Three-Dimensional Coordinate Systems

There are three common coordinate systems in three dimensions used in multivariate calculus.
Rectangular coordinates are the natural extension of the familiar used in two dimensions. The point is at a distance from the -plane, from the -plane, and from the - plane.
Cylindrical coordinates extend the polar coordinate system in two dimensions. The coordinates are the polar coordinates of the projection of the point in the - plane, so is the distance from the origin to the projection of the point in the - plane, is the angle of rotation around the axis from the positive axis, and is the distance from the - plane.
Spherical coordinates have no counterpart in two dimensions. A point in spherical coordinates is at the distance from the origin, is the angle between the positive axis and the line from the origin to the point, and is the same as in cylindrical coordinates, the rotation about the axis from the positive axis.


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