Change of Variables and the Jacobian Determinant

Core Idea

For transformation (u, v) = T(x, y), the Jacobian J = ∂(x, y)/∂(u, v) = det([∂x/∂u, ∂x/∂v; ∂y/∂u, ∂y/∂v]) scales area. Thus ∬_D f(x, y) dA = ∬_S f(x(u, v), y(u, v)) |J| du dv. Cylindrical and spherical coordinates are special cases.

Explainer

From your work with cylindrical and spherical coordinates, you have already used the change-of-variables idea in specific cases: in polar coordinates, dA becomes r dr dθ, and in spherical coordinates, dV becomes ρ² sin φ dρ dφ dθ. The extra factors r and ρ² sin φ are not magic — they measure how much area or volume is stretched or compressed by the coordinate transformation. The Jacobian determinant is the general tool that computes this stretching factor for any smooth change of coordinates.

Think about what a coordinate transformation does locally. Near any point, a smooth map T(u, v) = (x(u, v), y(u, v)) looks approximately linear. A small rectangle of area du dv in (u, v)-space gets mapped to a small parallelogram in (x, y)-space. The area of that parallelogram is |J| du dv, where J is the determinant of the 2×2 matrix of partial derivatives: J = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u). This matrix — the Jacobian matrix of the transformation — encodes the local linear approximation, and its determinant encodes the signed area scaling factor. Taking the absolute value |J| gives the unsigned area ratio, which is what you need to correctly account for how area changes under the map.

The change-of-variables formula is then: ∬_D f(x, y) dA = ∬_S f(x(u,v), y(u,v)) |J(u,v)| du dv, where D is the region in (x, y)-space, S is the corresponding region in (u, v)-space, and f is expressed in the new coordinates. You choose the transformation to simplify either the region S or the integrand — ideally both. For polar coordinates, x = r cos θ, y = r sin θ, and computing the Jacobian gives J = r, recovering the familiar factor. The formula is not a separate rule for polar coordinates; polar coordinates are simply one instance of the general theorem.

In three dimensions, the Jacobian becomes a 3×3 determinant and |J| du dv dw replaces dA. For cylindrical coordinates (x = r cos θ, y = r sin θ, z = z), J = r. For spherical (x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ), J = ρ² sin φ. Both results you used in your work with triple integrals are now derivable from first principles rather than accepted as formulas. The deeper principle: whenever a region or integrand is naturally described in some non-Cartesian coordinate system, compute the Jacobian of the transformation and substitute — the geometry will simplify, even if the algebra of computing J takes some effort.