PX275 - C1e - examples of curl

1

consider water flowing with a velocity, $\vec{v}$ , in the $\hat{i}$ direction: $\vec{v} = x \hat{i}$
the curl:

\vec{\nabla} \times \vec{v} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x & 0 & 0 \end{matrix} | = 0

interpretation: if there is a raft on the water, it will not rotate

2

now, the water is flowing in the $y$ -direction, but the velocity increases in the $x$ -direction: but $\vec{v} = x \hat{j}$
the curl:

\vec{\nabla} \times \vec{v} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 0 & x & 0 \end{matrix} | = \hat{k}

interpretation: a raft will tend to spin