%%
%% This is the file `mathlib.sty' generated on <1994/04/24>
%% by Andrew Lewis.
%%
%% This file contains some macros for mathematical typing.  If
%% they are not loaded prior to running this package, then the
%% packages `amsmath' and `euscript' are loaded.
%%
\NeedsTeXFormat{LaTeX2e}
\def\filedate{1994/04/24}
\ProvidesPackage{mathlib}[1994/04/24]
\RequirePackage{amsmath,amssymb,euscript}

\newcommand\real{\mathbb{R}}
\newcommand\complex{\mathbb{C}}
\newcommand\integer{\mathbb{Z}}
\renewcommand\natural{\mathbb{N}}
\newcommand\rot{\mathbb{S}}
\newcommand\rational{\mathbb{Q}}
\newcommand\torus{\mathbb{T}}

\newcommand\eqdef{\triangleq}
\def\big#1{{\hbox{$\left#1\vcenter to1.428\ht\strutbox{}\right.\n@space$}}}
\def\Big#1{{\hbox{$\left#1\vcenter to2.142\ht\strutbox{}\right.\n@space$}}}
\def\bigg#1{{\hbox{$\left#1\vcenter to2.857\ht\strutbox{}\right.\n@space$}}}
\def\Bigg#1{{\hbox{$\left#1\vcenter to3.571\ht\strutbox{}\right.\n@space$}}}
\newcommand\scirc{\raise1pt\hbox{$\,\scriptstyle\circ\,$}}
\newcommand\sscirc{\hbox{$\,\scriptscriptstyle\circ\,$}}
\newcommand\slnorm{\left|}
\newcommand\srnorm{\right|}
\newcommand\snorm[1]{\slnorm #1\srnorm}
\newcommand\bigsnorm[1]{\big| #1\big|}
\newcommand\Bigsnorm[1]{\Big| #1\Big|}
\newcommand\biggsnorm[1]{\bigg| #1\bigg|}
\newcommand\Biggsnorm[1]{\Bigg| #1\Bigg|}
\newcommand\dlnorm{\left|\!\left|}
\newcommand\drnorm{\right|\!\right|}
\newcommand\dnorm[1]{\dlnorm #1\drnorm}
\newcommand\bigdnorm[1]{\big|\!\big|\! #1\!\big|\!\big|}
\newcommand\Bigdnorm[1]{\Big|\!\Big|\! #1\!\Big|\!\Big|}
\newcommand\biggdnorm[1]{\bigg|\!\bigg|\! #1\!\bigg|\!\bigg|}
\newcommand\Biggdnorm[1]{\Bigg|\!\Bigg|\! #1\!\Bigg|\!\Bigg|}
\newcommand\tlnorm{\left|\!\left|\!\left|}
\newcommand\trnorm{\right|\!\right|\!\right|}
\newcommand\tnorm[1]{\tlnorm #1\trnorm}
\newcommand\angbrack[1]{\left< #1 \right>}
\newcommand\bigangbrack[1]{\big\langle #1\big\rangle}
\newcommand\Bigangbrack[1]{\Big\langle #1\Big\rangle}
\newcommand\biggangbrack[1]{\bigg\langle #1\bigg\rangle}
\newcommand\Biggangbrack[1]{\Bigg\langle #1\Bigg\rangle}
\newcommand\inprod[2]{\left< #1,#2\right>}
\newcommand\natpair[2]{\left< #1;#2\right>}
\newcommand\disjointunion[1]{\bigcup_{#1}^{\sscirc}}
\newcommand\transverse{\pitchfork\kern-0.855em\raise0.45em\hbox{--}
  \kern 0.37em}
\newcommand\transversejon{\overline{\pitchfork}}
\newcommand\maprightu[1]{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\newcommand\maprightd[1]{\smash{\mathop{\longrightarrow}\limits_{#1}}}
\newcommand\mapdownr[1]{\Big\downarrow\rlap
  {$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\newcommand\mapdownl[1]{\llap{$\vcenter{\hbox{$\scriptstyle#1$}}$}
  \Big\downarrow}
\newcommand\mapsel[1]{\llap{$\vcenter{\hbox{$\scriptstyle#1$}}$}\searrow}
\newcommand\mapswl[1]{\llap{$\vcenter{\hbox{$\scriptstyle#1$}}$}\swarrow}
\newcommand\mapser[1]{\searrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\newcommand\mapswr[1]{\swarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\newcommand\mapdefa[3]{#1\colon&#2\rightarrow#3\cr}
\newcommand\mapdef{\@ifstar{\@mapdef}{\@@mapdef}}
\newcommand\@mapdef[5]{\begin{align*}#1\colon&\begin{aligned}[t]#2\end{aligned}
  \rightarrow\begin{aligned}[t]#3\end{aligned}\\&
  \begin{aligned}[t]#4\end{aligned}\mapsto
  \begin{aligned}[t]#5\end{aligned}\end{align*}}
\newcommand\@@mapdef{\@ifnextchar [{\@@mapdefl}{\@@mapdefnl}}
\def\@@mapdefl[#1]#2#3#4#5#6{\begin{equation}\label{#1}\begin{split}
  #2\colon&\begin{aligned}[t]#3\end{aligned}\rightarrow
  \begin{aligned}[t]#4\end{aligned}\\&\begin{aligned}[t]#5\end{aligned}\mapsto
  \begin{aligned}[t]#6\end{aligned}\end{split}\end{equation}}
\newcommand\@@mapdefnl[5]{\begin{equation}\begin{split}
  #1\colon&\begin{aligned}[t]#2\end{aligned}\rightarrow
  \begin{aligned}[t]#3\end{aligned}\\&\begin{aligned}[t]#4\end{aligned}
  \mapsto\begin{aligned}[t]#5\end{aligned}\end{split}\end{equation}}
\newcommand\map[3]{#1\colon#2\rightarrow#3}

\newcommand\osubset{\subset\kern-1.0em\lower0.02em\hbox{\scirc}\kern0.35em}
\newcommand\osubseteq{\subseteq\kern-1.0em\raise0.09em\hbox{\scirc}
  \kern0.35em}

\newcommand\tenf[3]{\mathfrak{T}^{#1}_{#2}(#3)}
\newcommand\vf[1]{\mathfrak{X}(#1)}
\newcommand\forms{\@ifnextchar[{\@kform}{\@form}}
\def\@kform[#1]#2{\Omega^{#1}(#2)}
\newcommand\@form[1]{\Omega(#1)}
\newcommand\func[1]{C^\infty(#1)}
\let\oldsec\sec
\renewcommand\sec[1]{\EuScript{#1}}
\typeout{MATHLIB: Redefining \string\sec\space to \string\oldsec}

\newcommand\range{\operatorname{range}}
\newcommand\coker{\operatorname{coker}}
\newcommand\id{\operatorname{id}}
\newcommand\rank{\operatorname{rank}}

\newcommand\der{\boldsymbol{d}}
\newcommand\intprod[2]{#1\hook\,#2}
\newcommand\hook{\mathbin{\hbox{\vrule height .5pt width 3.5pt depth 0pt
\vrule height 6pt width .5pt depth 0pt}}}
\newcommand\lieder[2]{\boldsymbol{\EuScript{L}}_{#1}#2}
\newcommand\deriv[2]{\frac{d #1}{d #2}}
\newcommand\tderiv[2]{\tfrac{d #1}{d #2}}
\newcommand\dderiv[2]{\dfrac{d #1}{d #2}}
\newcommand\pderiv[2]{\frac{\partial #1}{\partial #2}}
\newcommand\tpderiv[2]{\tfrac{\partial #1}{\partial #2}}
\newcommand\dpderiv[2]{\dfrac{\partial #1}{\partial #2}}

\newcommand\tb{\@ifnextchar[{\@tbarg}{\@tb}}
\def\@tbarg[#1]#2{T_{#1}#2}
\newcommand\@tb[1]{T#1}

\newcommand\ctb{\@ifnextchar[{\@ctbarg}{\@ctb}}
\def\@ctbarg[#1]#2{T^*_{#1}#2}
\newcommand\@ctb[1]{T^*#1}

\newcommand\ttb{\@ifnextchar[{\@ttbarg}{\@ttb}}
\def\@ttbarg[#1]#2{T_{#1}T#2}
\newcommand\@ttb[1]{TT#1}

\newcommand\vb{\@ifnextchar[{\@vbarg}{\@vb}}
\def\@vbarg[#1]#2{V_{#1}#2}
\newcommand\@vb[1]{V#1}

\newcommand\cvb{\@ifnextchar[{\@cvbarg}{\@cvb}}
\def\@cvbarg[#1]#2{V^*_{#1}#2}
\newcommand\@cvb[1]{V^*#1}

\newcommand\hb{\@ifnextchar[{\@hbarg}{\@hb}}
\def\@hbarg[#1]#2{H_{#1}#2}
\newcommand\@hb[1]{H#1}

\newcommand\nb{\@ifnextchar[{\@nbarg}{\@nb}}
\def\@nbarg[#1]#2{N_{#1}#2}
\newcommand\@nb[1]{N#1}

\newcommand\chb{\@ifnextchar[{\@chbarg}{\@chb}}
\def\@chbarg[#1]#2{H^*_{#1}#2}
\newcommand\@chb[1]{H^*#1}

\newcommand\tf{\@ifnextchar[{\@tfarg}{\@tf}}
\def\@tfarg[#1]#2{T_{#1}#2}
\newcommand\@tf[1]{T#1}

\newcommand\ctf{\@ifnextchar[{\@ctfarg}{\@ctf}}
\def\@ctfarg[#1]#2{T^*_{#1}#2}
\newcommand\@ctf[1]{T^*#1}

\newcommand\vp{\mathit{vert}}
\newcommand\hp{\mathit{hor}}
\newcommand\lift{\mathit{lift}}

\newcommand\GL[2]{\mathit{GL}(#1,#2)}

\newcommand\Lie[1]{\mathit{#1}}
\newcommand\lie[1]{\mathfrak{#1}}
\newcommand\dual[1]{#1^*}

\endinput
