Difficult Vector Field Integral

[Daniel Sellers]

<Moderator's note: Image substituted by text.>

1. Homework Statement
Given the following vector field,
$$
\dfrac{2(x-1)\,dy - 2(y+1)\,dx}{(x-1)^2+(y+1)^2}
$$
how do I integrate :
The integral over the curve x^4 + y^4 = 1
x^4 + y^4 = 11

x^4 + y^4 = 21

x^4 + y^4 = 31

Homework Equations

Green's theorem and related equations for line integrals.

The Attempt at a Solution

None of the techniques I know seem to work for this problem and if there's a shortcut or trick I'm not seeing it.

There are multiple incorrect solutions available online, but no correct ones. I know that the integral over the first curve is 0 because one solution said they should all be 0 (because F is conservative, which it is not).

How do I parameterize this curve in a way that I can integrate the result?

 

[fresh_42]

Maybe this example here is of help:
https://www.physicsforums.com/threads/why-the-terms-exterior-closed-exact.871875/#post-5474443

 

[Ray Vickson]

<Moderator's note: Image substituted by text.>

1. Homework Statement
Given the following vector field,
$$
\dfrac{2(x-1)\,dy - 2(y+1)\,dx}{(x-1)^2+(y+1)^2}
$$
how do I integrate :
The integral over the curve x^4 + y^4 = 1
x^4 + y^4 = 11

x^4 + y^4 = 21

x^4 + y^4 = 31

Homework Equations

Green's theorem and related equations for line integrals.

The Attempt at a Solution

None of the techniques I know seem to work for this problem and if there's a shortcut or trick I'm not seeing it.

There are multiple incorrect solutions available online, but no correct ones. I know that the integral over the first curve is 0 because one solution said they should all be 0 (because F is conservative, which it is not).

How do I parameterize this curve in a way that I can integrate the result?

You need to show us more of what you have tried; just saying that "none of the techniques work" is not sufficient. How far did you get? Where do the tried techniques fail?

 

[ehild]

<Moderator's note: Image substituted by text.>

1. Homework Statement
Given the following vector field,
$$
δF=\dfrac{2(x-1)\,dy - 2(y+1)\,dx}{(x-1)^2+(y+1)^2}
$$
how do I integrate :
The integral over the curve x^4 + y^4 = 1
x^4 + y^4 = 11

x^4 + y^4 = 21

x^4 + y^4 = 31

Check if δF is an exact differential. How do you do it?