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| -rw-r--r-- | FuVar.tex | 87 |
1 files changed, 81 insertions, 6 deletions
@@ -13,30 +13,60 @@ %% Language configuration \usepackage{polyglossia} -\setdefaultlanguage[variant=swiss]{german} +\setdefaultlanguage{english} %% License configuration \usepackage[ type={CC}, modifier={by-nc-sa}, version={4.0}, - lang={german}, + lang={english}, ]{doclicense} +%% Math +\usepackage{amsmath} +\usepackage{amsthm} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Metadata \course{Elektrotechnik} \module{FuVar} -\semester{Fr\"uhlingssemester 2021} +\semester{Spring Semseter 2021} \authoremail{naoki.pross@ost.ch} \author{\textsl{Naoki Pross} -- \texttt{\theauthoremail}} -\title{\texttt{\themodule} Zusammenfassung} +\title{\texttt{\themodule} Notes} \date{\thesemester} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Macros and settings + +%% number sets +\newcommand\Nset{\mathbb{N}} +\newcommand\Zset{\mathbb{Z}} +\newcommand\Qset{\mathbb{Q}} +\newcommand\Rset{\mathbb{R}} +\newcommand\Cset{\mathbb{C}} + +%% Theorems +\newtheoremstyle{fuvarzf} % name of the style to be used + {\topsep} + {\topsep} + {} + {0pt} + {\bfseries} + {.} + { } + { } + +\theoremstyle{fuvarzf} +\newtheorem{theorem}{Theorem} +\newtheorem{definition}{Definition} +\newtheorem{lemma}{Lemma} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Document \begin{document} @@ -44,8 +74,53 @@ \maketitle \tableofcontents - -\section{Lizenz} +\section*{License} \doclicenseThis +\section{Scalar Fields} + +\begin{definition}[Scalar Field] + We call a function \(f\) a \emph{scalar field} when it maps values from + \(\Rset^n \to \Rset\). +\end{definition} + +\begin{definition}[Partial derivative of a scalar field] + Let \(f: \Rset^n \to \Rset\), the \emph{partial} derivative of \(f\) with + respect to \(x_k\), (\(0 < k < n\)), is defined as + \[ + \frac{\partial f}{\partial x_k} := + \lim_{h \to 0} \frac{f(x_1, \dots, x_k + h, \dots, x_n) + - f(x_1, \dots, x_k, \dots, x_n)}{h} + = \partial_{x_k} f(x, y) + \] + That is, we keep all variables of \(f\) fixed, except for \(x_k\). +\end{definition} + +\begin{definition}[Tangent plane] + For a scalar field \(f(x,y)\) we define the \emph{tangent plane} \(p(x,y)\) + at coordinates \((x_0, y_0)\) to be: + \[ + p(x, y) = + f(x_0, y_0) + + \partial_x f(x_0, y_0) (x - x_0) + + \partial_y f(x_0, y_0) (y - y_0) + \] +\end{definition} + +The above can be used to calculate the one dimensional derivative of an implicit curve. + +\begin{lemma}[Implicit derivative] + The slope \(m\) of an implicit curve \(f(x,y)\) at the point \((x_0, y_0)\) is given by + \[ + m = \partial_x f(x_0, y_0) / \partial_y f(x_0, y_0) + \] + of course only if \(\partial_y f(x_0, y_0) \neq 0\). +\end{lemma} + +\begin{definition}[Total derivative] + \[ + \dd{f} + \] +\end{definition} + \end{document} |