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# Orthogonal Curvilinear Coordinates

Let , , be a set of standard right-handed Cartesian coordinates. Furthermore, let , , be three independent functions of these coordinates which are such that each unique triplet of , , values is associated with a unique triplet of , , values. It follows that , , can be used as an alternative set of coordinates to distinguish different points in space. Since the surfaces of constant , , and are not generally parallel planes, but rather curved surfaces, this type of coordinate system is termed curvilinear.

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.,

 (1595)

In this case, , , are said to constitute an orthogonal coordinate system. Suppose, further, that
 (1596)

at all points in space, so that , , also constitute a right-handed coordinate system. It follows that
 (1597)

Finally, a general vector , associated with a particular point in space, can be written
 (1598)

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

 (1599)

Hence, from (1595), an element of length (squared) in the , , coordinate system is written
 (1600)

Here, the , which are generally functions of position, are known as the scale factors of the system. Elements of area that are normal to , , and , at a given point in space, take the form , , and , respectively. Finally, an element of volume, at a given point in space, is written , where
 (1601)

Note that [see Equation (1481)]

 (1602)

and
 (1603)

The latter result follows from Equations (1480) and (1481) because , etc. Finally, it is easily demonstrated from (1595) and (1597) that
 (1604) (1605)

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

 (1606)

Hence, the components of in the , , coordinate system are
 (1607)

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
 (1609)

We can write

 (1610)

where use has been made of Equations (1483), (1602), and (1606). It follows from (1605) that
 (1611)

Hence, the components of in the , , coordinate system are
 (1612)

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

 (1613)

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

 (1614)

Making use of Equations (1607), (1609), and (1612), as well as the easily demonstrated results
 (1615) (1616)

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:
 (1617)

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)

In an orthogonal curvilinear coordinate system, the above expression generalizes to
 (1619)

It thus follows from (1617), and the relation , that
 (1620)

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

 (1621)

Making use of Equations (1609), (1612), and (1613), as well as (1615) and (1616), and the tensor identity (1500), the above equation reduces (after a great deal of tedious algebra) to the following expression for the components of in the , , coordinate system:
 (1622)

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

Next: Cylindrical Coordinates Up: Non-Cartesian Coordinates Previous: Introduction
Richard Fitzpatrick 2012-04-27