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# Cylindrical Coordinates

In the cylindrical coordinate system, , , and , where , , and , , are standard Cartesian coordinates. Thus, is the perpendicular distance from the -axis, and 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 - plane and the -axis. (See Figure C.1.)

A general vector is written

 (C.29)

where , , and . Of course, the unit basis vectors , , and are mutually orthogonal, so , et cetera.

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

 (C.30)

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

 (C.31) (C.32) (C.33)

Thus, surface elements normal to , , and are written

 (C.34) (C.35) (C.36)

respectively, whereas a volume element takes the form

 (C.37)

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

 (C.38) (C.39) (C.40)

respectively. Here, is a general scalar field, and a general vector field.

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

 (C.41)

Moreover, from Equation (C.23), the components of in the cylindrical coordinate system are

 (C.42) (C.43) (C.44)

Let us define the symmetric gradient tensor

 (C.45)

Here, the superscript denotes a transpose. Thus, if the element of some second-order tensor is then the corresponding element of is . According to Equation (C.26), the components of in the cylindrical coordinate system are

 (C.46) (C.47) (C.48) (C.49) (C.50) (C.51)

Finally, from Equation (C.28), the components of in the cylindrical coordinate system are

 (C.52) (C.53) (C.54)

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