\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{\sigmainv}{{\sigma^{-1}}}
\newcommand{\tauinv}{{\tau^{-1}}}

\newcommand{\veps}{\varepsilon}
\newcommand{\cle}{\preccurlyeq}
\newcommand{\vecA}{\mathbf{A}}
\newcommand{\vecB}{\mathbf{B}}
\newcommand{\vecE}{\mathbf{E}}
\newcommand{\vecF}{\mathbf{F}}
\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{\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{\veceps}{\mathbf{\varepsilon}}
\newcommand{\veclambda}{\mathbf{\lambda}}

\newcommand{\uhat}{\hat{\mathbf{u}}}
\newcommand{\vhat}{\hat{\mathbf{v}}}
\newcommand{\what}{\hat{\mathbf{w}}}
\newcommand{\ahat}{\hat{\mathbf{a}}}

\newcommand{\adju}{\mathrm{adju}}
\newcommand{\curl}{\mathrm{curl}}
\newcommand{\ord}{\mathrm{ord}}
\newcommand{\rGL}{\mathrm{GL}}
\newcommand{\rSL}{\mathrm{SL}}
\renewcommand{\div}{\mathrm{div}}
\newcommand{\Alt}{\mathrm{Alt}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\sgn}{\mathrm{sgn}}

\newcommand{\vecnq}{\mathbf{n}_\mathbf{q}}
\newcommand{\vecuq}{\mathbf{u}_\mathbf{q}}
\newcommand{\vecvq}{\mathbf{v}_\mathbf{q}}

\newcommand{\mcA}{\mathcal{A}}
\newcommand{\mcC}{\mathcal{C}}
\newcommand{\mcD}{\mathcal{D}}
\newcommand{\mcL}{\mathcal{L}}
\newcommand{\mcO}{\mathcal{O}}
\newcommand{\mcS}{\mathcal{S}}
\newcommand{\mcT}{\mathcal{T}}
\newcommand{\mcV}{\mathcal{V}}

\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{\grad}{\nabla}
\newcommand{\tensor}{\otimes}
\newcommand{\into}{\hookrightarrow}
\newcommand{\longto}{\longrightarrow}

\newcommand*{\slashD}[1]{\partial/\partial #1}
\newcommand*{\fracD}[1]{\frac{\partial}{\partial #1}}
\newcommand*{\slashDtwo}[2]{\partial #1/\partial #2}
\newcommand*{\fracDtwo}[2]{\frac{\partial #1}{\partial #2}}

\newcommand{\slashDx}{\partial/\partial x}
\newcommand{\slashDy}{\partial/\partial y}
\newcommand{\slashDz}{\partial/\partial z}

\newcommand{\fracDx}{\frac{\partial}{\partial x}}
\newcommand{\fracDy}{\frac{\partial}{\partial y}}
\newcommand{\fracDz}{\frac{\partial}{\partial z}}

\newcommand{\fracDfx}{\frac{\partial f}{\partial x}}
\newcommand{\fracDfy}{\frac{\partial f}{\partial y}}
\newcommand{\fracDfz}{\frac{\partial f}{\partial z}}
\newcommand{\fracDgx}{\frac{\partial g}{\partial x}}
\newcommand{\fracDgy}{\frac{\partial g}{\partial y}}
\newcommand{\fracDgz}{\frac{\partial g}{\partial z}}


\newcommand*{\slashd}[1]{d/d #1}
\newcommand*{\fracd}[1]{\frac{d}{d #1}}
\newcommand*{\slashdtwo}[2]{d #1/d #2}
\newcommand*{\fracdtwo}[2]{\frac{d #1}{d #2}}

\newcommand*{\vvectwo}[2]{\begin{pmatrix} #1 \\ #2 \end{pmatrix}}
\newcommand*{\pickpipess}[1]{\bigg|_{\scriptsize{#1}}}
\newcommand*{\pickpipet}[1]{\bigg|_{\tiny{#1}}}
\newcommand*{\pickpipe}[1]{\pickpipet{#1}}

\newcommand*{\rowvectwo}[2]{\begin{pmatrix}#1 & #2 \end{pmatrix}}
\newcommand*{\rowvecthree}[3]{\begin{pmatrix}#1 & #2 & #3\end{pmatrix}}
\newcommand*{\colvectwo}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}}
\newcommand*{\colvecthree}[3]{\begin{pmatrix}#1 \\ #2 \\ #3\end{pmatrix}}

%\newcommand{\uscore}{\textunderscore}
\newcommand{\uscore}{\_}
\newcommand{\cupprod}{\smallsmile}

\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}

\newcommand{\TqM}{T_\vecq M}
\newcommand{\TqsM}{T^*_\vecq M}
