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
In calculus, the Jacobian determinant appears in the change of variables for integration. When transforming variables
For invertibility, det J_F(x0) ≠ 0 implies, by the inverse function theorem, that F is locally invertible
Examples help intuition. A rotation in the plane has J equal to the rotation matrix, whose determinant