Mathematical interlude

The mathematical interlude that follows is a justification of the formulae in Box 1. If you are interested only in using neural networks, not the background mathematics, you may want to skip this section. 

MATHEMATICAL INTERLUDE. MORE PROPERTIES OF EXACT DIFFERENTIALS. THE CYCLIC RULE... 

Before proceeding, we take time out for a rather mathematical interlude. Stereodynamics deals with vectors, and these have botii a magnitude and a direction. Ideally, we want to know the complete distribution of the possible directions of the vector as, for example, in an angular distribution after a collision, where we want to know how the final velocity is distributed with respect to the initial direction of approach. Orientation and alignment are the technical terms that are used to describe deviations from a purely random direction in space. In this section we want to express more precisely how to characterize orientation and ahgmnent. This treatment leads naturally to multipole moments, which serve as a basis in which the distribution of oriented and aligned molecules can be expanded. 

Interlude 1.1 Mathematical Essentials The Euler Angles A description of... 

Interlude 2.1 The Abstract Concept of Hyperspace We have introduced hyperspace as a QN multidimensional space. The independent coordinates are the ZN position variables and the ZN momentum variables for the N total molecules in the system. It is impossible to draw such a system in three dimensions, so we must think of hyperspace in abstract or mathematical terms. 

Using scaling analysis and perturbation methods, we have been able to derive approximate expressions for the momentum and energy flux in dilute gases and liquids. These methods physically involve formal expansions about local equilibrium states, and the particular asymptotic restrictions have been formally obtained. The flux expressions now involve the dependent transport variables of mass or number density, velocity, and temperature, and they can be utilized to obtain a closed set of transport equations, which can be solved simultaneously for any particular physical system. The problem at this point becomes a purely mathematical problem of solving a set of coupled nonlinear partial differential equations subject to the particular boundary and initial conditions of the problem at hand. (Still not a simple matter see interlude 6.2.)... 

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