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-rw-r--r--FuVar.tex87
1 files changed, 81 insertions, 6 deletions
diff --git a/FuVar.tex b/FuVar.tex
index ec5367e..7658499 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -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}