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-rw-r--r--FuVar.tex68
1 files changed, 46 insertions, 22 deletions
diff --git a/FuVar.tex b/FuVar.tex
index 5d1fdb9..aa4ba54 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -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]