DescriptionJacobian determinant and distortion.svg
English: A diagram that describes the first-order approximation of the area of a rectangle by the jacobian determinant. A nonlinear map {\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2}} sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.
Deutsch: Eine nichtlineare Abbildung {\displaystyle {\vec {f}}\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2}} transformiert ein Rechteck (links, in rot) zu einer verzerrten Fläche (rechts, in rot). Die Jacobi-Matrix in einem Punkt ist die beste lineare Approximation der Funktion um diesen Punkt herum {\displaystyle {\vec {f}}({\vec {x}})\approx {\vec {f}}({\vec {x}}_{0})+Df({\vec {x}}_{0})({\vec {x}}-{\vec {x}}_{0}))}: Das originale Rechteck würde durch die Jacobi-Matrix in das weiße durchsichtige Parallelogramm überführt. Die Funktionaldeterminante ist das Verhältnis der Flächen des approximierenden Parallelogramms und dem originalen Rechteck.
Українська: Нелінійне відображення {\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2}} перетворює невеликий квадрат (червоний, ліворуч) до деформованого паралелограма (червоний, праворуч).
Якобіан у точці дає найкраще лінійне наближення деформованого паралелограма в околі цієї точки (напівпрозорий білий, праворуч), а визначник Якобі дає відношення площі апроксимаційного паралелограма до площі заданого квадрата.
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