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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

\begin{displaymath}
{\bf\delta r} = \delta R{\bf e}_{R} + R\delta \phi {\bf e}_{\phi} + \delta z {\bf e}_{z},
\end{displaymath}

and, as indicated in Figure 1.16, the infinitessimal volume element is
\begin{displaymath}
dV = R dRd\phi dz.
\end{displaymath} (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 $\delta z$.
\includegraphics {fundfig18.eps}

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

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

The other important expressions in cylindrical coordinates are

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


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

and

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


next up previous
Next: General Orthogonal Coordinates Up: Non-Cartesian Coordinates Previous: 2D Cylindrical Polars
Prof. Alan Hood
2000-11-06