3.7: Green's Theorem
- Page ID
- 50593
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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).
If the vector field \(F = (M, N)\) is defined and differentiable on \(R\) then
\[\oint_{C} M\ dx + N\ dy = \int \int_R N_x - M_y\ dA. \nonumber \]
In vector form this is written
\[\oint_{C} F \cdot dr = \int \int_{R} \text{curl} F\ dA. \nonumber \]
where the curl is defined as \(\text{curl} F = (N_x - M_y)\)