PlanetMath: Green's theorem

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Green's theorem

(Theorem)

Green's theorem provides a connection between path integrals over a well-connected region in the plane and the area of the region bounded in the plane. Given a closed path bounding a region with area , and a vector-valued function $\vec{F}=(f(x,y),g(x,y))$ over the plane,

$\displaystyle \oint_P\vec{F}\cdot d\vec{x} = \int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA$

where

is the derivative of

with respect to the

$\includegraphics[width=2.694444in]{greensthm}$

The closed path integral over a gradient of a function with continuous partial derivatives is always zero. Thus, gradients are conservative vector fields. The smooth function is called the potential of the vector field.

The corollary states that

$\displaystyle \oint_P\vec{\nabla}_h\cdot d\vec{x} = 0$

We can easily prove this using Green's theorem.

$\displaystyle \oint_P\vec{\nabla}_h\cdot d\vec{x} = \int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA$

But since this is a gradient...

$\displaystyle \int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA = \int\!\!\!\int_{\!\!R} [h_{21}(x,y) - h_{12}(x,y)] dA$

Since $h_{12}=h_{21}$ for any function with continuous partials, the corollary is proven.

"Green's theorem" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Keywords:

path integrals, evaluating path integrals, curl of a vector field

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26B20 (Real functions :: Functions of several variables :: Integral formulas )

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