Begin2.DVI

9-20.

A right circular cone intersects a sphere of radius

r as illustrated. Find the solid angle subtended

by this cone.

Right circular cone

intersecting sphere.

9-21. Evaluate

∫∫

S

r ·dS ,
where Sis a closed surface having a volume V.

9-22. In the divergence theorem

∫∫∫

V

∇· F dV
 =

∫∫

S

F
·eˆndS let F
=φ(x, y, z)C

where C
is a nonzero constant vector and show

∫∫∫

V

∇φ dV =

∫∫

S

φˆendS

9-23. Assume F
is both solenoidal and irrotational so that F
is the gradient of a

scalar function Φ(a) Show Φis a solution of Laplace’s equation and (b) Show the

integral of the normal derivative of Φover any closed surface must equal zero.

9-24. Let
r =xˆe 1 +yeˆ 2 +zˆe 3 and show ∇|
r |ν=ν|
r |ν−^2
r =ν|
r |ν−^1 ˆer where eˆr =
r

|
r |

is a unit vector in the direction
r.

9-25. For
r =xˆe 1 +yeˆ 2 +zˆe 3 and
r 0 =x 0 ˆe 1 +y 0 ˆe 2 +z 0 ˆe 3 show that

∂|
r −
r 0 |

∂x

=x−x^0

|
r −
r 0 |

∂|
r −
r 0 |

∂y =

y−y 0

|
r −
r 0 |

∂|
r −
r 0 |

∂z =

z−z 0

|
r −
r 0 |

9-26. Let
r =xˆe 1 +yˆe 2 +zˆe 3 denote the position vector to the variable point (x, y, z)

and let
r 0 =x 0 ˆe 2 +y 0 ˆe 2 +z 0 ˆe 3 denote the position vector to the fixed point (x 0 , y 0 , z 0 ).

(a) Show ∇|
r −
r 0 |ν=ν|
r −
r 0 |ν−^1 ˆer−r 0 where ˆer−r 0 is a unit vector in the direction

r −
r 0.

(b) Show ∇^2 |
r −
r 0 |ν=ν(ν+ 1)|
r −
r 0 |ν−^2

(c) Write out the results from part (b) in the special cases ν=− 1 and ν= 2.