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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)

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)

Next: Conic sections Up: Useful mathematics Previous: Precession
Richard Fitzpatrick 2016-03-31