Cylindrical

Cylindrical coordinates are often given as $P = (s,\phi,z)$ so that $\phi$ is azimuthal in the same sense as spherical polar, and so that $s$ is differentiated from $r$ . However, many other similar conventions are used. For example, $P = (r,\theta,z)$ or $P = (r,\phi,z)$ or $P = (\rho,\theta,z)$ are not uncommon. We will use $(s,\phi,z)$ in this review to avoid as much confusion as possible with spherical polar coordinates.

• Vectors:
  $$\vr = r\ \hr + z \hz$$

  But Note Well: $\hr$ is now a function of $(\theta)$ ! Similarly:

  $$\vA = A_r\ \hr + A_z \hz$$

  with $\hr$ a function of the angle $A_\theta = \theta$ that defines the direction of $\hr$ . Specifically:

• Unit vectors (relative to Cartesian $\hx,\hy,\hz$ :
  $$\hs = \cos\phi\ \hx + \sin\phi\ \hy$$
  $$\hphi = -\sin\phi \hx + \cos\phi \hy$$
  $$\hz = \hz$$

• Direct Length
  $$d\vell = ds\ \hs + s d\phi\ \hphi + dz\ \hz$$

• Directed Area
  $$dA = s d\phi dz$$
  $$dA\ \hn = d\va = s\ d\phi\ dz\ \hs = dA\ \hs$$
  And again, there are many other possible $d\va$ 's, for example:
  $$dA\ \hn = d\va = s\ d\phi\ ds\ \hz$$
  for an end cap of a cylindrical volume.

• Volume Element
  $$dV = d\tau = s\ d\phi\ ds\ dz = d\va \cdot dz\hz$$
  for the second of these area elements.
• Gradient:
  $$\grad f = \partialdiv{f}{s} \hs + \frac{1}{s} \partialdiv{f}{\phi} \hphi + \partialdiv{f}{z} \hz$$

• Divergence; The divergence is constructed by the same argument that proves the divergence theorem in a general curvilinear coordinate system given above.
  $$\deldot \vB = \frac{1}{s}\partialdiv{\left(s B_s\right)}{s} + \frac{1}{s} \partialdiv{\left(B_\phi\right)}{\phi} + \partialdiv{B_z}{z}$$

• Curl The curl is evaluated in exactly the same way from the expression above, but it ends up being much more complex:
  $$\curl{B} = \left[\frac{1}{s} \partialdiv{B_z}{\phi} - \partialdiv... ...rac{1}{s} \left[ \partialdiv{(sB_\phi)}{s} - \partialdiv{B_s}{\phi}\right] \hz$$

• Laplacian The Laplacian follows by applying the divergence rule to the gradient rule and simplifying:
  $$\lapl f = \frac{1}{s} \partialdiv{}{s}\left( s\partialdiv{f}{s}\r... ...ac{1}{s^2}\left(\ppartialdiv{f}{\phi}\right) + \left(\ppartialdiv{f}{z}\right)$$