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Diffstat (limited to '')
| -rw-r--r-- | FuVar.tex | 58 |
1 files changed, 54 insertions, 4 deletions
@@ -3,7 +3,7 @@ % !TeX root = FuVar.tex %% TODO: publish to CTAN -\documentclass[twocolumn]{tex/hsrzf} +\documentclass[twocolumn, margin=normal]{tex/hsrzf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages @@ -27,6 +27,9 @@ \usepackage{amsmath} \usepackage{amsthm} +% Layout +\usepackage{enumitem} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Metadata @@ -66,6 +69,15 @@ \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma} +\setlist[description]{ + align = right, labelwidth = 2cm, leftmargin = !, + format = { \normalfont\itshape } +} + +\setlist[itemize]{ + align = right, labelwidth = 5mm, leftmargin = ! +} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Document @@ -74,8 +86,38 @@ \maketitle \tableofcontents -\section*{License} -\doclicenseThis +\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} @@ -86,7 +128,7 @@ \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 + respect to \(x_k\), (\(0 < k \leq n\)), is defined as \[ \frac{\partial f}{\partial x_k} := \lim_{h \to 0} \frac{f(x_1, \dots, x_k + h, \dots, x_n) @@ -123,4 +165,12 @@ The above can be used to calculate the one dimensional derivative of an implicit \] \end{definition} + +\section*{License} +\doclicenseText + +\begin{center} + \doclicenseImage +\end{center} + \end{document} |