# 2.7: Undetermined Multipliers

- Page ID
- 8566

Let ψ(*x, y, z*) be some function of *x, y* and *z*. Then if *x, y* and *z* are independent variables, one would ordinarily understand that, where ψ is a maximum, the derivatives are zero:

\[ \frac{\partial \psi}{\partial x} = \frac{\partial \psi}{\partial y} = \frac{\partial \psi}{\partial z} = 0.\]

However, if x, y and z are not completely independent, but are related by some constraining equation such as *f(x, y, z)* = 0, the situation is slightly less simple. (In a thermodynamical context, the three variables may be, for example, three “intensive state variables”, *P, V* and *T*, and ψ might be the entropy, which is a function of state. However the intensive state variables may not be completely independent, since they are related by an “equation of state”, such as *PV = RT*.)

If we move by infinitesimal displacements *dx, dy, dz* from a point where ψ is a maximum, the corresponding changes in ψ and *f* will both be zero, and therefore both of the following equations must be satisfied.

\[ d \psi = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy + \frac{\partial \psi}{\partial z} dz = 0,\]

\[ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz = 0.\]

Consequently any linear combination of ψ and *f*, such as Φ = ψ + λ*f*, where λ is an arbitrary constant, also satisfies a similar equation. The constant λ is sometimes called an “undetermined multiplier” or a “Lagrangian multiplier”, although often some additional information in an actual problem enables the constant to be identified.

In summary, the conditions that ψ is a maximum (or minimum or saddle point), if *x, y* and *z* are related by a functional constraint f (*x, y, z*) = 0, are

\[ \begin{matrix} \frac{\partial \Phi}{\partial x} = 0 & \frac{\partial \Phi}{\partial y} = 0, & \frac{\partial \Phi}{\partial z} = 0, \end{matrix}\]

where

\[ \Phi = \psi + \lambda f.\]

Of course, if ψ is a function of many variables x_{1} , x_{2} , x_{3}..., and the variables are subjected to several constraints, such as f = 0, g = 0, h = 0, etc., where *f, g, h*, etc., are functions connecting all or some of the variables, the conditions for ψ to be a maximum (etc.) are

\[ \frac{\partial \psi}{\partial x_i} + \lambda \frac{\partial \psi}{\partial x_i} + \mu \frac{\partial \psi}{\partial x_i} + \nu \frac{\partial \psi}{\partial x_i} + ... = 0,~ i = 1,~2,~3\]