JacobianDeterminant

The Jacobian determinant, usually denoted det J or det(Df), is a scalar quantity associated with a differentiable map F from one Euclidean space to another. If F maps R^n to R^n and can be written as F(x) = (f1(x), f2(x), ..., fn(x)), the Jacobian matrix J(F)(x) is the n×n matrix of all first-order partial derivatives, with entries ∂fi/∂xj. The Jacobian determinant is the determinant of this matrix: det J(F)(x) = det( [∂fi/∂xj] ).

Interpretation and properties: The Jacobian determinant measures the local scaling factor of volume (or area in two dimensions) under the map F near a point x. In particular, for small regions around x, their image under F has volume approximately |det J(F)(x)| times the original volume, and the sign of det J(F)(x) indicates the local orientation of the mapping. If det J(F)(x) ≠ 0, the inverse function theorem guarantees that F is locally invertible near x, with a differentiable inverse.

Applications: The Jacobian determinant plays a central role in change of variables for multiple integrals. When transforming variables from x to u = F(x), the differential volume element transforms as dx1 dx2 ... dxn = |det J(F)(x)| du1 du2 ... dun. This factor ensures correct accounting for stretching or compression under the transformation and is essential in evaluating integrals after a substitution.

Example: For the mapping defined by u = x^2 − y^2, v = 2xy, the Jacobian matrix is [[2x, −2y], [2y, 2x]] and its determinant is 4(x^2 + y^2). The determinant vanishes at the origin, indicating a failure of local invertibility there.

In summary, the Jacobian determinant encodes local geometric information about a differentiable map and underpins techniques for variable changes in analysis and geometry.