12.7 Cylindrical and Spherical Coordinates

12.7 Cylindrical and Spherical Coordinates - Section 13.7...

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Section 13.7 Cylindrical and Spherical Coordinates “Non-Rectangular Coordinate Systems in 3-space” In Calculus II, we considered the polar coordinate system to help inte- grate functions whose graphs were circular regions. In this section, we consider two new coordinate systems for graphs in 3-space. 1. Cylindrical Coordinates The Frst coordinate system we consider is a generalization of polar coordinates - the basic idea is to take the polar coordinates in the xy - plane and then simply add the z -coordinate to determine the height of a point. They are particularly useful when describing cylinders. ±ormally, we deFne the cylindrical coordinate system as follows. Defnition 1.1. The cylindrical coordinates of a point P in 3-space is deFned to be ( r, ϑ, z ) where ( r, ϑ ) are the polar coordinates of the projection of P in the xy -plane and z is the z -coordinate of the plane where r g 0 and 0 l ϑ < 2 π . z (r,theta) Since cylindrical coordinates are so closely related to polar coordinates, it is easy to convert from rectangular coordinates in 3-space into cylin- drical and vice versa. Result 1.2. ( i ) The rectangular coordinates of the point ( r, ϑ, z ) in 3 space are x = r cos ( ϑ ), y = r sin ( ϑ ) and z = z . ( ii
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12.7 Cylindrical and Spherical Coordinates - Section 13.7...

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