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% !TeX program = xelatex
% !TeX encoding = utf8
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% Packages
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type={CC},
modifier={by-nc-sa},
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% Metadata
\course{Elektrotechnik}
\module{FuVar}
\semester{Spring Semseter 2021}
\authoremail{naoki.pross@ost.ch}
\author{\textsl{Naoki Pross} -- \texttt{\theauthoremail}}
\title{\texttt{\themodule} Notes}
\date{\thesemester}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Document
\begin{document}
\maketitle
\tableofcontents
\section{Fields and vector spaces}
\begin{definition}[Field]
A field is a set \(F\) with two binary operators \(+\) and \(\cdot\) that map
\(F\times F \to F\) and follow the \emph{field axioms} listed below. We let
\(a, (-a), b, b^{-1} \in F\) and \(\star\) stand for either \(\cdot\) or \(+\).
\begin{description}[leftmargin=2.5cm]
\item[Associativity] \((a \star b) \star c = a \star (b \star c)\)
\item[Commutativity] \(a \star b = b \star a\)
\item[Identities] \(0 + a = a\) and \(1\cdot a = a\)
\item[Inverses]
\begin{itemize}
\item \(a + (-a) = 0\) and
\item \(b \cdot b^{-1} = 1\) where \(b \neq 0\)
\end{itemize}
\item[Distributivity] \(a \cdot (b + c) = a\cdot b + a \cdot c\)
\end{description}
\end{definition}
\begin{theorem}
\(\Rset\) is a field.
\end{theorem}
\begin{definition}[Vector space]
A vector \(V\) space over a field \(F\)
\end{definition}
\begin{theorem}
\(\Rset^n\) is a vector space.
\end{theorem}
\begin{definition}[Row and column vectors]
\end{definition}
\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 \(x_k\), (\(0 < k \leq n\)), 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}
\section*{License}
\doclicenseText
\begin{center}
\doclicenseImage
\end{center}
\end{document}
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