Jacobian Matrix
A matrix of all first-order partial derivatives of a vector-valued multivariable function, representing local linear coordinate transformations.
Governing formula
J_ij = ∂f_i / ∂x_j, dV' = |det J| dV
SI unit
Coordinate Transformation Factor
In depth
The absolute determinant |det J| represents the local volume scaling factor during multivariable coordinate transformations (e.g. Cartesian to spherical coordinates).
Examples in the real world
Transforming differential integration volume elements dV = dx dy dz = r² sin θ dr dθ dφ using the spherical coordinate Jacobian.