Change of variables and Jacobian determinants

Near a point u, a differentiable map T behaves to first order like its derivative DT(u). A small coordinate box therefore becomes approximately a parallelotope whose edge vectors are the derivative's columns multiplied by the corresponding coordinate widths. The absolute determinant of that derivative is its volume scale. The determinant's sign records whether orientation is preserved or reversed; ordinary unsigned volume uses the magnitude. A negative determinant does not mean negative area, although it matters in oriented integration of differential forms.

For a standard sufficient version of the substitution theorem, take a C¹ diffeomorphism between open subsets of equal-dimensional Euclidean space, and integrate a nonnegative measurable function or an absolutely integrable function. The transformed integrand is the original function composed with T, multiplied by the absolute Jacobian determinant. Restricting to a measurable subregion gives its transformed version. Many familiar coordinate systems have exceptional boundaries or axes where they are not diffeomorphisms; those sets can be excluded when they have measure zero and the remaining hypotheses hold.

In the plane, T(r,θ)=(r cos θ,r sin θ). Its derivative columns point in the radial and angular directions. Their lengths are one and r, and they are perpendicular, giving determinant r. Thus a small angular interval has physical length approximately r times its angle width. Using an angular interval of length one full turn covers an annulus once except for a seam; using two full turns covers almost every point twice. The factor r corrects local scale but does not correct this global double coverage by itself.

For a disk, the origin and the chosen angular seam cause failures of the simplest diffeomorphism description. They have area zero, so one may work on the punctured disk with a seam removed and then restore those null sets. This explanation is more precise than claiming polar coordinates are globally one-to-one on the entire closed rectangle of radius and angle. If an integrand is singular at the origin, nullity of the point does not guarantee finite mass nearby; its transformed radial behavior must still be checked for integrability.

A substitution is valuable when it simplifies a region, an integrand, or both. For an ellipse, scaling a unit disk by its semiaxes converts a curved inequality into a round one and contributes a constant area factor. For a linear shear, the image can be visually slanted while its area remains unchanged when the determinant is one. This is why judging scale from the longest visible edge is unreliable: stretching one direction, shrinking another, and shear combine through the full determinant.

A reliable calculation writes the coordinate map, its valid region, the transformed bounds, and its determinant before integrating. Check dimensions and multiplicity separately. A map with a vanishing determinant cannot satisfy the ordinary diffeomorphism theorem at that point, but may still be usable after partitioning into regular regions or applying a more general result. The theorem is not permission to discard singular sets without analysis. In higher dimensions the same principle gives cylindrical and spherical volume factors, derived from their own Jacobian matrices.