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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. Because 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 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,
![$\displaystyle {\bf e}_i\cdot {\bf e}_j = \delta_{ij}.$](img7037.png) |
(C.1) |
In this case,
,
,
are said to constitute an orthogonal coordinate system.
Suppose, further, that
![$\displaystyle {\bf e}_1\cdot {\bf e}_2\times {\bf e}_3 = 1$](img7038.png) |
(C.2) |
at all points in space, so that
,
,
also constitute a right-handed
coordinate system. It follows that
![$\displaystyle {\bf e}_i\cdot{\bf e}_j\times {\bf e}_k = \epsilon_{ijk}.$](img7039.png) |
(C.3) |
Finally, a general vector
, associated with a particular point in space, can be written
![$\displaystyle {\bf A} = \sum_{i=1,3} A_i\,{\bf e}_i,$](img7040.png) |
(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
![$\displaystyle d {\bf x} = \frac{du_1}{\vert\nabla u_1\vert}\,{\bf e}_1 + \frac{...
...\frac{du_3}{\vert\nabla u_3\vert}\,{\bf e}_3=\sum_{i=1,3} h_i\,du_i\,{\bf e}_i.$](img7045.png) |
(C.5) |
Hence, from (C.1), an element of length (squared) in the
,
,
coordinate system is written
![$\displaystyle d{\bf x}\cdot d{\bf x}= \sum_{i=1,3} h_i^{\,2}\,du_i^{\,2}.$](img7046.png) |
(C.6) |
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
![$\displaystyle h = h_1\,h_2\,h_3.$](img7055.png) |
(C.7) |
It can be seen that [see Equation (A.176)]
![$\displaystyle \nabla\times \nabla u_i = 0,$](img7056.png) |
(C.8) |
and
![$\displaystyle \nabla\cdot \left(\frac{h_i^{\,2}}{h}\,\nabla u_i\right) = 0.$](img7057.png) |
(C.9) |
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
![$\displaystyle \nabla \phi = \sum_{i=1,3} \frac{\partial \phi}{\partial u_i}\,\nabla u_i=\sum_{i=1,3}\frac{1}{h_i}\,\frac{\partial \phi}{\partial u_i}\,{\bf e}_i.$](img7065.png) |
(C.12) |
Hence, the components of
in the
,
,
coordinate system are
![$\displaystyle (\nabla\phi)_i = \frac{1}{h_i}\,\frac{\partial \phi}{\partial u_i}.$](img7067.png) |
(C.13) |
Consider a vector field
. We can write
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
![$\displaystyle \nabla\cdot{\bf A} =\sum_{i=1,3} \frac{1}{h}\,\frac{\partial}{\partial u_i}\!\left(\frac{h}{h_i}\,A_i\right).$](img7072.png) |
(C.15) |
We can write
where use has been made of Equations (A.178), (C.8), and (C.12).
It follows from Equation (C.11) that
![$\displaystyle (\nabla\times {\bf A})_i = {\bf e}_i\cdot\nabla\times {\bf A} = \...
...,k=1,3}\epsilon_{ijk}\,\frac{h_i}{h}\,\frac{\partial (h_k\,A_k)}{\partial u_j}.$](img7076.png) |
(C.17) |
Hence, the components of
in the
,
,
coordinate system are
![$\displaystyle (\nabla\times{\bf A})_i = \sum_{j,k=1,3}\epsilon_{ijk}\,\frac{h_i}{h}\,\frac{\partial (h_k\,A_k)}{\partial u_j}.$](img7077.png) |
(C.18) |
Now,
[see Equation (A.172)], so Equations (C.12) and (C.15)
yield the following expression for
in the
,
,
coordinate system:
![$\displaystyle \nabla^2\phi =\sum_{i=1,3}\frac{1}{h}\,\frac{\partial}{\partial u_i}\!\left(\frac{h}{h_i^{\,2}}\,\frac{\partial\phi}{\partial u_i}\right).$](img7079.png) |
(C.19) |
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:
![$\displaystyle [({\bf A}\cdot\nabla)\,{\bf B}]_i= \sum_{j=1,3}\left(\frac{A_j}{h...
...ial u_i} + \frac{A_i\,B_j}{h_i\,h_j}\,\frac{\partial h_i}{\partial u_j}\right).$](img7088.png) |
(C.23) |
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
![$\displaystyle (\nabla {\bf A})_{ij} = \frac{\partial A_i}{\partial x_j}.$](img7092.png) |
(C.24) |
In an orthogonal curvilinear coordinate system, the previous
expression generalizes to
![$\displaystyle (\nabla{\bf A})_{ij} = [({\bf e}_j\cdot\nabla)\,{\bf A}]_i.$](img7093.png) |
(C.25) |
It thus follows from Equation (C.23), and the relation
, that
![$\displaystyle (\nabla{\bf A})_{ij} = \frac{1}{h_j}\,\frac{\partial A_i}{\partia...
...\delta_{ij}\sum_{k=1,3}\frac{A_k}{h_i\,h_k}\,\frac{\partial h_i}{\partial u_k}.$](img7095.png) |
(C.26) |
The vector identity (A.177) yields the
following expression for
that is valid in a general coordinate system:
![$\displaystyle \nabla^{\,2}{\bf A} = \nabla(\nabla\cdot{\bf A}) - \nabla\times(\nabla\times {\bf A}).$](img7097.png) |
(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
).
Next: Cylindrical Coordinates
Up: Non-Cartesian Coordinates
Previous: Introduction
Richard Fitzpatrick
2016-03-31