Lagrange Multipliers

Given some function $f(x,y,z)$ subject to some constraints

$$g_1(x,y,z)=0,\dots g_n(x,y,z)=0$$

the extrema of $f$ subject to the constraints satisfy the system

$$\nabla L=0$$

where

$$L(x,y,z,{\lambda }_1,\dots ,{\lambda }_n)=f(x,y,z)+{\lambda }_1{g}_1(x,y,z)+\cdots +{\lambda }_n{g}_n(x,y,z)$$

This function is called the Lagrangian of the problem and the variables ${\lambda }_i$ for $i\in \{1,2,\dots ,n\}$ are called Lagrange Multipliers.