A diagram that describes the first-order approximation of the area of a rectangle by the jacobian determinant. A nonlinear map f:R2→R2{\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.
Own work
Blacklemon67