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# 2.5 Just in case ...

... someone has forgotten how to find the extremum of a many-variable function under additional constraints. To remind you: the method of undetermined Lagrange multipliers waits to be applied here.

Let be the given function. Of course, this is the equation of a parabolid with its tip - or minimum - at the origin. However, let be the constraint equation, meaning that we don't search for the global minimum but for the minimum along the line . There are two ways to go about it. The simple but inconvenient way is to substitute in , thus rendering a function of only. Equating the derivative to zero we find the locus of the conditional minimum, and . The process of substitution is, in general, tedious.

A more elegant method is this: defining a (undetermined) Lagrange multiplier , find the minimum of the function according to (2.53) (2.54)

Eliminating we find the solution without substituting anything. In our case (2.55) (2.56)  , and from : .   Next: 2.6 Problems for Chapter Up: 2. Elements of Kinetic Previous: 2.4 Transport processes
Franz Vesely
2005-01-25