Curl

  • Calculate curl.

  • Understand physical interpretation and applications of curl.

\[\nabla \times \mathbf{V} = \left ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k }\right ) \times (v_{1} \mathbf{i}+ v_{2} \mathbf{j}+ v_{3}\mathbf{k})\]

Let us take same example of a variable vector.
e.g. if \(\bar{a} = x^3z\overline{i} + xy^2\overline{j} + yz\overline{k}\) \[\begin{split} \nabla \times \textbf{V} & = \begin{vmatrix} \overline{i} & \overline{j} & \overline{k}\\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ x^3z & xy^2 & yz \end{vmatrix}\\ \nabla \times \mathbf{V} & =((z-0) \overline{i} - (0-x^3) \overline{j} + (y^2-0) \overline{k}) \\ & = (z \overline{i} + x^3 \overline{j} + y^2 \overline{k}) \end{split}\]

If you want to calculate at some location, let’s say, at (1,2-1) then simply substitute the coordinates, \[\begin{aligned} \nabla \times \mathbf{V}\ \mathrm{at}\ (1,2,-1) &=(z \overline{i} + x^3 \overline{j} + y^2 \overline{k}) \\ &= (-1)( \overline{i} + (1)^3 \overline{j} + (2)^2 \overline{k}) \\ &= -\overline{i} + \overline{j} + 4\overline{k} \end{aligned}\]

Physical Interpretation of the Curl:

  • Consider a vector field \(\mathbf{F}\) that represents a fluid velocity, then the curl of \(\mathbf{F}\) at a point in a fluid is a measure of the rotation of the fluid.

  • If there is no rotation of fluid anywhere then \(\nabla \times F = 0\) Such a vector field is said to be irrotational or conservative.

  • For a 2D flow with \(\overline{F}\) represents the fluid velocity, \(\nabla \times \overline{F}\) is perpendicular to the motion and represents the direction of axis of rotation.

Solved Examples

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.

Correct Answer: C

Solved Example:

4-11-02

A vector is said to be irrotational when its:

Solution:
When the curl of a vector is zero, it is said to be irrotational.

Correct Answer: C