We will consider the vector field \(\vF = \langle 2y,3x^2 y\rangle\text{,}\) which is defined on the entire \(xy\)-plane. Suppose that we want to calculate the circulation of \(\vF\) around the circle \(C\) of radius \(2\text{,}\) centered at \((0,0)\text{,}\) and oriented counterclockwise.
Verify that \(\vF\) is not path-independent by calculating the circulation of \(\vF\) around the circle \(C\text{.}\) The SageMath cell below is set up to assist you with this, but you will need to supply a parametrization of \(C\) on line 4.
Recall from Section 13.7 that if \(\vF = \langle F_1(x,y),F_2(x,y)\rangle\) is a vector field, then the circulation density (in 2D) is given by
\begin{equation*}
\text{Circulation Density at the point }(a,b)= \frac{\partial F_2}{\partial x} (a,b)-\frac{\partial F_1}{\partial y} (a,b)
\end{equation*}
What is the circulation density of \(\vF = \langle 2y,3x^2 y\rangle\text{?}\)
Calculate the double integral of \(\displaystyle \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\) over the region inside the circle \(C\text{.}\) This integral is not the most fun to do by hand, so a SageMath cell has been provided to assist you.
What do you notice about your results to parts (a) and (d)? Do you think this will happen in general? Write a sentence or two about what you think is happening here.
