Computing the flux of the vector field F
Compute the flux of the vector field F = 5yi + 2j - 5xzk through the surface S, which is the surface y = x² + z², with x² + z² ≤ 9, oriented in the positive y-direction.
I understand that F = <5y, 2, -5xz>, but how do I find dA? and would the bounds for x and z be [0, 3]?
Re: Computing the flux of the vector field F
I would change to polar coordinates in the xz-plane. That is, , , with
Now the "position vector" for any point on that paraboid would be . The derivatives are and .
The cross product of those vectors is
so the "vector differential of surface area" is
But since we want "oriented in the positive y direction" we need to multiply throught by -1 to make the y-component positive:
(That is the same as swapping the two derivative vectors in the cross product.)
Since , and there is circular symmetry on the xz-plane, the limits of integration are r= 0 to 3 and to .