Updates from Overleaf

main.tex

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 \usepackage{amsmath}
 \usepackage[bottom]{footmisc}
 
+% Keywords command
+\providecommand{\keywords}[1]
+{
+  \small	
+  \-\ \-\ \-\ \textbf{\textit{Keywords --}} #1
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 % Slides 
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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.
 \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: