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| -rw-r--r-- | FuVar.tex | 68 |
1 files changed, 46 insertions, 22 deletions
@@ -52,6 +52,7 @@ \newcommand\Qset{\mathbb{Q}} \newcommand\Rset{\mathbb{R}} \newcommand\Cset{\mathbb{C}} +\newcommand\T{\mathrm{T}} %% Theorems \newtheoremstyle{fuvarzf} % name of the style to be used @@ -70,14 +71,9 @@ \newtheorem{lemma}{Lemma} \setlist[description]{ - align = right, labelwidth = 2cm, leftmargin = !, format = { \normalfont\itshape } } -\setlist[itemize]{ - align = right, labelwidth = 5mm, leftmargin = ! -} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Document @@ -87,20 +83,19 @@ \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] \(a \cdot (b + c) = a\cdot b + a \cdot c\) + \(a, b \in F\) 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} @@ -109,21 +104,50 @@ \end{theorem} \begin{definition}[Vector space] - A vector \(V\) space over a field \(F\) + 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} - \(\Rset^n\) is a vector space. + \(\Rset^n = \Rset\times\cdots\times\Rset\) 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} +\section{Scalar fields} -\begin{definition}[Scalar Field] - We call a function \(f\) a \emph{scalar field} when it maps values from - \(\Rset^n \to \Rset\). +\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] |