JacobianDeterminante

Jacobian determinant, sometimes referred to as JacobianDeterminante, is the determinant of the Jacobian matrix of a differentiable map. For a map F: R^n → R^n with components F1, ..., Fn, the Jacobian matrix J_F(x) has entries ∂Fi/∂xj. The scalar det J_F(x) is the Jacobian determinant at x.

It measures local volume distortion under the map. Near a point x, F expands or contracts n-dimensional volume by approximately |det J_F(x)|. A negative determinant indicates a local reversal of orientation.

In calculus, the Jacobian determinant appears in the change of variables for integration. When transforming variables from x = (x1, ..., xn) to u = (u1, ..., un) via F, the differential volume elements relate by dx = |det J_F(u)| du. Consequently, integrals transform as ∫∫_D g(x) dx = ∫∫_{D̃} g(F(u)) |det J_F(u)| du, where D̃ is the preimage region in the u-coordinates.

For invertibility, det J_F(x0) ≠ 0 implies, by the inverse function theorem, that F is locally invertible and a local diffeomorphism at x0. If det J_F(x0) = 0, the map has a singular point, and local invertibility may fail.

Examples help intuition. A rotation in the plane has J equal to the rotation matrix, whose determinant is 1. A uniform scaling F(x) = λx in R^2 has det J = λ^2. For a general linear map represented by a 2×2 matrix A, the Jacobian determinant is det A, equal to the area distortion factor of the map. In broader contexts, the concept generalizes to derivatives between manifolds, where the determinant reflects the induced volume change on oriented tangent spaces.