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main.tex
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main.tex
@ -98,7 +98,6 @@ George Green, the mathematician who would go on to postulate the now famous Gree
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\label{fig:my_label}
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\end{figure}
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\subsection{Definition}
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% here is some definitions stuffs
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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, positively-oriented curve and $D$ the region enclosed by said curve. Assume $P$ and $Q$ are, at least, singly differentiable. Then, the following relationship holds:
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@ -275,9 +274,12 @@ Using a pair of functions that satisfies $\frac{\partial Q}{\partial x} - \frac{
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\end{align*}
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Otherwise, in order to solve the area enclosed by the spiral, we would need to evaluate:
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\begin{align*}
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\text{Area} &= \int^{2\pi}_{0} t\cos{t}\cdot (t\sin{t})'\:dt \\
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&= \int^{2\pi}_{0}t\cos{t}\cdot (t\cos{t}+\sin{t})\:dt \\
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&= \int^{2\pi}_{0} t^{2}\cos^{2}{t}+t\sin{t}\cos{t}\:dt
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\text{Area} &= \int^{b}_{a}F(x)dx,\;\; dx = f'(t)\:dt \\
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&= \int^{b}_{a}F(f(t))f'(t)\:dt, \;\; F(f(t)) = g(t) = y \\
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&= \int^{\beta}_{\alpha}y\:dx\;\; \text{or} \int^{\alpha}_{\beta}y\:dx\\
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&= \int^{2\pi}_{0} t\sin{t}\cdot (t\cos{t})'\:dt \\
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&= \int^{2\pi}_{0}t\sin{t}\cdot (-t\sin{t}+\cos{t})\:dt \\
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&= \int^{2\pi}_{0} -t^{2}\sin^{2}{t}+t\sin{t}\cos{t}\:dt
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\end{align*}
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Which results in a far more complex series of steps to the solution.
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@ -287,7 +289,30 @@ A second example can be shown with a squashed ellipse, or what appears to be an
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y &= \sin^2{\theta}\cos{\theta} \\
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0 &< \theta < 2\pi \\
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\oint_{c}y\:dx &= \int^{2\pi}_{0}(\sin^2{\theta}\cos{\theta})(\cos{\theta})\:d\theta \\
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&= \int^{2\pi}_{0} \sin^2{\theta}\cos^2{\theta}\:d\theta \\
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&= \int^{2\pi}_{0} \frac{1}{4}\sin^2(2\theta)\:d\theta \;\;\; \text{(Product to sum formula)} \\
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&= \frac{1}{4}\int^{2\pi}_{0}(\frac{1}{2}-\frac{1}{2}\cos{4\theta})\:d\theta \\
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&= \frac{\pi}{4} - \frac{1}{8}\int^{2\pi}_{0}\cos{4\theta}\:d\theta \\
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&= \frac{\pi}{4}-\frac{1}{32}\sin{4\theta}\Big|^{2\pi}_{0} \\
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&= \frac{\pi}{4}
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\end{align*}
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If we attempt this case without Green's theorem, we would need to evaluate:
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\begin{align*}
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\text{Area} &= \int^{2\pi}_{0} \sin^2{\theta}\cos{\theta}\cdot (\sin{\theta})'\:d\theta\;\;\; \text{(This is the same as above)}
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\end{align*}
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In both of these cases we can see that it is often beneficial for us to use Green's theorem instead of directly finding and computing the area inside a space through the formula for the area under a parametric curve. In fact, because the Green's theorem can be seen as a simpler representation of the different permutations of the original formula, all that it does is allow for more options in doing the integral such that a simpler version of the calculation can be done. It can be seen from the second example that the original formula also happens to be a convenient formula that can be obtained from Green's theorem.
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Aside from the use cases of Green's theorem in Vector Calculus, a discrete version of Green's theorem seems to be an application in the discrete mathematics of theoretical computer science. A journal article takes a look at computing the area of a series of pixels in a two-dimensional plane for polyominoes.\footnote{Alain Chalifour, Fathallah Nouboud, Yvon Voisin, "Computation of the area in the discrete plane: Green’s theorem revisited," J. Electron. Imag. 26(6) 063022 (5 December 2017) https://doi.org/10.1117/1.JEI.26.6.063022}
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%Maybe pop this in here https://www.spiedigitallibrary.org/ContentImages/Journals/JEIME5/26/6/063022/WebImages/JEI_26_6_063022_f002.png
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\begin{figure}
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\centering
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\includegraphics[scale=0.8]{polyominoes.png}
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\caption{Discrete Green's theorem
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(https://www.spiedigitallibrary.org/journals/journal-of-electronic-imaging/volume-26/issue-06/063022/Computation-of-the-area-in-the-discrete-plane--Greens/10.1117/1.JEI.26.6.063022.full)}
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\label{fig:my_label}
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\end{figure}
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\end{document}
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