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First, please answer this simple question: How to write a constant (in all space) vector in spherical coordinates?

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"What is the probability that a particle's speed lies between $u$ and $u+du$.?" Short answer The probability that a particle's velocity has an $x$-component between $u_x$ and $u_x+du_x$, and similarly for $y$ and $z$, is proportional to the volume taken up by that small cubic chunk of velocity-space, $du_xdu_ydu_z.$ Given these three components of the ...

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The intuition is that $4\pi u^2$ is the area of a sphere of radius $u$, and now to find the volume of the thin shell between radius $u$ and radius $u+du$, you multiply the area of the surface of the shell by the thickness of the shell and find that its volume is $4\pi u^2du$. We can make this intuition more precise in a number of ways. Here are two: ...

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