2013年10月30日

Lagrange未定係数法

I'll write down the idea of "method of Lagrange multiplier" here for my note. For simplicity, the number of variables would be 3.

Supposedly, a function $\psi: \Omega \to \mathbb{R}$ differential for each variable, and satisfies a restriction $\psi(x,y,z)=0$.
If a function $f : \mathbb{R}^3 \to \mathbb{R}$ realizes extremal at $p=(x_0, y_0, z_0)\in \Omega \subseteq \mathbb{R}^3$, then the following equations would be true for some constant $\lambda\in\mathbb{R}$:

$\displaystyle \frac{\partial f}{\partial x}(p)-\lambda\frac{\partial \psi}{\partial x}(p) =\frac{\partial f}{\partial y}(p)-\lambda\frac{\partial \psi}{\partial y}(p)=\frac{\partial f}{\partial z}(p)-\lambda\frac{\partial \psi}{\partial z}(p)=0$