How to find the flux of a vector field through a hemisphere?
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?
Re: How to find the flux of a vector field through a hemisphere?
Yup, that's the nicer way...In theory.
Use the divergence theorem to compute the flux over the closed hemisphere. Say you get f0 as your answer. Then compute the flux directly over the closed disk. This should be easy. Your g(x,y) = z = 0. Your normal vector will be <0,0,-1>.
But integrating that sin(x^3) will give problems, even if you switch to polar coordinates. Are you sure that's not a typo?