Green’s Theorem (continued)
Green’s theorem can also be expressed in its “divergence form”.
Let n be the unit outward normal to the curve, C.
Here n = dx i - dy j
and F = P i + Q j
So F . n = P dx - Q dy
∫ F . n ds = ∫ P dx – Q dy = ∫ ∫ [∂Q/∂x + ∂P/∂y]
C C R
Finally ∫ F . n ds = ∫ ∫ div F dA
C R
Summary:
Green's Theorem in Standard Form:
∫ P dx + Q dy = ∫ ∫ [ ∂Q/∂x ˗ ∂P/∂y ] dA
Green's Theorem in Curl Form:
(vector form)
∫ F . dr = ∫ F . T ds = ∫ ∫ curlz F dA
∫ F . dr = ∫ ∫ (curl F ) · k dA
Green's Theorem in Divergence Form:
∫ F . n ds = ∫ ∫ div F dA
Click here for four examples involving Green’s Theorem.
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