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Solve using Green's Theorem

  • Thread starter brainslush
  • Start date
  • #1
26
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Homework Statement


Use Green’s theorem to find the integral
[itex]\oint_{\gamma} \frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy[/itex]
along two different curves γ: first where γ is the simple closed curve which goes along x = −y2 + 4 and x = 2, and second where γ is the square with vertices (−1, 0), (1, 0), (0, 1), (0, −1).


Homework Equations





The Attempt at a Solution


I'm bit confused b/c
[itex]d(\frac{-y}{x^2+y^2})/dy = \frac{y^2-x^2}{(x^2+y^2)^2}[/itex]
[itex]d(\frac{x}{x^2+y^2})/dx = \frac{y^2-x^2}{(x^2+y^2)^2}[/itex]

Then by Green's theorem one gets

[itex]\int_{A}\int (d(\frac{x}{x^2+y^2})/dx-d(\frac{-y}{x^2+y^2})/dy) dx dy = \int_{A}\int 0 dx dy = 0[/itex]

What am I missing?
 

Answers and Replies

  • #2
5
0
You are not missing anything. The answer IS 0.

Here you have a line integral of a conservative field (you can tell it's conservative from the equality of the partial derivatives).
A line integral of a conservative field will always be zero for any path that begins and ends at the same point, i.e any closed curve.
 

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