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Cylindrical Coordinates

In the cylindrical coordinate system, $ u_1=r$ , $ u_2=\theta$ , and $ u_3=z$ , where $ r=\sqrt{x^{\,2}+y^{\,2}}$ , $ \theta=\tan^{-1}(y/x)$ , and $ x$ , $ y$ , $ z$ are standard Cartesian coordinates. Thus, $ r$ is the perpendicular distance from the $ z$ -axis, and $ \theta $ the angle subtended between the projection of the radius vector (i.e., the vector connecting the origin to a general point in space) onto the $ x$ -$ y$ plane and the $ x$ -axis. (See Figure C.1.)

Figure C.1: Cylindrical coordinates.
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A general vector $ {\bf A}$ is written

$\displaystyle {\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_z\,{\bf e}_z,$ (C.29)

where $ {\bf e}_r=\nabla r/\vert\nabla r\vert$ , $ {\bf e}_\theta = \nabla\theta/\vert\nabla\theta\vert$ , and $ {\bf e}_z=\nabla z/\vert\nabla z\vert$ . Of course, the unit basis vectors $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_z$ are mutually orthogonal, so $ A_r = {\bf A}\cdot {\bf e}_r$ , et cetera.

As is easily demonstrated, an element of length (squared) in the cylindrical coordinate system takes the form

$\displaystyle d{\bf x}\cdot d{\bf x} = dr^{\,2} + r^{\,2}\,d\theta^{\,2} + dz^{\,2}.$ (C.30)

Hence, comparison with Equation (C.6) reveals that the scale factors for this system are

$\displaystyle h_r$ $\displaystyle = 1,$ (C.31)
$\displaystyle h_\theta$ $\displaystyle = r,$ (C.32)
$\displaystyle h_z$ $\displaystyle = 1.$ (C.33)

Thus, surface elements normal to $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_z$ are written

$\displaystyle dS_r$ $\displaystyle = r\,d\theta\,dz,$ (C.34)
$\displaystyle dS_\theta$ $\displaystyle = dr\,dz,$ (C.35)
$\displaystyle dS_z$ $\displaystyle = r\,dr\,d\theta,$ (C.36)

respectively, whereas a volume element takes the form

$\displaystyle dV = r\,dr\,d\theta\,dz.$ (C.37)

According to Equations (C.13), (C.15), and (C.18), gradient, divergence, and curl in the cylindrical coordinate system are written

$\displaystyle \nabla \psi$ $\displaystyle = \frac{\partial \psi}{\partial r}\,{\bf e}_r + \frac{1}{r}\frac{...
...{\partial\theta}\,{\bf e}_\theta + \frac{\partial \psi}{\partial z}\,{\bf e}_z,$ (C.38)
$\displaystyle \nabla\cdot{\bf A}$ $\displaystyle =\frac{1}{r}\,\frac{\partial}{\partial r}\,(r\,A_r) + \frac{1}{r}\,\frac{\partial A_\theta}{\partial\theta} + \frac{\partial A_z}{\partial z},$ (C.39)
$\displaystyle \nabla\times {\bf A}$ $\displaystyle = \left(\frac{1}{r}\,\frac{\partial A_z}{\partial \theta}-\frac{\...
...\,A_\theta) - \frac{1}{r}\,\frac{\partial A_r}{\partial\theta}\right){\bf e}_z,$    
(C.40)

respectively. Here, $ \psi({\bf r})$ is a general scalar field, and $ {\bf A}({\bf r})$ a general vector field.

According to Equation (C.19), when expressed in cylindrical coordinates, the Laplacian of a scalar field becomes

$\displaystyle \nabla^{\,2} \psi = \frac{1}{r}\,\frac{\partial}{\partial r}\left...
...,2} \psi}{\partial\theta^{\,2}} + \frac{\partial^{\,2} \psi}{\partial z^{\,2}}.$ (C.41)

Moreover, from Equation (C.23), the components of $ ({\bf A}\cdot\nabla)\,{\bf A}$ in the cylindrical coordinate system are

$\displaystyle [({\bf A}\cdot\nabla)\,{\bf A}]_r$ $\displaystyle ={\bf A}\cdot\nabla A_r - \frac{A_\theta^{\,2}}{r},$ (C.42)
$\displaystyle [({\bf A}\cdot\nabla)\,{\bf A}]_\theta$ $\displaystyle = {\bf A}\cdot\nabla A_\theta + \frac{A_r\,A_\theta}{r},$ (C.43)
$\displaystyle [({\bf A}\cdot\nabla)\,{\bf A}]_z$ $\displaystyle = {\bf A}\cdot\nabla A_z.$ (C.44)

Let us define the symmetric gradient tensor

$\displaystyle \widetilde{\nabla {\bf A}} = \frac{1}{2}\left[\nabla {\bf A} + (\nabla{\bf A})^T\right].$ (C.45)

Here, the superscript $ T$ denotes a transpose. Thus, if the $ ij$ element of some second-order tensor $ {\bf S}$ is $ S_{ij}$ then the corresponding element of $ {\bf S}^T$ is $ S_{ji}$ . According to Equation (C.26), the components of $ \widetilde{\nabla {\bf A}}$ in the cylindrical coordinate system are

$\displaystyle (\widetilde{\nabla {\bf A}})_{rr}$ $\displaystyle = \frac{\partial A_r}{\partial r},$ (C.46)
$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta\theta}$ $\displaystyle =\frac{1}{r} \frac{\partial A_\theta}{\partial \theta}+ \frac{A_r}{r},$ (C.47)
$\displaystyle (\widetilde{\nabla {\bf A}})_{zz}$ $\displaystyle = \frac{\partial A_z}{\partial z},$ (C.48)
$\displaystyle (\widetilde{\nabla {\bf A}})_{r\theta}=(\widetilde{\nabla {\bf A}})_{\theta r}$ $\displaystyle = \frac{1}{2}\left(\frac{1}{r}\,\frac{\partial A_r}{\partial\theta} + \frac{\partial A_\theta}{\partial r} - \frac{A_\theta}{r}\right),$ (C.49)
$\displaystyle (\widetilde{\nabla {\bf A}})_{rz}=(\widetilde{\nabla {\bf A}})_{zr}$ $\displaystyle =\frac{1}{2}\left(\frac{\partial A_r}{\partial z} + \frac{\partial A_z}{\partial r}\right),$ (C.50)
$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta z}=(\widetilde{\nabla {\bf A}})_{z\theta}$ $\displaystyle =\frac{1}{2}\left(\frac{\partial A_\theta}{\partial z} + \frac{1}{r}\frac{\partial A_z}{\partial \theta}\right).$ (C.51)

Finally, from Equation (C.28), the components of $ \nabla^{\,2}{\bf A}$ in the cylindrical coordinate system are

$\displaystyle (\nabla^{\,2}{\bf A})_r$ $\displaystyle =\nabla^{\,2} A_r - \frac{A_r}{r^{\,2}} - \frac{2}{r^{\,2}}\,\frac{\partial A_\theta}{\partial\theta},$ (C.52)
$\displaystyle (\nabla^{\,2}{\bf A})_\theta$ $\displaystyle = \nabla^{\,2} A_\theta + \frac{2}{r^{\,2}}\,\frac{\partial A_r}{\partial \theta}-\frac{A_\theta}{r^{\,2}},$ (C.53)
$\displaystyle (\nabla^{\,2}{\bf A})_z$ $\displaystyle =\nabla^{\,2} A_z.$ (C.54)


next up previous
Next: Spherical Coordinates Up: Non-Cartesian Coordinates Previous: Orthogonal Curvilinear Coordinates
Richard Fitzpatrick 2016-01-22