Green's Theorem proof using Stoke's theorem

Asked Apr 15 '14 at 19:50

Active Apr 15 '14 at 19:50

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I'm slightly confused with this proof, from Stoke's Theorem we have:

$$\int_C \underline{F} \cdot \ d \underline{r} = \int \int_S (\nabla \times \underline{F}) \cdot \underline{n} \ dS$$ so going by all the using notation in the image we have:

$$\int_C \underline{F} \cdot \ d \underline{r} = \int \int_S \left (\dfrac{\partial F_2}{\partial x} + \dfrac{\partial F_1}{\partial y} \right ) \ dS$$ How did we get from $$\int \int_S\left (\dfrac{\partial F_2}{\partial x} + \dfrac{\partial F_1}{\partial y} \right ) \ dS$$ to $$\int \int_R\left (\dfrac{\partial F_2}{\partial x} + \dfrac{\partial F_1}{\partial y} \right ) \ dx dy$$

asked Apr 15 '14 at 19:50

Warz

1,129

the problem restricted to the xy plane, so S is some region R in the plane with dS=dxdy. Should be a (-) sign between the partials at the end too. – Ryan Vitale Apr 15 '14 at 20:05
@rVitale thanks – Warz Apr 15 '14 at 20:34

Green's Theorem proof using Stoke's theorem

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