From 9b025ccd8162aa51f3c930b88be5b31f6e188c9c Mon Sep 17 00:00:00 2001 From: Thomas N Atkins Date: Wed, 4 Dec 2019 08:06:24 -0600 Subject: [PATCH] Updates from Overleaf --- main.tex | 8 ++++++++ 1 file changed, 8 insertions(+) diff --git a/main.tex b/main.tex index 967f45a..610e8eb 100644 --- a/main.tex +++ b/main.tex @@ -3,6 +3,12 @@ \usepackage{amsmath} \usepackage[bottom]{footmisc} +% Keywords command +\providecommand{\keywords}[1] +{ + \small + \-\ \-\ \-\ \textbf{\textit{Keywords --}} #1 +} % Slides % @@ -35,6 +41,8 @@ Green's Theorem\\ 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. \end{abstract} +\keywords{Green, Stoke, integration, vector calculus} + \section{Introduction} 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: