The Curl of a Vector Field

Remember that the circulation density of a vector field $\vF=\langle F_1,F_2\rangle$ at a point $(a,b)$ is calculated as . 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 densi…

In this exercise, we will look at how different forms of component function's dependence on our variables will change the curl of the vector field. For each of the vector fields given below, decide if the curl of the vector field must be zero (as a vector) and write a sentence or two to justify your decision.

Let $h(x,y,z)$ be a smooth function of three variables and let $\vF=grad(h)=\nabla{h}$.

1. Write out the component functions of $\vF$ in terms of partial derivatives of $h$.

2. Write the divergence of $\vF$ in terms of derivatives of $h$.

3. Is the divergence of $\vF$ necessarily zero? Explain why or why not.

4. Calculate $curl(\vF)$ in terms of derivatives of $h$.

5. Show that $curl(\vF)=\vec{0}$.