From 7bf913f9bc7ab9f8f03c5530d245cf95e1edb43e Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 6 Mar 2021 19:27:57 +0100
Subject: Expand knowledge base

---
 content/know/concept/lagrange-multiplier/index.pdc | 27 ++++++++++++++++++----
 1 file changed, 23 insertions(+), 4 deletions(-)

(limited to 'content/know/concept/lagrange-multiplier/index.pdc')

diff --git a/content/know/concept/lagrange-multiplier/index.pdc b/content/know/concept/lagrange-multiplier/index.pdc
index fffe85f..fc1319e 100644
--- a/content/know/concept/lagrange-multiplier/index.pdc
+++ b/content/know/concept/lagrange-multiplier/index.pdc
@@ -49,15 +49,14 @@ of all the partial derivatives.
 
 To help us solve this, we introduce a "dummy" parameter $\lambda$,
 the so-called **Lagrange multiplier**,
-which need not be constant,
 and contruct a new function $L$ given by:
 
 $$\begin{aligned}
     L(x, y, z) = f(x, y, z) + \lambda \phi(x, y, z)
 \end{aligned}$$
 
-Clearly, $\dd{L} = \dd{f} + \lambda \dd{\phi} = 0$,
-so now the problem is a single equation again:
+At the extremum, $\dd{L} = \dd{f} + \lambda \dd{\phi} = 0$,
+so now the problem is a "single" equation again:
 
 $$\begin{aligned}
     0 = \dd{L}
@@ -69,7 +68,21 @@ This choice represents satisfying the constraint,
 so now the remaining $\dd{x}$ and $\dd{y}$ are independent again,
 and we simply have to find the roots of $f_x + \lambda \phi_x$ and $f_y + \lambda \phi_y$.
 
-This generalizes nicely to multiple constraints or more variables:
+In effect, after introducing $\lambda$,
+we have four unknowns $(x, y, z, \lambda)$,
+but also four equations:
+
+$$\begin{aligned}
+    L_x = L_y = L_z = 0
+    \qquad \quad
+    \phi = C
+\end{aligned}$$
+
+We are only really interested in the first three unknowns $(x, y, z)$,
+so $\lambda$ is sometimes called the **undetermined multiplier**,
+since it is just an algebraic helper whose value is irrelevant.
+
+This method generalizes nicely to multiple constraints or more variables:
 suppose that we want to find the extrema of $f(x_1, ..., x_N)$
 subject to $M < N$ conditions:
 
@@ -103,3 +116,9 @@ $$\begin{aligned}
     0 = \dd{L}
     = \sum_{n = 1}^N \Big( f_{x_n} + \sum_{m = 1}^M \lambda_m \phi_{x_n} \Big) \dd{x_n}
 \end{aligned}$$
+
+
+## References
+1.  G.B. Arfken, H.J. Weber,
+    *Mathematical methods for physicists*, 6th edition, 2005,
+    Elsevier.
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