Let , , and . It follows that , , and are a set of unit basis vectors that are normal to surfaces of constant , , and , respectively, at all points in space. Note, however, that the direction of these basis vectors is generally a function of position. Suppose that the , where runs from 1 to 3, are mutually orthogonal at all points in space: that is,

In this case, , , are said to constitute an orthogonal coordinate system. Suppose, further, that

at all points in space, so that , , also constitute a right-handed coordinate system. It follows that

Finally, a general vector , associated with a particular point in space, can be written

(C.4) |

where the are the local basis vectors of the , , system, and is termed the th component of in this system.

Consider two neighboring points in space whose coordinates in the , , system are , , and , , . It is easily shown that the vector directed from the first to the second of these points takes the form

(C.5) |

Hence, from (C.1), an element of length (squared) in the , , coordinate system is written

Here, the , which are generally functions of position, are known as the

(C.7) |

It can be seen that [see Equation (A.176)]

and

The latter result follows from Equations (A.175) and (A.176) because , et cetera. Finally, it is easily demonstrated from Equations (C.1) and (C.3) that

Consider a scalar field . It follows from the chain rule, and the relation , that

Hence, the components of in the , , coordinate system are

Consider a vector field . We can write

(C.14) |

where use has been made of Equations (A.174), (C.9), and (C.10). Thus, the divergence of in the , , coordinate system takes the form

We can write

(C.16) |

where use has been made of Equations (A.178), (C.8), and (C.12). It follows from Equation (C.11) that

(C.17) |

Hence, the components of in the , , coordinate system are

Now, [see Equation (A.172)], so Equations (C.12) and (C.15) yield the following expression for in the , , coordinate system:

The vector identities (A.171) and (A.179) can be combined to give the following expression for that is valid in a general coordinate system:

Making use of Equations (C.13), (C.15), and (C.18), as well as the easily demonstrated results

and the tensor identity (B.16), Equation (C.20) reduces (after a great deal of tedious algebra) to the following expression for the components of in the , , coordinate system:

Note, incidentally, that the commonly quoted result is only valid in Cartesian coordinate systems (for which ).

Let us define the gradient of a vector field as the tensor whose components in a Cartesian coordinate system take the form

(C.24) |

In an orthogonal curvilinear coordinate system, the previous expression generalizes to

(C.25) |

It thus follows from Equation (C.23), and the relation , that

The vector identity (A.177) yields the following expression for that is valid in a general coordinate system:

(C.27) |

Making use of Equations (C.15), (C.18), and (C.19), as well as (C.21) and (C.22), and the tensor identity (B.16), the previous equation reduces (after a great deal of tedious algebra) to the following expression for the components of in the , , coordinate system:

Note, again, that the commonly quoted result is only valid in Cartesian coordinate systems (for which ).