mirror of
https://github.com/Nova38/Math147FinalProject
synced 20241013 13:21:59 +00:00
Updates from Overleaf
This commit is contained in:
parent
10549c4b98
commit
14f1d75ce2
33
main.tex
33
main.tex
@ 98,7 +98,6 @@ George Green, the mathematician who would go on to postulate the now famous Gree


\label{fig:my_label}


\end{figure}






\subsection{Definition}


% here is some definitions stuffs


Green's theorem is commonly defined as follows.\footnote{"Section 57: Green's Theorem"  Paul Dawkins, Lamar University  02222019. (http://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx)} Let $C$ be a simple, smooth, closed, positivelyoriented curve and $D$ the region enclosed by said curve. Assume $P$ and $Q$ are, at least, singly differentiable. Then, the following relationship holds:


@ 275,9 +274,12 @@ Using a pair of functions that satisfies $\frac{\partial Q}{\partial x}  \frac{


\end{align*}


Otherwise, in order to solve the area enclosed by the spiral, we would need to evaluate:


\begin{align*}


\text{Area} &= \int^{2\pi}_{0} t\cos{t}\cdot (t\sin{t})'\:dt \\


&= \int^{2\pi}_{0}t\cos{t}\cdot (t\cos{t}+\sin{t})\:dt \\


&= \int^{2\pi}_{0} t^{2}\cos^{2}{t}+t\sin{t}\cos{t}\:dt


\text{Area} &= \int^{b}_{a}F(x)dx,\;\; dx = f'(t)\:dt \\


&= \int^{b}_{a}F(f(t))f'(t)\:dt, \;\; F(f(t)) = g(t) = y \\


&= \int^{\beta}_{\alpha}y\:dx\;\; \text{or} \int^{\alpha}_{\beta}y\:dx\\


&= \int^{2\pi}_{0} t\sin{t}\cdot (t\cos{t})'\:dt \\


&= \int^{2\pi}_{0}t\sin{t}\cdot (t\sin{t}+\cos{t})\:dt \\


&= \int^{2\pi}_{0} t^{2}\sin^{2}{t}+t\sin{t}\cos{t}\:dt


\end{align*}


Which results in a far more complex series of steps to the solution.




@ 287,7 +289,30 @@ A second example can be shown with a squashed ellipse, or what appears to be an


y &= \sin^2{\theta}\cos{\theta} \\


0 &< \theta < 2\pi \\


\oint_{c}y\:dx &= \int^{2\pi}_{0}(\sin^2{\theta}\cos{\theta})(\cos{\theta})\:d\theta \\


&= \int^{2\pi}_{0} \sin^2{\theta}\cos^2{\theta}\:d\theta \\


&= \int^{2\pi}_{0} \frac{1}{4}\sin^2(2\theta)\:d\theta \;\;\; \text{(Product to sum formula)} \\


&= \frac{1}{4}\int^{2\pi}_{0}(\frac{1}{2}\frac{1}{2}\cos{4\theta})\:d\theta \\


&= \frac{\pi}{4}  \frac{1}{8}\int^{2\pi}_{0}\cos{4\theta}\:d\theta \\


&= \frac{\pi}{4}\frac{1}{32}\sin{4\theta}\Big^{2\pi}_{0} \\


&= \frac{\pi}{4}


\end{align*}


If we attempt this case without Green's theorem, we would need to evaluate:


\begin{align*}


\text{Area} &= \int^{2\pi}_{0} \sin^2{\theta}\cos{\theta}\cdot (\sin{\theta})'\:d\theta\;\;\; \text{(This is the same as above)}


\end{align*}




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.




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 twodimensional 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}




%Maybe pop this in here https://www.spiedigitallibrary.org/ContentImages/Journals/JEIME5/26/6/063022/WebImages/JEI_26_6_063022_f002.png


\begin{figure}


\centering


\includegraphics[scale=0.8]{polyominoes.png}


\caption{Discrete Green's theorem


(https://www.spiedigitallibrary.org/journals/journalofelectronicimaging/volume26/issue06/063022/ComputationoftheareainthediscreteplaneGreens/10.1117/1.JEI.26.6.063022.full)}


\label{fig:my_label}


\end{figure}






\end{document}



BIN
polyominoes.png
Normal file
BIN
polyominoes.png
Normal file
Binary file not shown.
After Width:  Height:  Size: 56 KiB 
Loading…
Reference in New Issue
Block a user