DetJ

detJ denotes the determinant of the Jacobian matrix J of a differentiable map F: R^n -> R^n. The Jacobian J(x) is the n-by-n matrix whose entries are the partial derivatives J_{ij} = ∂F_i/∂x_j. The determinant detJ(x) captures the local linear behavior of F: its absolute value gives the local volume (or area in 2D) scaling induced by F near x, and its sign indicates whether the map preserves or reverses orientation.

In change of variables, detJ appears in the Jacobian factor that adjusts measure when transforming integrals between coordinate systems. If y = F(x) and F is differentiable with invertible derivative at points, then the change of variables formula states that the volume element transforms as dy = |detJ(x)| dx. Consequently, integrals follow ∫_{F(U)} g(y) dy = ∫_{U} g(F(x)) |detJ(x)| dx.

Key properties include that detJ is the determinant of the derivative map, and thus det(D(g∘f))(x) = det(Dg|_{f(x)}) det(Df|_x) under composition. A point where detJ(x) = 0 indicates a locally singular map and non-invertibility; the inverse function theorem ensures local invertibility when detJ(x) ≠ 0.

Common applications span multivariable calculus, differential geometry, and numerical methods. In finite element analysis, for example, detJ is used to relate volumes of mapped elements to a reference element and to ensure element invertibility. In summary, detJ is a scalar invariant that encodes local stretching, orientation, and invertibility of a differentiable transformation.