Rotational and irrotational vector fields

If you mean a vector field $\vec F$, then rotational and irrotational are determined using the curl.

1. Rotational Vector Field

A vector field is called rotational if

$$\boxed{\nabla\times\vec F\neq0}$$

That means the field has some local tendency to rotate or spin.

For

$$\vec F=P\hat i+Q\hat j+R\hat k$$

the curl is

$$\nabla\times\vec F= \begin{vmatrix} \hat i&\hat j&\hat k\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ P&Q&R \end{vmatrix}$$

2. Irrotational Vector Field

A vector field is irrotational if

$$\boxed{\nabla\times\vec F=0}$$

So the main thing to remember is:

$$\boxed{\text{Curl}=0\Rightarrow\text{Irrotational}}$$

$$\boxed{\text{Curl}\neq0\Rightarrow\text{Rotational}}$$

Simple Example

Take

$$\vec F=-y\hat i+x\hat j$$

So

$$P=-y,\qquad Q=x,\qquad R=0$$

The curl is

$$\nabla\times\vec F=(0,0,\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})$$

$$=(0,0,1-(-1))=(0,0,2)$$

Since

$$\nabla\times\vec F\neq0$$

the field is rotational.

Exam shortcut: Given a vector field → calculate curl → check whether it is zero or not.

How to Calculate the Curl of a Vector Field

To calculate the curl of a vector field, use this formula.

Suppose

$$\vec F=P(x,y,z)\hat i+Q(x,y,z)\hat j+R(x,y,z)\hat k$$

Then:

$$\boxed{\nabla\times\vec F= \begin{vmatrix} \hat i&\hat j&\hat k\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ P&Q&R \end{vmatrix}}$$

Expanding it:

$$\boxed{\nabla\times\vec F= \left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\hat i -\left(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z}\right)\hat j +\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\hat k}$$

Example

Let

$$\vec F=(y,x,0)$$

So:

$$P=y,\qquad Q=x,\qquad R=0$$

Calculate each part:

$$\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}=0-0=0$$

$$\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z}=0-0=0$$

$$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=1-1=0$$

Therefore:

$$\boxed{\nabla\times\vec F=(0,0,0)}$$

So this field is irrotational.

Quick Formula to Memorize

For

$$\vec F=(P,Q,R)$$

just remember:

$$\boxed{\nabla\times\vec F=(R_y-Q_z,\;P_z-R_x,\;Q_x-P_y)}$$

where, for example,

$R_y=\frac{\partial R}{\partial y}$