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% !TeX program = xelatex
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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
\(a, b \in F\) and \(\star\) stands for \(\cdot\) or \(+\).
\begin{description}
\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:]
\(a + (-a) = 0\) and
\(b \cdot b^{-1} = 1\) iff \(b \neq 0\)
\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 space \(U\) over a field \(F\) is a set of objects called
\emph{vectors} equipped with two operations: \emph{addition}
\(+: U \times U \to U\) and \emph{scalar multiplication}
\(\cdot: F\times U \to U\), that respect the following axioms.
Let \(\vec{u}, \vec{v}, \vec{w} \in U\) and \(a, b \in F\).
\begin{description}
\item[Additive associativity:] \((\vec{u} + \vec{v}) + \vec{w}
= \vec{u} + (\vec{v} + \vec{w})\)
\item[Additive commutativity:] \(\vec{u} + \vec{v} = \vec{v} + \vec{u}\)
\item[Identities:] There is an element
\(\vec{0} \in U : \vec{u} + \vec{0} = \vec{u}\)
and \(1 \in F : 1 \cdot \vec{u} = \vec{u}\)
\item[Additive inverse:] \(\vec{u} + (\vec{-u}) = 0\)
\item[Compatibility of multiplication]
\(a\cdot (b \cdot \vec{u}) = (a\cdot b) \cdot \vec{u}\)
\item[Distributivity:]
\((a + b) \cdot \vec{u} = a\cdot\vec{u} + b\cdot\vec{u}\) and conversely
\(a \cdot (\vec{u} + \vec{v}) = a\cdot\vec{u} + a\cdot\vec{v}\)
\end{description}
And of course elements in \(F\) follow the field axioms.
\end{definition}
\begin{theorem}
\(\Rset^n = \Rset\times\cdots\times\Rset\) is a vector space.
\end{theorem}
\begin{definition}[Row and column vectors]
Although there is virtually no difference between the two, we need two type
of \(n\)-tuples that satisfy the vector space axioms. \emph{Row} vectors are
written horizontally and \emph{column} vectors vertically.
\end{definition}
\begin{definition}[Transposition]
Let \(\vec{u} \in \Rset^n\) be a row vector. The \emph{transpose} of
\(\vec{u}\) denoted with \(\vec{u}^\T\) is column vector with the same
components. Conversely if \(\vec{v}\) is a column vector then \(\vec{v}^\T\)
is a row vector.
\end{definition}
\section{Scalar fields}
\begin{definition}[Scalar field]
Confusingly we call a function \(f: \Rset^n \to \Rset\) a \emph{scalar
field}, but this is unrelated to the previously defined field.
\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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