Question

help with green's theorem please?

Answer 1

u mean green's thm regarding diff eq

Answer 2

???

Answer 3

post the question

Answer 4

i'll start from scratch. what is green's theorem?

Answer 5

Ok, are you familiar with concept of line integral?

Answer 6

green's theorem regarding flux and circulation

Answer 7

yes

Answer 8

green theorem is just an alternative way of evaluating line integral over vector field

Answer 9

ok, and using green's theorem, how do i find the flux and circulation of a curve?

Answer 10

\[\text{circulation}=\oint_{c}F . T ds=\oint_{c}M dx + N dy=\iint_{R}dN/dx - dM/dY dx dy\]

Answer 11

\[flux=\oint_{c}F \cdot n ds=\oint_{c}-N dx + M dy=\iint_{R}dM/dx + dN/dY dx dy\]

Answer 12

see how in both cases, we went from closed line integral to double integral? That's green theorem

Answer 13

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

Answer 14

circulation or flux?

Answer 15

ok so far i understand what you're saying, it's exactly what i have in my notes. but when i am solving the problems, like the example above, i get an answer that is circ=flux and the answer in the back gives different results

Answer 16

i need to find circ and flux

Answer 17

for circ, \[M=y ^{2}-x ^{2} ; N=x ^{2}+y ^{2}\]

Answer 18

2x- 2y

Answer 19

ok

Answer 20

dN/dx-dM/dy

Answer 21

circulation

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

Answer 22

=9

Answer 23

does order in integration matter?

Answer 24

*of

Answer 25

not if your limits are right

Answer 26

flux

dm/dx+ dn/dy

Answer 27

if i switch the order, my limits are \[\int\limits_{0}^{3} \int\limits_{0}^{y}\]

Answer 28

\[\text{ }\text{Flux}=\int _0^3\int _0^x(-2x+2y)dydx=-9\]

Answer 29

and my answer is -9

Answer 30

-9 for circ

Answer 31

i don't see what's wrong with my limits. if y=x, x=y

Answer 32

your limit is right

Answer 33

for finding the flux, it's \[\int\limits_{?}^{?} Mdy-Ndx\]

Answer 34

\[\text{ }\text{Flux}=\int _0^3\int _0^x(-2x+2y)dydx=-9\]

Answer 35

i was taught to change the original equation to fit this form. so the new form is \[(x ^{2}+y ^{2})-(-y ^{2}+x ^{2})\]
so now \[M=x ^{2}+y ^{2} ; N=-y ^{2}+x ^{2}\]

Answer 36

is it correct so far?

Answer 37

no, my flux is right. but back to circulation, the limit does seem to be wrong or something. when you integrated in respect to y first, your limits are 0 to x and the final result is 9. but when i integrate with respect to x first, limit is 0 to y and my final answer is -9