Jacobian determinant

The Jacobian determinant is the determinant of the matrix of first partial derivatives of a multivariable map. It is the local volume-scaling factor: zoom in close enough to a point and a smooth nonlinear map looks like a linear one (its derivative), and the determinant of that linear approximation tells you how much the map stretches or shrinks tiny volumes near that point — sign included for whether it flips orientation.

For a map F from R^n to R^n with components F_1, ..., F_n, the Jacobian MATRIX is J = [ partial F_i / partial x_j ], and the Jacobian determinant is det(J), often written det(DF) or the symbol with bars around partials. It varies from point to point because the linear approximation changes as you move; it is exactly the determinant-as-volume idea applied to the best linear model of F at each location.

It is the determinant theory meeting calculus, and its headline role is the change-of-variables formula for integrals: when you substitute new coordinates, the integrand picks up a factor |det(J)| because that is precisely how the substitution rescales the volume element dV. The Inverse Function Theorem likewise hinges on it — F is locally invertible near a point exactly when its Jacobian determinant is nonzero there, echoing 'det != 0 means invertible' from the linear world.

Caveat: use the ABSOLUTE value of the Jacobian in the change-of-variables integral (volumes are positive); the sign of det(J) separately records orientation and matters in oriented integration theory. And a nonzero Jacobian at a point gives only LOCAL invertibility, not global — a map can be locally one-to-one everywhere yet wrap around globally (polar coordinates are the standard cautionary example).