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The attached image shows the text I am following. I get that the 3-D pdf

Why must

*F*can only depend on the speed*v*. I also understand that if*f_x , f_y, f_z*are the pdfs of the individual components of velocity, then rotational invariance requires them to all be the same function*f_x = f_y = f_z = ϕ.*What seems to be missing is that F must be separable, that is F must satisfy*F = f_x f_y f_z*Why must

*F*be separable?