- #1

Granger

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## Homework Statement

Compute the work of the vector field ##F(x,y)=(\frac{y}{x^2+y^2},\frac{-x}{x^2+y^2})##

in the line segment that goes from (0,1) to (1,0).

## Homework Equations

3. The Attempt at a Solution [/B]

My attempt (please let me know if there is an easier way to do this)

I applied Green's theorem in the region between the square of vertices (1,0), (0,1), (-1,0), (0,-1), and the circumference centered in the origin with radius 1/2, both clockwise.

Since both lines are clockwise, and because F is field of class ##C^1## then

##\int_C F = \int_S F## (C circumference and S square).

C is then described by the path ##\gamma=(\frac{\cos t}{2},\frac{-\sin t}{2}) t\in]0,2\pi[##

We have ##F(\gamma (t)) \gamma ´(t)=1## so ##\int_C F = 2\pi = \int_S F##

Now because we want only the work in the line segment that goes from (0,1) to (1,0) we divide our result by 4 and obtain ##\frac{\pi}{2}##My doubts here is if this is correct, especially the final step... I also wonder if there was an easier way to approach the problem. I first thought of applying the fundamental theorem of calculus but we can't because F is not conservative. Then I tried the definition but we end up with a hard integral to compute. So I ended up with this...

Thanks for the help.

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