Mathematical Physics Vol 1

4.6 Examples

187

Solution a) Using the divergence theorem we obtain x S n · r r 3 d S = y V ∇ · r r 3 d V . Given that ∇ · r r 3

= 0 (Example 53 on p. 131) it follows that

x S

n · r r 3

d S = 0 .

b) If O is inside S , let us observe a sphere s with a radius a around the point O . Let τ be the region bounded by S and s . According to the divergence theorem x S + s n · r r 3 d S = x S n · r r 3 d S + x s n · r r 3 d S = = y τ ∇ · r r 3 d V = 0 , because r̸ = 0 in τ . From here it follows that x S n · r r 3 d S = − x s n · r r 3 d S . For the sphere s , r = a , and n = − r a which yields n · r r 3 = − r a · r r 3 = − r · r a 4 = − a 2 a 4 = − 1 a 2

and

x S

d S = − x s

d S = x s

n · r r 3

n · r r 3

1 a 2

d S =

1 a 2 x s

4 π a 2 a 2

= 4 π .

d S =

=

4.6.7 Various examples

Exercise 130

Calculate

I c ( 3 x + 4 y ) d x +( 2 x − 3 y ) d y , where c is a circle with a radius of two, and center at the coordinate origin.