Find the flux of F = <x^3/3, yz^2 + e^sqrt(xz), zy^2 + y + 2 + sin(x^3)> through upper hemisphere (z>=0) of x^2 + y^2 + z^2 = a^2
This is how I proceeded:
Let z = g(x,y)
=> g(x,y) = sqrt(a^2 - x^2 - y^2)
I found the partial differentials gx,gy
= integral F.<-gx,-gy,1> dA
After that I evaluated the dot product, replaced dA by r dr d(theta),
x by r cos(theta),
y by r sin(theta)
I took the limits of integrals as theta = 0 to 2pi, and r = 0 to a
But the integrals are becoming very complicated.
Is there a simpler way? Will finding flux through the disk which closes the hemisphere help?