Flux of \vec{F} Through S: Compute

[-EquinoX-]

Homework Statement

Compute the flux of the vector field, $$\vec{F}$$ , through the surface, S.
$$\vec{F} = y\vec{i} + 7\vec{j} - xz\vec{k}$$ and S is the surface $$y = x^2 + z^2$$ with $$x^2 + z^2 \leq 36$$ oriented in the positive y direction.

Homework Equations

The Attempt at a Solution

$$\int\limit_R ((x^2+x^2)\vec{i} + 7\vec{j} - zx\vec{k}) \cdot (-2x\vec{i} -2z\vec{k} + \vec{j})dA$$

 

[Billy Bob]

Are you trying to evaluate

∫∫_R ( (x²+z²) i + 7 j - zx k ) · (-2x i + j -2z k) dA

where R is the disc x² + z² ≤ 36 in the xz plane?

So take the dot product, then convert to polar coordinates in the xz plane, is this what you mean?

I think this is correct if the unit normal to S is pointing "right" (i.e., into the parabolic bowl), but negate it if the unit normal to S is pointing "left" (i.e., out of the bowl). Also, you are assuming the bowl has no "lid."

 

[-EquinoX-]

thanks!