If you use Stoke’s theorem be sure to note direction of integration.