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Section 11.5 Lagrange Multipliers using Differentials

Suppose first that you want to extremize a function \(f\) of two variables subject to the constraint \(g=\hbox{constant}\text{.}\) In two dimensions, level sets are curves, and we can introduce a coordinate \(v\) along the level curves of \(g\text{.}\) Then \(v\) and \(g\) can be used as coordinates, at least in a small region about any point. Thus,

\begin{equation} df = \Partial{f}{g}\,dg + \Partial{f}{v}\,dv .\tag{11.5.1} \end{equation}

The condition that \(f\) be extremized at a point \(P\) on a given level curve of \(g\) is precisely that

\begin{equation} \Partial{f}{v}\bigg|_P = 0 .\tag{11.5.2} \end{equation}

Thus, at \(P\text{,}\) we have

\begin{equation} df = \lambda \,dg\label{dlagrange}\tag{11.5.3} \end{equation}

where

\begin{equation} \lambda = \Partial{f}{g}\bigg|_P .\tag{11.5.4} \end{equation}

Equation (11.5.3) can be used to find the point(s) \(P\) on the level curve where \(f\) is extremized: Simply expand both sides in terms of the original variables, and realize that corresponding partial derivatives must therefore be proportional, that is

\begin{equation} \Partial{f}{x^i} = \lambda \Partial{g}{x^i} .\tag{11.5.5} \end{equation}

These two equations (together with the constraint condition) can be solved for \(x^i\) (and \(\lambda\)); this is the method of Lagrange multipliers.

It is straightforward to generalize this procedure to higher dimensions. If \(f\) is a funciton of \(n\) variables, and the constrain function is also, then the level sets of \(g\) are \((n-1)\)-dimensional surfaces, on which we can introduce the \(n-1\) coordinates \(v^i\text{;}\) \(\{g,v^i\}\) can therefore be used as (local) coordinates. Thus,

\begin{equation} df = \Partial{f}{g}\,dg + \sum\limits_i \Partial{f}{v^i}\,dv^i\tag{11.5.6} \end{equation}

and the condition that \(f\) be extremized at a point \(P\) on a given level surface of \(g\) is that

\begin{equation} \Partial{f}{v^i}\bigg|_P = 0\tag{11.5.7} \end{equation}

for each \(i\text{.}\) Thus, at \(P\text{,}\) we again have (11.5.3); only the number of “components” of this equation has changed.

We can further generalize this procedure to multiple constraints, involving constraint functions \(g^j\text{.}\) The intersection of \(k\) such constraint surfaces will in general be an \((n-k)\)-dimensional surface, on which we can introduce \(n-k\) coordinates \(v^i\text{,}\) and we must now use \(\{g^j,v^i\}\) as coordinates, leading to

\begin{equation} df = \sum\limits_j \Partial{f}{g^j}\,dg^j + \sum\limits_i \Partial{f}{v^i}\,dv^i\tag{11.5.8} \end{equation}

The condition that \(f\) be extremized on this intersection surface is the same as before, namely that the second term above vanish, but now we obtain

\begin{equation} df = \sum\limits_j \lambda^j dg^j\tag{11.5.9} \end{equation}

at the point \(P\text{.}\) Thus, the differential of \(f\) is a linear combination of the differentials of the constraint functions, a condition which must also hold for each partial derivative. The resulting system of equations can be solved for \(P\) (and \(\lambda^j\)), at least in principle.

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