## Lagrange Multiplier

Used to find the Extremum of subject to the constraint , where and are functions with continuous first Partial Derivatives on the Open Set containing the curve , and at any point on the curve (where is the Gradient). For an Extremum to exist,

 (1)

But we also have
 (2)

Now multiply (2) by the as yet undetermined parameter and add to (1),

 (3)

Note that the differentials are all independent, so we can set any combination equal to 0, and the remainder must still give zero. This requires that
 (4)

for all , ..., . The constant is called the Lagrange multiplier. For multiple constraints, , , ...,
 (5)

Arfken, G. Lagrange Multipliers.'' §17.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 945-950, 1985.