Question

Evaluate \[\int\limits_{C}^{}xy dy-y ^{2}dx\] where C is the closed curve consisting of the top half of the circle\[x^{2}+y ^{2}=4\] and the x-axis segment \[-2 \le x \le 2.\]

Answer 1

@Zarkon Any ideas?

Answer 2

yes

Answer 3

the more important question is ...do you have any ideas?

Answer 4

Yes. I know
\[C=C _{i}=\] \[i=1: x=2cost, y=2sint, 0 \le t \le \pi\] \[i=2: x=t, y=0, -2 \le t \le 2\] \[\int\limits_{C}^{} F \dot \times dr=\int\limits_{C _{1}}^{}F \dot \times dr _{1} +\int\limits_{C}^{}F \dot \times dr _{2}\]

Answer 5

The times with the dot over it is supposed to be the dot product, I couldn't find the button for it.

Answer 6

\[F\cdot dr\]

F\cdot dr

Answer 7

Oh thanks

Answer 8

replace x and y...then compute dy and dx and replace them too...then integrate

or you can use Green's theorem

Answer 9

\[\int\limits_{C_1}\left[2\cos(t)2\sin(t)\frac{d(2\sin(t))}{dt}-(2\sin(t)) ^{2}\frac{d(2\cos(t))}{dt}\right]dt\]

Answer 10

I've got that part, but what about the \[\cdot dr\] part of the integral?

Answer 11

you don't need to do that

Answer 12

your original question was not written in terms of vector fields

Answer 13

Why not?

Answer 14

So would the correct formula be just\[\int\limits_{C}^{}fds=\int\limits_{C _{1}}^{}fds _{1}+\int\limits_{C _{2}}^{}fds _{2}\]?

Answer 15

You are integrating a differential form over a manifold. So the idea is to come up with coordinate patches (parameterizations) that cover the manifold, pull the form back on those parameterizations and integrate the pulled back form (for each parameterization) and sum them together.

Answer 16

Huh??

Answer 17

lol

Answer 18

do the integral I suggested above.

Answer 19

for the upper half of the circle

\[\int\limits_{C}xy dy-y ^{2}dx=\int\limits_{0}^{\pi}\left[x(t)y(t) \frac{dy}{dt}-y(t) ^{2}\frac{dx}{dt}\right]dt\]

Answer 20

I guess differential geometry isn't really discussed in the class you are taking. Please ignore what I said and listen to Zarkon.