Difficult Vector Field Integral

  • #1
<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?
 
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Answers and Replies

  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722
<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?
 
  • #4
ehild
Homework Helper
15,543
1,913
<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?
 

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