%lauseymp�rist�t
\newtheorem{Thm}{Lause}
{\theorembodyfont{\normalfont} \newtheorem{Lause}[Thm]{Lause}}
{\theorembodyfont{\normalfont} \newtheorem{Prop}[Thm]{Propositio}}
{\theorembodyfont{\normalfont} \newtheorem{Lem}[Thm]{Lemma}}
{\theorembodyfont{\normalfont} \newtheorem{Kor}[Thm]{Korollaari}}
{\theorembodyfont{\normalfont} \newtheorem{Def}[Thm]{M��ritelm�}}
{\theorembodyfont{\normalfont} \newtheorem{Sop}[Thm]{Sopimus}}
{\theorembodyfont{\normalfont} \newtheorem{Exa}{Esimerkki}}

\newcommand{\loppu}{\ensuremath{\quad \square}}
\newenvironment{Tod}{\underline{Todistus}\ \,}{\loppu}
\newenvironment{Ratk}{\underline{Ratkaisu}\ \,}{\loppu}
\newenvironment{Ratk*}{\underline{Ratkaisu}\ \,}{}
\newcommand{\ratk}{\underline{Ratkaisu}\ \,}
\newcommand{\tod}{\underline{Todistus}\ \,}

%lukujoukot
\newcommand{\N}{\ensuremath{\mathbb N}}
\newcommand{\Z}{\ensuremath{\mathbb Z}}
\newcommand{\Q}{\ensuremath{\mathbb Q}}
\newcommand{\R}{\ensuremath{\mathbb R}}
\newcommand{\C}{\ensuremath{\mathbb C}}
\newcommand{\K}{\ensuremath{\mathbb K}}
\newcommand{\A}{\ensuremath{\mathbb A}}
\newcommand{\G}{\ensuremath{\mathbb G}}

%itseisarvo ja normi
\newcommand{\abs}[1]{\vert #1 \vert}
\newcommand{\norm}[1]{\vert \! \vert #1 \vert \! \vert}

%matriisien ja vektoreiden lyhennysmerkkej�
\newcommand{\mA}{\ensuremath{{\boldsymbol{A}}}}
\newcommand{\mB}{\ensuremath{{\boldsymbol{B}}}}
\newcommand{\mC}{\ensuremath{{\boldsymbol{C}}}}
\newcommand{\mD}{\ensuremath{{\boldsymbol{D}}}}
\newcommand{\mE}{\ensuremath{{\boldsymbol{E}}}}
\newcommand{\mF}{\ensuremath{{\boldsymbol{F}}}}
\newcommand{\mI}{\ensuremath{{\boldsymbol{I}}}}
\newcommand{\mL}{\ensuremath{{\boldsymbol{L}}}}
\newcommand{\mU}{\ensuremath{{\boldsymbol{U}}}}
\newcommand{\mX}{\ensuremath{{\boldsymbol{X}}}}
\newcommand{\mH}{\ensuremath{{\boldsymbol{H}}}}
\newcommand{\mJ}{\ensuremath{{\boldsymbol{J}}}}
\newcommand{\mV}{\ensuremath{{\boldsymbol{V}}}}
\newcommand{\mR}{\ensuremath{{\boldsymbol{R}}}}

\newcommand{\va}{\ensuremath{{\boldsymbol{a}}}}
\newcommand{\vb}{\ensuremath{{\boldsymbol{b}}}}
\newcommand{\vc}{\ensuremath{{\boldsymbol{c}}}}
\newcommand{\vd}{\ensuremath{{\boldsymbol{d}}}}
\newcommand{\ve}{\ensuremath{{\boldsymbol{e}}}}
\newcommand{\vf}{\ensuremath{{\boldsymbol{f}}}}
\newcommand{\vg}{\ensuremath{{\boldsymbol{g}}}}
\newcommand{\vh}{\ensuremath{{\boldsymbol{h}}}}
\newcommand{\vi}{\ensuremath{{\boldsymbol{i}}}}
\newcommand{\vj}{\ensuremath{{\boldsymbol{j}}}}
\newcommand{\vk}{\ensuremath{{\boldsymbol{k}}}}
\newcommand{\vl}{\ensuremath{{\boldsymbol{l}}}}
\newcommand{\vm}{\ensuremath{{\boldsymbol{m}}}}
\newcommand{\vn}{\ensuremath{{\boldsymbol{n}}}}
\newcommand{\vo}{\ensuremath{{\boldsymbol{o}}}}
\newcommand{\vp}{\ensuremath{{\boldsymbol{p}}}}
\newcommand{\vq}{\ensuremath{{\boldsymbol{q}}}}
\newcommand{\vr}{\ensuremath{{\boldsymbol{r}}}}
\newcommand{\vs}{\ensuremath{{\boldsymbol{s}}}}
\newcommand{\vt}{\ensuremath{{\boldsymbol{t}}}}
\newcommand{\vu}{\ensuremath{{\boldsymbol{u}}}}
\newcommand{\vv}{\ensuremath{{\boldsymbol{v}}}}
\newcommand{\vw}{\ensuremath{{\boldsymbol{w}}}}
\newcommand{\vx}{\ensuremath{{\boldsymbol{x}}}}
\newcommand{\vy}{\ensuremath{{\boldsymbol{y}}}}
\newcommand{\vz}{\ensuremath{{\boldsymbol{z}}}}

\newcommand{\Nabla}{\ensuremath{{\boldsymbol{\nabla}}}}

%sis�tulo
\newcommand {\scp}[2]{\langle #1, #2 \rangle}

%kompleksiluvut
\newcommand{\pointer}{\ensuremath{\vec}}
\newcommand{\vkulma}[1]{\ensuremath{\angle \!
\raisebox{0.15cm}{$\underline{\phantom{\varphi} \negmedspace \! #1}$}}}
\newcommand{\re}{\ensuremath{\text{Re}\;}}
\newcommand{\im}{\ensuremath{\text{Im}\;}}

%trigonometristen k��nteisfunktioiden p��haarat, areafunktiot
\DeclareMathOperator{\Arcsin}{\overline{arc}sin}
\DeclareMathOperator{\Arccos}{\overline{arc}cos}
\DeclareMathOperator{\Arctan}{\overline{arc}tan}
\DeclareMathOperator{\Arccot}{\overline{arc}cot}
\DeclareMathOperator{\arsinh}{arsinh}
\DeclareMathOperator{\arcosh}{arcosh}
\DeclareMathOperator{\artanh}{artanh}

%sijoitusmerkki
\newcommand{\sijoitus}[2]{\operatornamewithlimits{\Bigl|}_{#1}^{#2}}

%sekalaista
\newcommand{\eps}{\ensuremath{\varepsilon}}