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√(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((\partial g)/(\partial x)\right)^2 + \left((\partial g)/(\partial y)\right)^2} \, dx \, dy \\\\ ~~~~~~~~+ \iint_(S^-) f(x,y) \sqrt{1 + \left((\partial g)/(\partial x)\right)^2 + \left((\partial g)/(\partial y)\right)^2} \, dx \, dy`