2D Cylindrical Polars $(r, \theta )$

To study a new coordinate system we need to obtain the relevent scale factors. To this we, as above, analyse how the point $(r, \theta )$ moves when

$$r \qquad \rightarrow \qquad r + \delta r,$$

and

$$\theta \qquad \rightarrow \qquad \theta + \delta \theta.$$

Thus, from Figure 1.15, we see that ${\bf r}$ moves to ${\bf r} + {\bf\delta r}$, where

$${\bf\delta r} = \delta r {\bf e}_{r} + r\delta \theta {\bf e}_{\theta}.$$ (1.30)

Figure 1.15: Increasing $r$ and $\theta$ by the respetive amounts $\delta r$ and $\delta \theta$ gives the change in position, ${\bf \delta r}$. Note that the distance round the circle is $r \delta \theta$.

This time the scale factors are (the coefficients of $\delta r$ and $\delta \theta$) $1$ and $r$. The infinitesimal change in area is

$$dS = r dr d\theta.$$ (1.31)

The gradient operator is now

$$\nabla f = {\partial f\over \partial r}{\bf e}_{r} + {1\over r}{\partial f \over \partial \theta}{\bf e}_{\theta},$$ (1.32)

while

$$\nabla \cdot {\bf A} = {1\over r}{\partial \over \partial r... ...ht ) + {1\over r}{\partial A_{\theta}\over \partial \theta}.$$ (1.33)

Note that the dimensions must be correct. $\theta$ has no dimensions. Finally, for the 2D case considered here,

$$\nabla \times {\bf A} = \left ({1\over r}{\partial \over \p... ...r}{\partial A_{r}\over \partial \theta}\right ) {\bf e}_{z}.$$ (1.34)