Method of Lagrange multipliers

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:

d ni

Taking the differentials of V and W:

dV = Vnidnj + vn2dn2 + = 0 and dW = wnjdni + wn2dn2 + = 0 (9.65)

where

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

Solar Panel Basics

Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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