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Normalization of the probability mass function and the geometric series giveThe tail is another geometric series:

Expanding the cross product givesTaking its divergence component by component,

Complete the square in the velocity variable:Since $m>0$, the final term is nonpositive and vanishes at exactly one velocity. ThereforeThis is the Legendre transform in mechanics that constructs the Hamiltonian mechanics energy from the kinetic-minus-potential Lagrangian.

For $n=0$, the change of variables $s=\sqrt{a}x$ reduces the integral to the given Gaussian integral:For $n=1$, the integrand is an integrable odd function, so its contributions from the two half-lines cancel:The condition $a>0$ ensures convergence in both cases.

For this first-order linear differential equation, the integrating factor is $e^x$. Thusand the general solution is

The boundary order fixes the orientation. Two edges issuing from $A$ are $\overrightarrow{AB}=(0,0,1)$ and $\overrightarrow{AD}=(1,1,0)$. Their cross product gives the oriented vector area:Its norm is $\sqrt{2}$, the ordinary rectangle area. The sign agrees with traversing $A\to B\to C\to D\to A$ by the right-hand orientation rule.