Green's theorem GreensTheorem 2002-02-02 21:51:16 2004-04-30 12:11:32 Theorem path integrals evaluating path integrals curl of a vector field % 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. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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.