- #1
Liferider
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Homework Statement
Use greens theorem to solve the closed curve line integral:
[itex]\oint[/itex](ydx-xdy)
The curve is a circle with its center at origin with a radius of 1.
Homework Equations
x^2 + y^2 = 1
The Attempt at a Solution
Greens theorem states that:
Given F=[P,Q]=[y, -x]=yi-xj
[itex]\oint[/itex]F*dr=[itex]\oint[/itex]Pdx+Qdy=[itex]\int[/itex][itex]\int[/itex]([itex]\frac{dQ}{dx}[/itex]-[itex]\frac{dP}{dy}[/itex])dA
From the circle equation i find:
x=[itex]\sqrt{1-y^2}[/itex]
y=[itex]\sqrt{1-x^2}[/itex]
Which means that:
[itex]\frac{dQ}{dx}[/itex]=0 and [itex]\frac{dP}{dy}[/itex]=0
Obviously, I am doing something wrong... but which rules am I breaking??
I did find the answer the "normal" way, without greens, which was -2[itex]\pi[/itex].
I think a lot of my difficulties originates from lack of knowledge about different coordinate systems and the conversion between these.
One could write:
x=cos t and y=sin t, but do I have to? (btw, i used that for solving the normal way).