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'Green's theorem'
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| Title of object: |
Green's theorem |
| Canonical Name: |
GreensTheorem |
| Type: |
Theorem |
| Created on: |
2002-02-02 21:51:16 |
| Modified on: |
2004-04-30 11:33:08 |
| Classification: |
msc:26B20 |
| Keywords: |
path integrals, evaluating path integrals, curl of a vector field |
Revision comment (for changes between this and next version):
| Changes for correction #3969 ('broken'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
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 $P$ bounding a region $R$ with area $A$, and a vector-valued function $\vec{F}=(f(x,y),g(x,y))$ over the plane,
$$\oint_P\vec{F}\cdot d\vec{x} = \int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA$$
where $a_n$ is the derivative of $a$ with respect to the $n$th variable.
\begin{center}
\includegraphics[width=2.694444in]{greensthm}
\end{center}
\paragraph{Corollary:}
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.
\paragraph{Proof:}
The corollary states that
$$\oint_P\vec{\nabla}_h\cdot d\vec{x} = 0$$
We can easily prove this using Green's theorem.
$$\oint_P\vec{\nabla}_h\cdot d\vec{x} = \int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA$$
But since this is a gradient...
$$\int\!\!\!\int_{\!\!R} [g_1(x,y) - f_2(x,y)] dA = \int\!\!\!\int_{\!\!R} [h_{21}(x,y) - h_{12}(x,y)] dA$$
Since $h_{12}=h_{21}$ for any function with continuous partials, the corollary is proven. |
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