In order to understand the new computational method, a mathematical detour into the theory of Lagrange multipliers is necessary.
Consider a function F(n1, n2, ). The values of n1, n2, at which F is a minimum are to be determined. The system is subject to the following constraints:
Where F, V and W are specified functions of the mole numbers of all species, ni. The differential of F is:
The differential is a minimum so dF is set equal to zero. The coefficients of dni are:
Taking the differentials of V and W:
dV = Vnidnj + vn2dn2 + = 0 and dW = wnjdni + wn2dn2 + = 0 (9.65)
Multiply dV by Xv and dW by Xw, the Lagrange multipliers, and add to dF:
(fni + XvVnl + XwWnl)dni + (fn2 + XvVn2 + XwWn2)dn2 +...= 0 (9.67)
Since all dni are arbitrary, their coefficients must be zero:
with constraints given by Equation (9.62). These equations are solved for Xv, Xw and n1, n2
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Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.