Mathematical Physics Vol 1

4.6 Examples

189

Exercise 134

Calculate

x S

F · n d S ,

where F = 2 xy i + yz 2 j + xz k , and S is: a) the parallelepiped bounded by x = 0, y = 0, z = 0, x = 2, y = 1 i z = 3, b) the surface bounding the region given by x = 0, y = 0, z = 0, y = 3and x + 2 z = 6.

Solution a) 30, b) 351 / 2.

Exercise 135 If H = rot A , prove that

x S

H · n d S = 0 ,

for a closed surface S .

Exercise 136

Prove that

y V

= x S

r · n r 2

d V r 2

d S .

Exercise 137

Prove that

x S

r 5 n d S = y V

5 r 3 r d V .

Exercise 138

Prove that

x S

n d S = 0 ,

for any surface S .

Exercise 139

Calculate

x S

( ∇ × A ) · n d S ,