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/index.md')

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}
-- 
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