Fundamental equation geometrical meaning

Equation (C.4) has a very straightforward geometrical meaning, discussed in connection with the fundamental equation in Chapter 4 ( 4.6). Thermodynamics commonly deals with continuous changes in multivariable systems. For this reason, total differentials are frequently used, and it is essential to have a clear idea of their meaning. 

The nature of the dual vector ( ) can be deduced without using any equation of motion, but the dual 4-vector is a fundamental geometric property in the four dimensions of spacetime. The complete description of the electromagnetic field in 0(3) electrodynamics must therefore involve boosts, rotations, and spacetime translations, meaning that is a fundamental geometric property of spacetime. The unit 4-vector i M is orthogonal to the unit 4-vector... 

For an actual system having the nuclear and geometric properties assumed above, only one of the solutions (8.283) is physically acceptable. This solution is the fundamental mode o(r). It is clear that, for n > 0, the steady-state spatial distribution would require that i4(r, ) < 0 in certain regions of the reactor. The determination of the neutron-flux spatial distribution in a finite system by means of the integral-equation formulation was demonstrated in Sec. 5.7c for the case of an infinite slab. Those results may be applied directly to the multiplying medium problem. In the case of the infinite-slab reactor, of width 2a, Eq. (8.283) takes the form... 

---