Hi all,

This is my first post on MHF. I have been trying to improve my basic calculus skills, working through an (excellent) textbook called "div, grad, curl and all that" by H. M. Schey.

I have just finished reading the second chapter on surface integrals and the divergence and am working on the problems at the end of the chapter. Problem II-4 has me stumped.

It is as follows:

The distribution of mass on the hemispherical shell,

$\displaystyle z = (R^{2}-x^{2}-y^{2})^{1/2}$

is given by

$\displaystyle \sigma (x,y,z) = (\frac{\sigma_{0}}{R^{2}})(x^{2}+y^{2})$

where $\displaystyle \sigma_{0}$ is a constant. Find an expression in terms of $\displaystyle \sigma_{0}$ and R for the total mass of the shell.

I know the solution is $\displaystyle 4 \pi R^{2}\frac{\sigma_{0}}{3}$ but am not sure how to get there.

I started off with a surface integral

$\displaystyle \iint_{S} \sigma (x,y,z) dS$

where S is the surface mentioned above.

Now I know to evaluate this, I need to recast the problem in terms of an integral over the projection of the surface on a plane, i.e. the xy-plane. This projected surface (a circle) will be called Q.

I think the first step towards achieving this is the following:

$\displaystyle \iint_{Q} \sigma [x,y,z(x,y)] \left( 1 + \frac{\partial z(x,y)}{\partial x} + \frac{\partial z(x,y)}{\partial y} \right)^{1/2} dQ$

This is where I am stumped. For a start, $\displaystyle \sigma$ doesn't actually seem to be a function of z -- it only depends on x and y. I'm also not sure what role (if any) R plays in this. I assume this is the radius of the hemisphere, but am not sure.

I'd be really grateful if someone could point me in the right direction with what to do next here.

Many thanks!