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Curvilinear coordinates
In the cylindrical coordinate system, the standard 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.2.) A general vector
is thus written

(A.97) 
where
and
. (See Figure A.2.) The unit vectors
,
, and
are mutually orthogonal.
Hence,
, and so on. The
volume element in this coordinate system is
.
Moreover, the gradient of a general scalar field
takes the form

(A.98) 
Figure A.2:
Cylindrical coordinates.

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.3.)
Note that
and
in the spherical system are not the same as their counterparts in the cylindrical system.
A general vector
is written

(A.99) 
where
,
, and
. The unit
vectors
,
, and
are mutually
orthogonal. Hence,
, and so on.
The
volume element in this coordinate system is
.
Moreover, the gradient of a general scalar field
takes the form

(A.100) 
Figure A.3:
Spherical coordinates.

Next: Conic sections
Up: Useful mathematics
Previous: Precession
Richard Fitzpatrick
20160331