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Diffstat (limited to 'FuVar.tex')
| -rw-r--r-- | FuVar.tex | 31 |
1 files changed, 24 insertions, 7 deletions
@@ -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: |