Surface Integral: Evaluating Double Integral of f.n ds on Sphere

[anand]

Homework Statement

Evaluate [double integral]f.n ds where f=xi+yj-2zk and S is the surface of the sphere x^2+y^2+z^2=a^2 above x-y plane.

The Attempt at a Solution

I know that the sphere's orthogonal projection has to be taken on the x-y plane,but I'm having trouble with the integration.

 

[HallsofIvy]

Of course, you will have to do upper and lower hemispheres separately. One way to get the projection into the xy-plane is to find the gradient of x²+ y²+z², 2xi+ 2yj+ 2zk, and "normalize by dividing by 2z: (x/z)i+ (y/z)j+ k. Then n dS is (x/z)i+ (y/z)j+ k dxdy.
f.n dS is ((x²/z)+ (y²/z)- 2z) dxdy. I think I would rewrite that as ((x²/z)+ (y²/z)+ z- 3z) dxdy= ((x²+ y²+ z²)/z- 3z) dxdy= (a^2/z- 3z)dxdy. Now, for the upper hemisphere, $z= \sqrt{a^2- x^2- y^2}$ while for the negative hemisphere it is the negative of that. Because your integrand is an odd function of z, I think the symmetry of the sphere makes this obvious.

Finally, do you know the divergence theorem?
$$\int\int_T\int (\nabla \cdot \vec{v}) dV= \int\int_S (\vec{v} \cdot \vec{n}) dS$$
where S is the surface of the three dimensional region T. Here $\nabla\cdot f$ is very simple and, in fact, you don't have to do an integral at all! I wouldn't be surprized to see this as an exercise in a section on the divergence theorem.

 

[anand]

I know this is a bit embarrassing for me,but how d'you integrate
(a^2/z- 3z)dxdy .After having substituted for z,and converted to polar co-ordinates,I get zero in the denominator!

This is the expression:
[double integral]a^2/(sqrt(a^2-x^2-y^2)) dx dy.
For conversion to polar co-ords,if I substitute x=a cos(theta) and
y=a sin(theta),the denominator becomes zero.

(Thanks a lot for the help anyway )

 

[HallsofIvy]

Why bother to find an anti-derivative? The function is odd in z and the region of integration is symmetric about the origin- the integral is 0.

My point about the divergence theorem is that [itex]\nabla \cdot (x\vec{i}+y\cdot\vec{j}-2z\vec{k}) = 1+ 1- 2= 0[/tex]! The integral of that over any region is 0![/itex]

 

[anand]

Thanks a lot.
(The question did ask not to use divergence theorem,by the way)

 

[HallsofIvy]

Did it say you couldn't?