微分形式
d y 1 ∧ d y 2 = ( ∂ y 1 ∂ x 1 d x 1 + ∂ y 1 ∂ x 2 d x 2 ) ∧ ( ∂ y 2 ∂ x 1 d x 1 + ∂ y 2 ∂ x 2 d x 2 ) = ∂ y 1 ∂ x 1 ∂ y 2 ∂ x 1 d x 1 ∧ d x 1 + ∂ y 1 ∂ x 1 ∂ y 2 ∂ x 2 d x 1 ∧ d x 2 + ∂ y 1 ∂ x 2 ∂ y 2 ∂ x 1 d x 2 ∧ d x 1 + ∂ y 1 ∂ x 2 ∂ y 2 ∂ x 2 d x 2 ∧ d x 2 = ∂ y 1 ∂ x 1 ∂ y 2 ∂ x 2 d x 1 ∧ d x 2 + ∂ y 1 ∂ x 2 ∂ y 2 ∂ x 1 d x 2 ∧ d x 1 = ( ∂ y 1 ∂ x 1 ∂ y 2 ∂ x 2 − ∂ y 1 ∂ x 2 ∂ y 2 ∂ x 1 ) d x 1 ∧ d x 2 = ∂ ( y 1 , y 2 ) ∂ ( x 1 , x 2 ) d x 1 ∧ d x 2 {\displaystyle {\begin{aligned}\mathrm {d} y_{1}\wedge \mathrm {d} y_{2}&=\left({\frac {\partial y_{1}}{\partial x_{1}}}\,\mathrm {d} x_{1}+{\frac {\partial y_{1}}{\partial x_{2}}}\,\mathrm {d} x_{2}\right)\wedge \left({\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{1}+{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{2}\right)\\&={\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{1}+{\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\\&\qquad {}+{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{2}\wedge \mathrm {d} x_{1}+{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{2}\wedge \mathrm {d} x_{2}\\&={\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{2}}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}+{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{1}}}\,\mathrm {d} x_{2}\wedge \mathrm {d} x_{1}\\&=\left({\frac {\partial y_{1}}{\partial x_{1}}}{\frac {\partial y_{2}}{\partial x_{2}}}-{\frac {\partial y_{1}}{\partial x_{2}}}{\frac {\partial y_{2}}{\partial x_{1}}}\right)\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\\&={\frac {\partial (y_{1},y_{2})}{\partial (x_{1},x_{2})}}\,\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\end{aligned}}}