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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
 (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 -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

 (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)

Next: Exercises Up: Vector Algebra and Vector Previous: Grad Operator
Richard Fitzpatrick 2011-03-31