Vector field ##\vec F= 4x \hat i+4y \hat j +3 \hat k##
Let S be the open surface above the xy-plane defined by ##z=4-x^2-y^2##
a. Evaluate normal outward flux of F through S.
b. Use Stokes' theorem to evaluate the normal outward flux of ##∇ \times \vec F## through S.
c. Use Gauss' theorem to evaluate the volume integral $$\iiint_V (∇\times \vec F)\,. dxdydz$$ where V is the volume enclosed by S and the xy-plane.
vector field, curl, div, flux formula, etc.
The Attempt at a Solution
First, i found the unit normal vector, ##\hat n## and then i used the formula:
$$\iint_S \vec F. \hat n .dS$$
Is that correct? I calculated the integrand and then projected on the xy-plane for the surface area.
Part (b): I used the following formula:
$$\iint_S ∇\times F. \hat n .dS$$
Is that correct? If yes, then for this part i got the curl F = 0, so i didn't calculate the rest as the double integral of 0 is 0.
Part (c): I used the formula:
$$\iiint_V ∇. F .dxdydz$$
I found div F, then used cylindrical coordinates to find the volume. I'm quite sure that i got this part right.