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Curvilinear Coordinates
In the cylindrical coordinate system, the Cartesian coordinates
and
are replaced by
and
.
Here,
is the perpendicular distance from the
-axis, and
the angle subtended between the perpendicular radius vector and the
-axis--see
Figure A.113. A general vector
is thus written
![\begin{displaymath}
{\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_z\,{\bf e}_z,
\end{displaymath}](img3488.png) |
(1372) |
where
and
--see Figure A.113. Note that the unit vectors
,
, and
are mutually orthogonal.
Hence,
, etc. The
volume element in this coordinate system is
.
Moreover, the gradient of a general scalar field
takes the form
![\begin{displaymath}
\nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r
+ \frac{...
...a}\,{\bf e}_\theta
+ \frac{\partial V}{\partial z}\,{\bf e}_z.
\end{displaymath}](img3494.png) |
(1373) |
In the spherical coordinate system, the Cartesian coordinates
,
, and
are replaced by
,
,
and
. Here,
is the radial distance from the origin,
the angle subtended between the radius vector and the
-axis,
and
the angle subtended between the projection of the radius vector
onto the
-
plane and the
-axis--see Figure A.114.
Note that
and
in the spherical system are not the same as their counterparts in the cylindrical system.
A general vector
is written
![\begin{displaymath}
{\bf A} = A_r\,{\bf e}_r + A_\theta\,{\bf e}_\theta+ A_\phi\,{\bf e}_\phi,
\end{displaymath}](img3498.png) |
(1374) |
where
,
, and
. The unit
vectors
,
, and
are mutually
orthogonal. Hence,
, etc.
The
volume element in this coordinate system is
.
Moreover, the gradient of a general scalar field
takes the form
![\begin{displaymath}
\nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r
+ \frac{...
...\,\sin\theta}\,\frac{\partial V}{\partial \phi}\,{\bf e}_\phi.
\end{displaymath}](img3502.png) |
(1375) |
Figure A.114:
Spherical polar coordinates.
![\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixA/figA.26.eps}}
\end{figure}](img3503.png) |
Next: Exercises
Up: Vector Algebra and Vector
Previous: Grad Operator
Richard Fitzpatrick
2011-03-31