Question

Greens Theorum:

Answer 1

Integrate h(x, y) = yi - xj over the triangle with vertices (-1, 0), (1, 0) and (0, 1) traversed counterclockwise.

Answer 2

\[\oint (\frac{ dP }{ dy }-\frac{ dQ }{ dx })dA\] or something like this.

So I set up:
\[\int\limits\limits_{-1}^{0}\int\limits\limits_{0}^{x}-1-1dydx\]

Answer 3

The answer is -1 but im coming up with -1

Answer 4

lol what?

Answer 5

lol greens theorum zep. comon! I dropped a negative, should be 1.

Anyway its asking me to integrate this triangle...
just figured it out I had my bounds wrong. Thanks zep!

Answer 6

oh ok XD

Answer 7

wait actually,

what would the y bounds be for my double integral if i was integrating this triangle?

Answer 8

Hmm I was thinking of it like this I guess...

In the x-direction you're bounded by the curves x=-y on the left,
and x=y on the right.
\(\rm -y\le x\le y\)

And then your y is bound by the bottom and top,
\(\rm 0\le y\le1\)

But ummm.. you can probably do it your way...
lemme see if that makes sense..

Answer 9

Oh yes, if you try to do it in the other direction,
you run into a problem right?

Because you have to split it into two separate integrals.
The upper boundary `changes` in the middle! :O

Answer 10

With the symmetry I could just multiply it by two and integrate from 0 to 1 or -1 to 0 I think then multiply by 2, but that still would get me a different answer.