?This exercise demonstrates a connection between the curl vector and rotations. Let B be

Chapter 14, Problem 39

(choose chapter or problem)

This exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector \(\mathbf{w}=\omega \mathbf{k}\), where \(\omega\) is the angular speed of B, that is, the tangential speed of any point P in B divided by the distance d from the axis of rotation. Let \(\mathbf{r}=\langle x, y, z\rangle\) be the position vector of P.

(a) By considering the angle \(\theta\) in the figure, show that the velocity field of B is given by \(\mathbf{v}=\mathbf{w} \times \mathbf{r}\).

(b) Show that \(\mathbf{v}=-\omega y \mathbf{i}+\omega x \mathbf{j}\).

(c) Show that \(\operatorname{curl} \mathbf{v}=2 \mathbf{w}\).