Question

Explain how to compute the surface integral of a scalar-valued function f over a sphere using an explicit description of the sphere.

Answer 1

The sphere will have an equation of the form

$(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$

so that its center is at the point $(a,b,c)$ and its radius is $r$.

You would solve for either half of the hemisphere,

$z = \pm\sqrt{r^2 - (x - a)^2 - (y - b)^2}$

and denote this by $z=g(x,y)$. Then the surface integral of $f(x,y)$ over either half of the sphere (call them $S^+$ and $S^-$ for upper and lower hemispheres, respectively) would be

$\displaystyle \iint_{S^+\cup S^-} f(x,y) \, ds \\\\ ~~~~ = \iint_{S^+} f(x,y) \sqrt{1 + \left(\frac{\partial g}{\partial x}\right)^2 + \left(\frac{\partial g}{\partial y}\right)^2} \, dx \, dy \\\\ ~~~~~~~~+ \iint_{S^-} f(x,y) \sqrt{1 + \left(\frac{\partial g}{\partial x}\right)^2 + \left(\frac{\partial g}{\partial y}\right)^2} \, dx \, dy$