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54 lines
2.0 KiB
54 lines
2.0 KiB
\documentclass[11pt,English]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath}
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\usepackage[bottom]{footmisc}
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% Keywords command
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\providecommand{\keywords}[1]
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{
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\small
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\-\ \-\ \-\ \textbf{\textit{Keywords --}} #1
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}
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% Slides
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%
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% 1) Authors names and affiliations
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% 2) Title abstract keywords
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% 3) 147 why its important
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\title{
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Green's Theorem\\
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\large Historical Origins and Analytical Applications\\
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\small MATH 147 Final Project\\
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\small University of Kansas, Dept. of Mathematics
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}
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\author{
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Atkins, Thomas\\
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\texttt{thomas.atkins@ku.edu}
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\and
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Mills, Garrett\\
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\texttt{glmdev@ku.edu}
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\and
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Weng, QiTao\\
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\texttt{wengqt@ku.edu}
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}
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\date{December 2019}
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\begin{document}
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\maketitle
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\begin{abstract}
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In which the authors investigate the historical origins and several mathematical applications of the commonly known Green's theorem. Discovered by George Green in the late 1820s, this theorem provides a relationship between the line integral of a particular curve and the surface integral of its enclosed region. Green's theorem is closely related to the divergence theorem, and is simply a specific case of the more general Stoke's theorem. Beyond basic applications to flux and surface integrals, Green's theorem can be reverse applied to calculate difficult-to-evaluate area calculations. It also plays an integral role (pun intended) in the proof of other important theorems such as Cauchy's.
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\end{abstract}
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\keywords{Green, Stoke, integration, vector calculus}
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\section{Introduction}
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Green's theorem is commonly defined as follows.\footnote{"Section 5-7: Green's Theorem" - Paul Dawkins, Lamar University - 02-22-2019. (http://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx)} Let $C$ be a simple, smooth, closed, positive curve and $D$ the region enclosed by said curve. Assume $P'$, $Q'$ are continuous. Then, the following relationship holds:
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$$
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\int_C{ P dx + Q dy } = \iint_D{ \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA }
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$$
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\end{document}
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