Activity 13.8.2

Remember that the circulation density of a vector field $\vF=\langle F_1,F_2\rangle$ at a point $(a,b)$ is calculated as $$\begin{gathered} \text{Circulation Density at the point }(a,b)= \frac{\partial F_2}{\partial x} (a,b)-\frac{\partial F_1}{\partial y} (a,b) \end{gathered}$$. The circulation density will be positive when a small spinner placed at $(a,b)$ will spin counterclockwise, negative when the spinner will move clockwise, and zero when the spinner does not rotate.

For each of the vector fields given below, answer the following questions:

1. What is the formula for the circulation density of the given vector field?
2. For what points will you have a positive circulation density?
3. For what points will you have a negative circulation density?
4. For what points will you have a zero circulation density?

1. $\langle -y,x\rangle$

2. $\langle x,y\rangle$

3. $\langle 1-x,xy\rangle$