Velocity space spherical polar coordinates

Figure 9 Spherical polar coordinates in velocity space.

The speed is the magnitude of the velocity. All velocities that have the same magnitude but different directions correspond to the same speed. In order to obtain a formula for the probability distribution of speeds, we must add up the probabilities of all velocities corresponding to the same speed. To do this conveniently, we change to spherical polar coordinates in velocity space. The first coordinate is the speed v. The second coordinate is 6, the angle between the positive axis and the velocity vector. The third coordinate... 

Figure 9.10 Spherical Polar Coordinates in Velocity Space, with a Volume Element.

As shown in Fig. 4.1, we use a system of spherical polar coordinates in the velocity space with Cx, Cy, c -axes. The shaded spherical shell in Fig. 4.1 has a radius c and thickness dc. With spherical symmetry, the increment of volume in velocity space is ... 

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