Line element in plan polar co-ordinate system:

Let us consider two points \( P \) and \( Q \) in a plane polar co-ordinate system with respect to the origin O as shown in Fig. 1.

The polar co-ordinate of \( P \) is \( (r,\theta) \)
and the polar co-ordinate of \( Q \) is \( (r+dr,\theta+d\theta) \).

Now \( \vec{OP}=\vec{r} \) and \( \vec{OQ}=\vec{r}+\vec{dr} \).

Therefore,  \( \vec{PQ}=\vec{OQ}-\vec{OP} \)
or, \( \vec{PQ}=(\vec{r}+\vec{dr})-\vec{r}=\vec{dr} \)

Here the vector \( \vec{PQ} \) is a line element.

If \( \hat{r} \) be the unit vector along the direction of the vector \( \vec{r} \).

Then we can write, \( \vec{r}= |\vec{r}|\hat{r}=r\hat{r} \), where  \( |\vec{r}|=r \) is the magnitude of the vector \( \vec{r} \).

So, \( \displaystyle{\frac{d\vec{r}}{dt}=\frac{d}{dt}(r\hat{r})} \)

or, \( \displaystyle{\frac{d\vec{r}}{dt}=\frac{dr}{dt}\hat{r}+r\frac{d\hat{r}}{dt}} \)

\(or,\ \displaystyle{\frac{d\vec{r}}{dt}=\frac{dr}{dt}\hat{r}+r\frac{d\hat{r}}{d\theta}\frac{d\theta}{dt}\tag{i}} \)

If \( \hat{\theta} \) be the unit vector along the direction of increasing \( {\theta} \)

Then the unit vectors \( \hat{r} \) and \( \hat{\theta} \) of plane polar co-ordinate system can be expressed in terms of unit vectors \( \hat{i} \) and \( \hat{j} \) of Cartesian co-ordinate system,

\( \hat{r}=\cos\theta\ \hat{i}+\sin\theta\ \hat{j} \)

and, \( \hat{\theta}=-\sin\theta\ \hat{i}+\cos\theta\ \hat{j} \)

[ To know the derivation of the above equations (CLICK HERE) ]

Now, \( \frac{d\hat{r}}{d\theta}=-\sin\theta\ \hat{i}+\cos\theta\ \hat{j}=\hat{\theta} \)

Putting the value of \( \frac{d\hat{r}}{d\theta} \) in equation (i) we get,

\( \frac{\vec{dr}}{dt}=\frac{dr}{dt}\hat{r}+r\ \hat{\theta}\ \frac{d\theta}{dt} \)

Therefore, the line element \( \vec{dr}=dr\ \hat{r}+r\ d\theta\ \hat{\theta} \).