Added graph

main.tex

@@ -10,6 +10,24 @@
   \-\ \-\ \-\ \textbf{\textit{Keywords --}} #1
 }
 
+% Plot set up --Thomas--
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.16,scale=1.25}
+
+\pgfplotsset{
+    vector/.style={
+        % Axis Line setup
+        axis lines=center,
+        axis line style={latex-latex},
+        % View Set up, to rotate the graph
+        view={0}{90},
+    }
+}
+
+
+
+
+
 % Slides 
 % 
 % 1) Authors names and affiliations
@@ -81,7 +99,48 @@ $$
 \section{Verification of Green's Theorem}
 Green's theorem is commonly used in applications of vector calculus to other fields of study, particularly physics in the plane. One example of this is calculating the circulation of vector fields along a closed boundary. For our purposes, circulation refers to the magnitude of the vector field that \emph{passes through} or \emph{pushes against} the closed boundary.\footnote{"Vector Calculus: Understanding Circulation and Curl" - Editors of BetterExplained, BetterExplained, Vector Calculus - n.d.} 
 
-==== Illustration Image of Circulation Here ====
+% ==== Illustration Image of Circulation Here ====
+
+\begin{figure}[]
+    \centering
+        \begin{tikzpicture}
+            \begin{axis}[vector,
+                title={},
+                domain=-2.5:2.5,
+                ymin = -1.5,
+                ymax = 2.5,
+                xlabel=$x$,
+                ylabel=$y$,
+            ]
+                \addplot[
+                    patch,
+                    patch type=triangle,
+                    color = white,
+                    line width=2pt,
+                    faceted color=red,
+                ] 
+                    coordinates {
+                        (-1,0)
+                        (1,0)
+                        (0,2)
+                    };
+                \addplot3[
+                    cyan,
+                    quiver={
+                        u={-y},
+                        v={x^2+1},
+                        scale arrows=0.1
+                    },
+                    -stealth,
+                    samples=15]
+                    {x};
+            \end{axis}
+        \end{tikzpicture}
+    \caption{Illustration of Circulation}
+    \label{Circulation}
+\end{figure}
+
+
 
 Take, for example, the field and boundary pictured above. There is some vector field in the plane, $\vec{E}$, that emanates from some point $P$. Inset in that field is some simple, closed boundary $\partial S$. The circulation of the field $\vec{E}$ with respect to the boundary $\partial S$ is the magnitude $E$ that is incident on $\partial S$. That is, the \emph{amount} of the field that passes through the boundary.
 
@@ -120,14 +179,53 @@ So we can calculate the various $\Phi_i$ using the formula above. Then, we can f
     &= \int_0^1 \left<-2t, (1-t)^2 +1\right> \cdot \left<-1,2\right> dt \\
     &= \int_0^1 (2t + 2(1-t)^2 +2) \; dt \\
     &= \int_0^1 (2t^2 -2t + 4) \; dt \\
+    &= \frac{2}{3}t - t^2 + 4t \vert_0^1 \\
     &= \frac{11}{3}
 \end{align*}
 
+\begin{align*}
+    \Phi_3 &= \int_{C_3} \vec{E}(\vec{r}_3) \cdot d\vec{r}_3 \\
+    &= \int_0^1 \left<2t-2, t^2 +1\right> \cdot \left<-1, -2\right> \\
+    &= \int_0^1 -2t-2t^2 \; dt \\
+    &= -t^2 -\frac{2}{3}t^3 \mid_0^1 \\
+    &= -\frac{5}{3}
+\end{align*}
+
+We can then compute the total $\Phi_{net}$ as the sum of the fluxes across the three boundary-components. That is, $\Phi_net = \Phi_1 + \Phi_2 + \Phi_3 = 0 + \frac{11}{3} - \frac{5}{3} = 2$. This result indicates that the flux of the electric field has a magnitude of 2 across the described boundary. Typically, the units for electric flux are Volt-meters ($Vm$).
+
+\subsection{Using Green's Theorem}
+As mentioned above, Green's theorem relates the circulation of a vector field over a boundary to the area of the curl of that field over the surface enclosed by the boundary. We can verify the result gained above, then, by calculating the following double-integral:
+\begin{align*}
+    \int_C \vec{E}(\vec{r}) \cdot d\vec{r} = \iint_D \nabla \times \vec{E} \cdot dA
+\end{align*}
+
+We begin by computing the curl of $\vec{E}$ as the virtual cross-product of the nabla-operator against the components of $\vec{E}$:
+\begin{align*}
+    \nabla \times \vec{E} &=
+    \begin{vmatrix}
+        \hat{i} & \hat{j} & \hat{k} \\
+        \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
+        -y & x^2 + 1 & 0
+    \end{vmatrix} \\
+    &= \left<0,0,2x+1\right>
+\end{align*}
+
+Now, we can compute the related area by dotting this result with the differential area. Because the boundary lies in the xy-plane, we can take the z-component for our integrand. Thus, the circulation across the boundary can be calculated as such:
+\begin{align*}
+    \Phi_{net} &= \iint_D \nabla \times \vec{E} \cdot dA \\
+    &= \int_0^2 \int_{y/2-1}^{1-y/2} 2x+1 \; dx \; dy \\
+    &= \int_0^2 2-y \; dy \\
+    &= 2
+\end{align*}
+
+Thus, the circulation across the boundary is $2 \; Vm$, which is consistent with the result we found above.
+
 \section{Applications}
-Calculating parametric curves using Green's theorem instead of directly computing a line integral can often be much simpler. Calculating the area of a seashell:
+Calculating parametric curves using Green's theorem instead of directly computing a line integral can often be much simpler. Calculating the area of a seashell-like spiral:
 \begin{align*}
-    x(t) &= t\cos{t} \\
-    y(t) &= t\sin{t} \\ 
+    x(t) = t\cos{t} \\
+    y(t) = t\sin{t}  \\
+    0<t<2\pi \\
 \end{align*}
 
 \end{document}