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-rw-r--r--FuVar.tex31
1 files changed, 24 insertions, 7 deletions
diff --git a/FuVar.tex b/FuVar.tex
index 5843d02..13b0d2d 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -10,7 +10,6 @@
%% TODO: publish to CTAN
\usepackage{tex/hsrstud}
-\usepackage{mathtools}
%% Language configuration
\usepackage{polyglossia}
@@ -27,10 +26,14 @@
%% Math
\usepackage{amsmath}
\usepackage{amsthm}
+\usepackage{mathtools}
-% Layout
+%% Layout
\usepackage{enumitem}
+%% Nice drwaings
+\usepackage{tikz}
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Metadata
@@ -147,12 +150,14 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
A function \(f(\vec{x})\) has a directional derivative in direction
\(\vec{r}\) (with \(|\vec{r}| = 1\)) given by
\[
- \frac{\partial f}{\partial\vec{r}} = \nabla_\vec{r} f = \vec{r} \dotp \grad f
+ \frac{\partial f}{\partial\vec{r}}
+ = \nabla_\vec{r} f = \vec{r} \dotp \grad f
\]
\end{definition}
\begin{theorem}
- The gradient vector always points towards \emph{the direction of steepest ascent}.
+ The gradient vector always points towards \emph{the direction of steepest
+ ascent}.
\end{theorem}
\section{Methods for maximization and minimization problems}
@@ -272,8 +277,8 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
Search for a constrained extremum in higher dimensions,
method of Lagrange multipliers]
We wish to find the extrema of \(f: D \subseteq \mathbb{R}^m \to \mathbb{R}\)
- under \(k < m\) constraints \(n_1 = 0, \cdots, n_k = 0\). For that we consider
- the following points:
+ under \(k < m\) constraints \(n_1 = 0, \cdots, n_k = 0\). To find the extrema
+ we consider the following points:
\begin{itemize}
\item Points on the boundary \(\vec{u} \in \partial D\) that satisfy
\(n_i(\vec{u}) = 0\) for all \(1 \leq i \leq k\),
@@ -293,11 +298,22 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
n_i(\vec{u}) = 0 & \text{ for } 1 \leq i \leq k
\end{dcases}
\]
- The \(\lambda\) values are known as \emph{Lagrange multipliers}.
+ The \(\lambda\) values are known as \emph{Lagrange multipliers}. The same
+ calculation can be written more compactly by defining the
+ \(m+k\) dimensional \emph{Lagrangian}
+ \[
+ \mathcal{L}(\vec{u}, \vec{\lambda})
+ = f(\vec{u}) - \sum_{i = 0}^k \lambda_i n_i(\vec{u})
+ \]
+ where \(\vec{\lambda} = \lambda_1, \ldots, \lambda_k\) and then
+ evaluating \(\grad \mathcal{L}(\vec{u}, \vec{\lambda}) = \vec{0}\).
\end{itemize}
\end{method}
\section{Integration}
+\begin{remark}
+
+\end{remark}
\section*{License}
@@ -308,3 +324,4 @@ typesetting may trick you into thinking it is rigorous, but really, it is not.
\end{center}
\end{document}
+% vim:ts=2 sw=2 et spell: