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Mathematics LibreTexts

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).

    003 - (3.8 - Green s theorem).svg
    Figure \(\PageIndex{1}\): Examples of piecewise smooth and piecewise smooth regions. (CC BY-NC; Ümit Kaya)

    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 \(\text{curl} F = (N_x - M_y)\)