3-D Cylindrical Polar Coordinates $(R, \phi , z)$

When the coordinates $R, \phi, z$ are increased by $\delta R, \delta \phi, \delta z$ the point $P (R, \phi, z)$ moves by the amount

$${\bf\delta r} = \delta R{\bf e}_{R} + R\delta \phi {\bf e}_{\phi} + \delta z {\bf e}_{z},$$

and, as indicated in Figure 1.16, the infinitessimal volume element is

$$dV = R dRd\phi dz.$$ (1.35)

Figure 1.16: Illustrating how the volume element is related to the increases in the coordinates by $\delta R, \delta \phi$ and $\delta z$. Notice that the lengths of the sides are $\delta R$, $R\delta \theta$ and $\delta z$.

The scale factors are 1 (in the $R$ direction) $R$ (in the $\phi$ direction) and 1 (in the $z$ direction).

So the gradient operator is

$$\nabla f = {\partial f\over \partial R}{\bf e}_{R} + {1\ove... ...hi}{\bf e}_{\phi} + {\partial f\over \partial z}{\bf e}_{z}.$$ (1.36)

The other important expressions in cylindrical coordinates are

$$\nabla \cdot {\bf A} = {1\over R}{\partial \over \partial R... ...phi}\over \partial \phi} + {\partial A_{z}\over \partial z},$$

$$\nabla^{2}f = \nabla \cdot \nabla f = {1\over R}{\partial \... ...ver \partial \phi^{2}} + {\partial^{2}f\over \partial z^{2}}$$

and

$$\nabla \times {\bf A} = \left ({1\over R}{\partial A_{z}\ov... ...er R}{\partial A_{R}\over \partial \phi}\right ){\bf e}_{z}.$$