Uses of Rectangular, Cylindrical, and Spherical Coordinates    (continued)

 

Spherical Coordinates of a point ,P, in space are:   ( ρ, θ, φ )

 

Let    ρ  =  magnitude of vector from the origin, O,  out to the point P

 

        θ  =  angle between the  x-axis and the line formed by the projection of  ρ 

            on to the x-y plane  NOTE: This projection is the same as  r  in cylindrical

            coordinates so  θ  has the same meaning for both spherical and cylindrical

            coordinates

 

        φ  =  angle between the z-axis and   ρ 

            

 

So the rectangular coordinates (x, y, z) of  P in spherical coordinates are:

               x =  ρ sin φ cos θ ,  y =  ρ sin φ sin θ ,  z = ρ cos φ

 

 

 

An example that illustrates uses of rectangular, cylindrical and spherical

coordinates systems in evaluating an integral is calculating the volume of a sphere.

 

Note which of these options simplifies the determination of the limits of integration

and/or evaluation of the integral.

 

 

 

Click here for this example.

 



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