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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 axissee
Figure A.113. A general vector is thus written

(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

(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 axissee 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

(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

(1375) 
Figure A.114:
Spherical polar coordinates.

Next: Exercises
Up: Vector Algebra and Vector
Previous: Grad Operator
Richard Fitzpatrick
20110331