Area element in plane polar co-ordinate system:

Let us consider a point \( P \) having plane polar co-ordinates \( (r,\theta) \) with respect to the origin \( O \) as shown in the figure 1.

Now the vector \( \vec{OP} \) is increased to \( \vec{OP'} \) through a small increment \( dr=PP' \), in the direction of unit vector \( \hat{r} \). here the angular increment is \( d\theta \) over the angle \( \theta \) in the directon of \( \hat{\theta} \).

As a result, \( PP' \) traces out an area element \( PP'Q'Q \).

Here, \( \vec{PP'}=dr\ \hat{r} \) and \( \vec{PQ}=r\ d\theta\ \hat{\theta} \)

Therefore the area \( \vec{dA}=\vec{PP'}\times\vec{PQ}=dr\hat{r}\times{r}\ d\theta\ \hat{\theta} \)

or, \( \vec{dA}=r\ dr\ d\theta[\hat{r}\times\hat{\theta}] \)

or, \( \vec{dA}=r\ dr\ d\theta\ \hat{n} \)

where, \( \hat{n} \) is the unit vector normal to the plane containing the unit vectors \( \hat{r} \) and \( \hat{\theta} \), this means that normal to the plane containing the area.