Let C be the triangle with vertices $(0,0),(1,0)$ and $(0,4),$ and let $F(x,y)=($:$x^2+e^y,x+y$:$).$ We want to study the flux of F outward from this triangle.

Plot the vector field, and either plot or sketch the triangle on your plot. $($Make sure you can see the vector field throughout the triangle.$)$ Based on your picture, make a conjecture about whether the flux will be positive or negative.

On the plot of the vector field, based on the idea that divergence measures "spread" of the vector field, or how much more fluid is leaving a point than entering, make some guesses about where the divergence of the vector field would be positive or negative.

Find the divergence $P_x+Q_y$ of the vector field $F$ and make a contour plot of this function, so you can see where the divergence is positive and negative.

Now calculate the flux directly by parametrizing the triangle.

Note you will need three integrals, one for each side of the triangle. To save time, you can use technology to evaluate the integrals $($Desmos$,$ Wolfram Alpha, Sage, etc$).$

$5.$ Find the flux using Green's theorem $($flux form$).$