still having a problem with finding the flux..

Question

anonymous · Answer

$$F=(y ^{2}-x ^{2})i + (x ^{2}+y ^{2})$$
C: triangle bounded by y=0, x=3, y=x

anonymous · Answer

u can find it by taking its surface intigral

anonymous · Answer

$$\int\limits_{?}^{?} Mdy-Ndx$$ so...
$$(x ^{2}+y ^{2})-(-y ^{2}+x ^{2})$$
the limits were established in an earlier post
$$\int\limits_{0}^{3} \int\limits_{y}^{3} 2x+(-2y)dxdy$$

anonymous · Answer

the answer is 9 but should be -9

anonymous · Answer

$$\int _0^3\int _0^x(2x-2y)dydx$$

anonymous · Answer

$$N_x- M_y=2x-2y$$

anonymous · Answer

so should i avoid swapping? it's just a way the instructor does it but it seems like he only swaps it to get the final answer to be the same. i don't know why you would need the circulation to equal flux

anonymous · Answer

you don't need to do it as far as I know

anonymous · Answer

they are different quantity

anonymous · Answer

after the swap, that is what i have 2x-2y but it's the limit again. if i do dxdy i should use the limit we finally found again

anonymous · Answer

oh yeah, you can do it my way or

$$\int\limits_{0}^{3} \int\limits_{y}^{3} 2x-2ydxdy$$

anonymous · Answer

$$\int _0^3\int _0^x(2x-2y)dydx =\int _0^3\int _y^3(2x-2y)dxdy$$

anonymous · Answer

i don't have a problem doing it dydx but i need to know why i cn't get the right answer with dxdy. it would confuse me in the future cuz i wouldn't know which way i should integrate

anonymous · Answer

you do get the right answer, if you use the right limit

anonymous · Answer

well if i do it dxdy i get the same answer as circulation

anonymous · Answer

dn/dx - dm/dy is circulation

anonymous · Answer

flux in dm/dx + dn/dy

anonymous · Answer

both=2x-2y

anonymous · Answer

flux
$$\int _0^3\int _0^x(-2x+2y)dydx =\int _0^3\int _y^3(-2x+2y)dxdy$$

anonymous · Answer

that's correct if i don't do the swap. so now i'm confused as to why i would ever need to swap

anonymous · Answer

yeah, forget about swap good source on this stuff http://omega.albany.edu:8008/calc3/div-curl-dir/lec5.html

anonymous · Answer

it would seem like i'd do the swap to get circ=flux to prove green's theorem??

anonymous · Answer

ok i'll check it out thank you

anonymous · Answer

@ yeah , I think that's what you are trying to do