- #1

DryRun

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## Homework Statement

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.

## Homework Equations

vector field, curl, div, flux formula, etc.

## The Attempt at a Solution

Part (a):

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.