Let
,
, and
. It follows that
,
, and
are
a set of unit basis vectors which 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: *i.e.*,

at all points in space, so that , , also constitute a

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

(1598) |

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

(1599) |

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

(1601) |

Note that [see Equation (1481)]

The latter result follows from Equations (1480) and (1481) because ,

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

Consider a vector field
. We can write

(1608) |

where use has been made of Equations (1479), (1603), and (1604). Thus, the divergence of in the , , coordinate system takes the form

We can write

(1610) |

where use has been made of Equations (1483), (1602), and (1606). It follows from (1605) that

(1611) |

Now,
[see (1477)], so Equations (1606) and (1609)
yield the following expression for in the , , coordinate system:

The vector identities (1476) and (1484) can be combined to give the
following expression for
that is valid in a general coordinate system:

Making use of Equations (1607), (1609), and (1612), as well as the easily demonstrated results

and the tensor identity (1500), Equation (1614) 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

(1618) |

(1619) |

The vector identity (1482) yields the
following expression for
that is valid in a general coordinate system:

(1621) |

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