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.
