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# 3.7: Green's Theorem

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Ingredients: $$C$$ a simple closed curve (i.e. no self-intersection), and $$R$$ the interior of $$C$$.

$$C$$ must be piecewise smooth (traversed so interior region $$R$$ is on the left) and piecewise smooth (a few corners are okay).

## Theorem $$\PageIndex{1}$$: Green's Theorem

If the vector field $$F = (M, N)$$ is defined and differentiable on $$R$$ then

$\oint_{C} M\ dx + N\ dy = \int \int_R N_x - M_y\ dA.$

In vector form this is written

$\oint_{C} F \cdot dr = \int \int_{R} \text{curl} F\ dA.$

where the curl is defined as $$\text{curl} F = (N_x - M_y)$$