### Curl

• Calculate curl.

• Understand physical interpretation and applications of curl.

### Solved Example:

#### 4-11-01

The vector field is F = xi - yj (where i and j are unit vector) is:

Solution:
$F = xi - yj$ . First check divergency, for divergence, \begin{align*} \nabla \cdot F &=\left[ \dfrac {\partial }{0x}\overline {i}+\dfrac {\partial }{dy}\overline {j}+\dfrac {\partial }{\partial z}\overline {k}\right] \left[ x\overline {i}-y\overline {j}\right]\\ &=1-1 =0 \end{align*} So we can say that F is divergence free. Now checking the irrotionality. For irrotation the curl F must be 0. \begin{align*} \nabla \times F &=\left[ \dfrac {\partial }{\partial x}\overline {i}+\dfrac {\partial }{\partial y}\overline {j}+\dfrac {\partial }{\partial z}\overline {k}\right] \times \left( x\overline {i}-y\overline {j}\right)\\ &= \begin{vmatrix} i & j & k \\ \dfrac{\partial}{\partial x} &\dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z}\\ x & -y & 0 \end{vmatrix}\\ &= 0 \end{align*} So, vector field is irrotational. We can say that the vector field is divergence free and irrotational.