Find the total flux of the vector field
$$ F = (3x, xy,1) $$
across the boundary of the box $$ D = {|x| \leq 1 , |y| \leq 2, |z|\leq 3} $$
Somehow, I think I know what this looks like very clearly. It's just that I'm not sure how to set up the integral for this.
So I would do something like:
$$ \int_A f(x) \cdot \vec{n} dA $$
over each surface of my box and add them together.
Now, I'm not sure how to get n
Best Answer
Your surface has six sides, each of which has a different ${\bf n}$. For instance, on the side with $x=\pm 1$, you'll have ${\bf n}=\pm{\bf e}_x$. Draw a picture if this is not clear to you.
Alternately, you can do this problem with the divergence theorem. You have $${\rm div}\ {\bf F}=3$$ and $$\int {\bf F}\cdot d{\bf A}=\int dV\ {\rm div}\ {\bf F}=3({\rm volume})=144.$$