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Loser66
 one year ago
Let C be the circle \(x^2+y^2 =1\) oriented counterclockwise in the xyplane. What is the value of the line integral
\(\oint_C(2xy)dx +(x+3y)dy \)
Loser66
 one year ago
Let C be the circle \(x^2+y^2 =1\) oriented counterclockwise in the xyplane. What is the value of the line integral \(\oint_C(2xy)dx +(x+3y)dy \)

This Question is Closed

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2you don't wanto use green's thm ?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1ok, let me try. Actually, I didn't know what the notation \(\oint\) mean. I never see it before. :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2that just means the curve is a closed loop

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1I am working on it, will tag you to check it later, ok?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2ok, just need to find the curl and setup double integral

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2knw how to find the curl ?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1yes, I divide it into 2 parts, \(y = \pm \sqrt{1x^2}\) ,hence the limit for the first part will go from 0 to 1, right?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1oh, We talk about 2 different things. ha!!

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2lol yeah actually we don't need to do much work here, find the curl, you will know why :)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1ok, give me your way, please. hehehe..

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\(\large Mdx + Ndy\) curl = \(N_x  M_y\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\(\large (2xy)dx +(x+3y)dy\) curl = ?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1It looks like differential equation part? finding exactness, right?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\(\large (2xy)dx +(x+3y)dy\) \(M = 2xy\) \(N = x+3y\) \(N_x = 1\) \(M_y = 1\) curl = \(N_x  M_y = 1(1) = 2\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Easy.. just take partials and subtract

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1I know, but don't know why we have to do that.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2because we want to use green's thm

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1I was taught that I have to find parametric equations for x, y and replace and take a loooooooooong steps to get the answer. This is somehow different.

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1But that is the reason i post the problem here to learn the shorter way. :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\[\oint_C(2xy)dx +(x+3y)dy ~~=~~ \iint_R~2 dxdy = 2\iint_R~1 dxdy = ?\]

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1You still use x, y , not r and theta?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1oh, that is perimeter of the circle?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2I just applied green's theorem to convert line integral into double integral

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Ok, I got you. Thanks a lot. Need practice more. :)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1If the curve is not a circle, we must define the limits of x,y to put into the double integral, right?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2green's theorem works only if the curve is a closed loop

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Yes, Again, don't we have to change to polar form?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2for all other cases you need to work it by parameterizing

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2you can but its not really needed here if you recall the fact that \(\iint_R ~1 ~dxdy\) represents the area of the region.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\[\oint_C(2xy)dx +(x+3y)dy ~~=~~ \iint_R~2 dxdy = 2\color{red}{\iint_R~1 dxdy }= ?\] that red part represents the area of the circular unit disk

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1hey, on the previous comment (and you delete it), you stated the result is 4pi, ha!! now it turns to 2pi??

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2that red part is 2pi final answer is 4pi

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1area of unit circle is pi

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Oops! you're right haha

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1hihihi... ok, got you now. Much appreciate for being patient to me.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2np :) maybe for practice, work it by parameterizing also
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