A velocity-space generalization of this lemma is used to take moments of the Boltzmann equation, bec
A velocity-space generalization of this lemma is used to take moments of the Boltzmann equation, because it trivially shows that the velocity-space integral of the distribution function under the namesake operation in velocity space vanishes. A special case of the generalized Stokes theorem for a Hodge-starred exterior derivative, it can be used to reduce by one the dimensionality of the set over which a scalar field is integrated as long as that scalar field can be expressed as a vector field under the namesake operation. FTP, name this fundamental theorem from vector calculus; a multi-dimensional analogue of integration by parts sometimes called Guass' theorem and named for the operation symbolized by del-dot.
A velocity-space generalization of this lemma is used to take moments of the Boltzmann equation, because it trivially shows that the velocity-space integral of the distribution function under the namesake operation in velocity space vanishes.
divergence theorem (accept Gauss' theorem before it's mentioned; accept Ostrogradsky's theorem or Ostrogradsky-Gauss theorem or Gauss-Ostrogradsky theorem)