Cylindrical Coordinates

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

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