The Jacobian determinant is the scaling factor that relates an element of area (or volume) in one coordinate system to the corresponding element of area in a transformed coordinate system when performing a change of variables in multiple integration. For a mapping (x, y) = f(u, v) that is one-to-one and continuously differentiable, an infinitesimal region ΔA in the original plane is approximately a parallelogram whose area equals the magnitude of the cross product of the partial-derivative vectors of x and y with respect to u and v; this magnitude is the absolute value of the determinant of the Jacobian matrix, ∂(x,y)/∂(u,v). This belongs to multivariable calculus, specifically the theory of change of variables in double (and multiple) integrals, generalizing single-variable u-substitution (where the one-dimensional "Jacobian" is simply dx/du) to higher dimensions.
The Jacobian determinant is the scaling factor that relates an element of area (or volume) in one coordinate system to the corresponding element of area in a transformed coordinate system when perfor…