For upper hemisphere S: x^2 + y^2 + z^2 = 1 , with z≥0, find the area element dS and unit normal vector N. Compute the total area of the hemisphere, ∫∫dS over S.
Unit normal to surface f(x,y,z) = const
N(hat) = grad(f)/|grad(f)|
Surface area element of surface z = f(x,y)
dS = sqrt(1 + (df/dx)^2 + (df/dy)^2)*dx*dy
The Attempt at a Solution
N(hat) = (x/sqrt(x^2+y^2+z^2), y/sqrt(x^2+y^2+z^2), z/sqrt(x^2+y^2+z^2))
dS = dx*dy/sqrt(1-x^2-y^2)
So, I computed the gradient of f(x,y,z) and normalized it to get N(hat). I also used z = +sqrt(1-x^2-y^2) to find dS. Now that I have these things I do not know how to use them to integrate over the surface. I have a feeling I need to use Stokes Theorem to solve this, but I don't know how. Am I even on the right track? Thanks.