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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

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1 Answer 1

up vote 1 down vote accepted

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 $$\int {\bf F}\cdot d{\bf A}=\int dV\ {\rm div}\ {\bf F}=3({\rm volume})=144.$$

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