\({\displaystyle \iint f(x,y)dxdy=\iint f(\phi (u,v),\psi (u,v))|J|dudv}\)
ここで
\({\displaystyle J={\frac {\partial (x,y)}{\partial (u,v)}}=\det \left({\begin{matrix}{\frac {\partial {x}}{\partial {u}}}&{\frac {\partial {x}}{\partial {v}}}\\{\frac {\partial {y}}{\partial {u}}}&{\frac {\partial {y}}{\partial {v}}}\end{matrix}}\right)}\)
これは形式的に \(dxdy=|J|dudv\) と書ける。