From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Tue, 20 Dec 2022 20:11:25 +0100 Subject: More improvements to knowledge base --- source/know/concept/lagrange-multiplier/index.md | 18 ++++++++++++++++-- 1 file changed, 16 insertions(+), 2 deletions(-) (limited to 'source/know/concept/lagrange-multiplier') diff --git a/source/know/concept/lagrange-multiplier/index.md b/source/know/concept/lagrange-multiplier/index.md index a0b22aa..ce5418f 100644 --- a/source/know/concept/lagrange-multiplier/index.md +++ b/source/know/concept/lagrange-multiplier/index.md @@ -127,8 +127,22 @@ about the interdependence of a system of equations then $$\lambda$$ is not even given an expression! Hence it is sometimes also called an *undetermined multiplier*. -This method generalizes nicely to multiple constraints or more variables. -Suppose that we want to find the extrema of $$f(x_1, ..., x_N)$$ +This does not imply that $$\lambda$$ is meaningless; +it often represents a quantity of interest. +In general, defining $$h \equiv g + c$$ so that the constraint is $$h(x, y) = c$$, +we see that the Lagrange multiplier represents the rate of change of $$\mathcal{L}$$ +with respect to the value being constrained: + +$$\begin{aligned} + \mathcal{L}(x, y, \lambda) + = f(x, y) + \lambda (h(x, y) - c) + \qquad \implies \qquad + -\pdv{\mathcal{L}}{c} = \lambda +\end{aligned}$$ + +The method of Lagrange multipliers +generalizes nicely to more constraints or more variables. +Suppose we want to find extrema of $$f(x_1, ..., x_N)$$ subject to $$M < N$$ conditions: $$\begin{aligned} -- cgit v1.2.3