Flux of Vector Field Across Surface

[cheeee]

Homework Statement

Compute the flux of vector field (grad x F) where F = (xz+x^2y + z, x^3yz + y, x^4z^2)
across the surface S obtained by gluing the cylinder z^2 + y^2 = 1 (x is > or eq to 0 and < or eq to 1) with the hemispherical cap z^2 + y^2 (x-1)^2 = 1 (x > or eq to 1) oriented in such a way that the unit normal at (1,0,0) is given by i.

My approach to find the flux is to take the triple integral of the gradient of F * dV. I found the gradient to be <z + 2xy, x^3z +1, 2zx^4> and I set up the integral but I am a bit confused as to what the limits should be. Should I be even taking a triple integral??

 

[cheeee]

also... can I use cylindrical coordinates for the volume of the cylinder then use spherical coordinates for the volume of the cap and add them together?

 

[HallsofIvy]

How are you taking the triple integral? The "gradient of F" is a vector quantity and dV is not. I thought perhaps you were using the divergence theorem but you say nothing about the divergence of grad F (which would be \nabla^2 F).

If you are doing this directly (not using the divergence theorem), you need to integrate grad F over the surface of the figure. Yes, you can do the cylinder and hemisphere separately.