@ -98,7 +98,6 @@ George Green, the mathematician who would go on to postulate the now famous Gree
\label{fig:my_label}
\label{fig:my_label}
\end{figure}
\end{figure}
\subsection{Definition}
\subsection{Definition}
% here is some definitions stuffs
% here is some definitions stuffs
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:
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:
@ -275,9 +274,12 @@ Using a pair of functions that satisfies $\frac{\partial Q}{\partial x} - \frac{
\end{align*}
\end{align*}
Otherwise, in order to solve the area enclosed by the spiral, we would need to evaluate:
Otherwise, in order to solve the area enclosed by the spiral, we would need to evaluate:
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*}
\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 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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