# Green's theorem

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• Apr 14th 2010, 06:59 AM
Tekken
Green's theorem
$\displaystyle \int_{C}[f(x,y)dy - g(x,y)dx] = \int_{A}\int[\frac{\partial f}{\partial x} , \frac{\partial g}{\partial {y}}] dA$

Im trying to verify this theorem for the following line integral

$\displaystyle \int_{C}(x + y^{2})dy + (xy^{2}-y)dx$ where $\displaystyle C$ is the triangle with vertices $\displaystyle (1,1), (3,1), (3,3)$

I had no problem calculating the right hand side of green's theorem. However im struggling on the left hand side of the theorem.

Im thinking i need to get the line integral in terms of either x or y only ...?
• Apr 14th 2010, 07:52 AM
dedust
Quote:

Originally Posted by Tekken
$\displaystyle \int_{C}[f(x,y)dy - g(x,y)dx] = \int_{A}\int[\frac{\partial f}{\partial x} , \frac{\partial g}{\partial {y}}] dA$

Im trying to verify this theorem for the following line integral

$\displaystyle \int_{C}(x + y^{2})dy + (xy^{2}-y)dx$ where $\displaystyle C$ is the triangle with vertices $\displaystyle (1,1), (3,1), (3,3)$

I had no problem calculating the right hand side of green's theorem. However im struggling on the left hand side of the theorem.

Im thinking i need to get the line integral in terms of either x or y only ...?

Let $\displaystyle A=(1,1), B=(3,1), D=(3,3)$
along the line segment AB, $\displaystyle y=1$ and $\displaystyle x=t$ with $\displaystyle 1 \leq t \leq 3$, hence $\displaystyle dy = 0$ and $\displaystyle dx = dt$, calculate the integral.

do the same thing for the line segment BD and DA

regard

DD
• Apr 14th 2010, 09:28 AM
Tekken
Quote:

Originally Posted by dedust
Let $\displaystyle A=(1,1), B=(3,1), D=(3,3)$
along the line segment AB, $\displaystyle y=1$ and $\displaystyle x=t$ with $\displaystyle 1 \leq t \leq 3$, hence $\displaystyle dy = 0$ and $\displaystyle dx = dt$, calculate the integral.

do the same thing for the line segment BD and DA

regard

DD

Thanks alot,

I think i may have calculated the right hand side of green's theorem incorrectly.

i let $\displaystyle f(x,y) = x + y^{2}$ and $\displaystyle g(x,y) = xy^{2} - y$

this gave me $\displaystyle \frac{\partial f}{\partial x} = 1$and $\displaystyle \frac{\partial g}{\partial y} = 2xy - 1$

so filling these into the equation i got

$\displaystyle \int^{3}_{1}\int^{3}_{1}[(x+y^{2})-(xy^{2}-y)]dx.dy$

However im not sure if these limits are correct, aren't the inside limits of a double integral normally $\displaystyle x's$ or $\displaystyle y's$?

If you could answer this, it would be great