The Lagrange method of undetermined multipliers is a technique for finding the maximum or minimum value of a function subject to one or more constraints.
Consider the function =-x^2-y^2+5)
. The total differential of the function is
=\left(\frac{\partial f}{\partial x}\right)dx+\left(\frac{\partial f}{\partial y}\right)dy)
Since x and y are independent variables (see below diagram), the maximum value of the function occurs at the point where =0)
and =0)
, i.e.
dx+\left(\frac{\partial f}{\partial y}\right)dy=0\;\;\;\;\;\;\;\;(10))
In general, the extremum of a function with i-independent variables )
is at the point where all =0)
.
If we want to find the extremum of )
subject to a constraint, e.g. 
, we will realise that one of the variables, e.g. y, in is no longer independent. This is because we can substitute the constraint in the function to eliminate the dependent variable y and then let the 1st derivative of the function with respect to the remaining independent variable x be zero. However, this simple method of substitution becomes complicated when the problem involves more than two variables and more than one constraint.
In 1804, Joseph-Louis Lagrange devised another procedure to handle such problems. For the above example, the procedure involves transforming the constraint into the function 
, where 
. The total differential of g is
=\left(\frac{\partial g}{\partial x}\right)dx+\left(\frac{\partial g}{\partial y}\right)dy=\sum_{i=1}^2\left(\frac{\partial g}{\partial n_i}\right)dn_i)
where 
and 
.
Since , 
and dn_i=0)
. This means that we can multiply throughout by a factor 
and add the result to (or subtract from) eq10 to obtain the total differential of a new function:
where =f(n_1,n_2)+\lambda g(n_1,n_2))
.
The extremum of this function occurs when
)
If there is a 
, where 
, then it renders the dependent variable term 
, and we are left with the independent term 
. Solving the equations , and the constraint simultaneously, we can determine x, y and . This procedure is called the Lagrange method of undetermined multipliers. The factor is the multiplier, which is undetermined because it need not be solved to find the extremum of the function.
To evaluate the extremum of the n-independent variables function )
subject to m constraints, where 
, eq11 becomes
where =f(n_1,n_2,\cdots,n_n)+\sum_{l=1}^m \lambda_l g_l(n_1,n_2,\cdots,n_n))
.
If we can find a set of values of 
that renders all the m dependent variable terms zero, we have m equations of
)
leaving us with 
independent variable terms and equations of
)
This implies that each 
in eq12 can now vary arbitrarily. The values of all the variables and multipliers corresponding to the extremum are finally obtained by solving eq13, eq14 and all the constraint equations simultaneously. The Lagrange method of undetermined multipliers is used in the derivations of the Boltzmann distribution, the Hartree self-consistent field method, the Hartree-Fock method and the Hartree-Fock-Roothaan method.