# closed path integrals

Watch
Announcements
#1
I have this question to do but we never covered how to go about closed path integrals over a triangular region, absolutely stuck please help.

Closed path integral of P(x, y) dx + Q(x, y) dy = Greens theorem integral(∂Q/∂x − ∂P/∂y )dx dy.
By evaluating the integrals on both sides verify that the theorem holds in the case of the triangular region R with vertices at (0, 0), (−1, −1) and (1, −1) and the functions
P and Q defined by P(x, y) = xy and Q(x, y) = xy^2.
0
4 weeks ago
#2
(Original post by Kaia19)
I have this question to do but we never covered how to go about closed path integrals over a triangular region, absolutely stuck please help.

Closed path integral of P(x, y) dx + Q(x, y) dy = Greens theorem integral(∂Q/∂x − ∂P/∂y )dx dy.
By evaluating the integrals on both sides verify that the theorem holds in the case of the triangular region R with vertices at (0, 0), (−1, −1) and (1, −1) and the functions
P and Q defined by P(x, y) = xy and Q(x, y) = xy^2.
You need to integrate along each of the sides of the triangle, going in a counter clockwise direction.

So we have that;

where is the closed triangular path, and;

is the path from (0,0) to (-1.-1)

is the path from (-1,-1) to (1,-1)

is the path from (1,-1) to (0,0)
Last edited by RDKGames; 4 weeks ago
0
#3
(Original post by RDKGames)
You need to integrate along each of the sides of the triangle, going in a counter clockwise direction.

So we have that;

where is the closed triangular path, and;

is the path from (0,0) to (-1.-1)

is the path from (-1,-1) to (1,-1)

is the path from (1,-1) to (0,0)
thats what I did the first time but was unsure if it was right because there are 2 sloped sides so do they need to be handled differently than a strictly vertical or horizontal side? thanks.
0
4 weeks ago
#4
(Original post by Kaia19)
thats what I did the first time but was unsure if it was right because there are 2 sloped sides so do they need to be handled differently than a strictly vertical or horizontal side? thanks.
For strictly horizontal/vertical sides you have that either dx or dy is zero so your LHS simplifies nicely.

For the sloped side, e.g. let us take the path for .

Then the LHS for this side becomes

Last edited by RDKGames; 4 weeks ago
0
X

new posts
Back
to top
Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

### Poll

Join the discussion

#### Should there be a new university admissions system that ditches predicted grades?

No, I think predicted grades should still be used to make offers (715)
33.87%
Yes, I like the idea of applying to uni after I received my grades (PQA) (904)
42.82%
Yes, I like the idea of receiving offers only after I receive my grades (PQO) (399)
18.9%
I think there is a better option than the ones suggested (let us know in the thread!) (93)
4.41%