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Mathematics LibreTexts

3.7: Green's Theorem

( \newcommand{\kernel}{\mathrm{null}\,}\)

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 3.7.1: Examples of piecewise smooth and piecewise smooth regions. (CC BY-NC; Ümit Kaya)

Theorem 3.7.1: Green's Theorem

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

CM dx+N dy=RNxMy dA.

In vector form this is written

CFdr=RcurlF dA.

where the curl is defined as curlF=(NxMy)


This page titled 3.7: Green's Theorem is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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