\newcommand{\Dn}{\mathcal{D}_n}
\newcommand{\Sn}{\mathcal{S}_n}
\newcommand{\An}{\mathcal{A}_n}
\newcommand{\Vf}{\mathcal{V}_4}
\newcommand{\Qe}{\mathcal{Q}_8}
\newcommand*{\pig}[1]{\langle #1 \rangle}

\newcommand*{\bfidx}[1]{\index{#1}\textbf{#1}}
\newcommand*{\bfidxtwo}[2]{\textbf{#1}\index{#2}}
\newcommand*{\bfidxaux}[2]{\index{#1}\index{#2}\textbf{#1}}
\newcommand*{\emphidx}[1]{\index{#1}\emph{#1}}
\newcommand*{\plainidx}[1]{\index{#1}#1}

\newcommand{\cle}{\preccurlyeq}
\newcommand{\vecA}{\mathbf{A}}
\newcommand{\vecB}{\mathbf{B}}
\newcommand{\vecE}{\mathbf{E}}
\newcommand{\vecF}{\mathbf{F}}
\newcommand{\vecL}{\mathbf{L}}
\newcommand{\veca}{\mathbf{a}}
\newcommand{\vecb}{\mathbf{b}}
\newcommand{\vecc}{\mathbf{c}}
\newcommand{\vecd}{\mathbf{d}}
\newcommand{\vece}{\mathbf{e}}
\newcommand{\vecf}{\mathbf{f}}
\newcommand{\vecg}{\mathbf{g}}
\newcommand{\veci}{\mathbf{i}}
\newcommand{\vecj}{\mathbf{j}}
\newcommand{\veck}{\mathbf{k}}
\newcommand{\vecl}{\mathbf{\ell}}
\newcommand{\vecn}{\mathbf{n}}
\newcommand{\vecp}{\mathbf{p}}
\newcommand{\vecq}{\mathbf{q}}
\newcommand{\vecr}{\mathbf{r}}
\newcommand{\vecu}{\mathbf{u}}
\newcommand{\vecv}{\mathbf{v}}
\newcommand{\vecw}{\mathbf{w}}
\newcommand{\vecx}{\mathbf{x}}
\newcommand{\vecy}{\mathbf{y}}
\newcommand{\vecz}{\mathbf{z}}
\newcommand{\myvecx}{\mathbf{x}}
\newcommand{\veclambda}{\mathbf{\lambda}}

\newcommand{\hatf}{\hat{f}}
\newcommand{\ovW}{\overline{W}}

\newcommand{\mcA}{\mathcal{A}}
\newcommand{\mcC}{\mathcal{C}}
\newcommand{\mcD}{\mathcal{D}}
\newcommand{\mcF}{\mathcal{F}}
\newcommand{\mcL}{\mathcal{L}}
\newcommand{\mcS}{\mathcal{S}}
\newcommand{\mcT}{\mathcal{T}}

\newcommand{\msC}{\mathscr{C}}
\newcommand{\msD}{\mathscr{D}}

\newcommand{\D}{\partial}
\newcommand{\DA}{\partial A}
\newcommand{\DB}{\partial B}
\newcommand{\DC}{\partial C}
\newcommand{\DX}{\partial X}
\newcommand{\Dc}{\partial c}
\newcommand{\Df}{\partial f}
\newcommand{\Dg}{\partial g}
\newcommand{\Dh}{\partial h}
\newcommand{\DM}{\partial M}
\newcommand{\Dx}{\partial x}
\newcommand{\Dy}{\partial y}
\newcommand{\Dz}{\partial z}

\newcommand{\tensor}{\otimes}
\newcommand{\into}{\hookrightarrow}
\newcommand{\longto}{\longrightarrow}
\newcommand{\upto}{\uparrow}
\newcommand{\downto}{\downarrow}

\newcommand{\qand}{\quad\textrm{and}\quad}
\newcommand{\qqand}{\qquad\textrm{and}\qquad}
\newcommand{\qor}{\quad\textrm{or}\quad}
\newcommand{\qqor}{\qquad\textrm{or}\qquad}
\newcommand{\qie}{\quad\textrm{i.e.}\quad}
\newcommand{\qqie}{\qquad\textrm{i.e.}\qquad}
