Jacobian Change of Variables

Why a stretch factor is needed

In one dimension, u-substitution carries a derivative: ∫ f(g(u)) g'(u) du. The factor g'(u) rescales the differential — if g stretches the line by 2× near a point, du there represents 2× as much "length" as it would un-mapped.

The same thing happens in higher dimensions, only now the rescaling is a matrix and the determinant is the volume scale. A unit square in the (u,v)-plane gets mapped to some parallelogram in the (x,y)-plane. The area of that parallelogram is the absolute value of the determinant of the matrix of partial derivatives — the Jacobian determinant.

J = ∂(x,y)/∂(u,v) = det [[∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v]].

The change-of-variables formula then reads

∬_R f(x,y) dx dy = ∬_S f(x(u,v), y(u,v)) |J| du dv

where R is the original region and S is its preimage in (u,v) space.

Worked derivation: polar coordinates

The polar transformation is x = r cosθ, y = r sinθ. Compute partials:

• ∂x/∂r = cosθ, ∂x/∂θ = −r sinθ
• ∂y/∂r = sinθ, ∂y/∂θ = r cosθ

The Jacobian matrix is [[cosθ, −r sinθ], [sinθ, r cosθ]], with determinant

r cos²θ + r sin²θ = r.

So dx dy = r dr dθ. The geometric picture: a thin polar wedge at radius r with width dr and angle dθ has arc length r dθ, so its area is exactly r dr dθ. The classic example is

∫_{−∞}^∞ e^{−x²} dx = √π,

which is impossible in Cartesian coordinates but trivial in polar:

(∫_{−∞}^∞ e^{−x²} dx)² = ∬ e^{−x²−y²} dx dy = ∫₀^{2π} ∫₀^∞ e^{−r²} r dr dθ = π.

The factor of r is what made the radial integral elementary.

Worked derivation: spherical coordinates

Spherical: x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ. The 3×3 Jacobian matrix has rows (∂/∂ρ, ∂/∂φ, ∂/∂θ) of (x, y, z). Working through the determinant — or expanding along the bottom row — gives

|J| = ρ² sinφ.

Geometrically, a small spherical box with sides dρ, ρ dφ, ρ sinφ dθ has volume ρ² sinφ dρ dφ dθ. The ρ² accounts for the fact that the surface area of a sphere of radius ρ scales as ρ²; the sinφ accounts for circumferences of latitude lines shrinking near the poles.

Common changes of variables

Transform	Equations	|J|	Best for
Polar (2D)	x = r cosθ, y = r sinθ	r	Disks, annuli, e^{−x²−y²}
Cylindrical (3D)	x = r cosθ, y = r sinθ, z = z	r	Cylinders, paraboloids
Spherical (3D)	x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ	ρ² sinφ	Balls, shells, central forces
Linear x = Au	x = a u + b v, y = c u + d v	|ad − bc|	Tilted parallelograms, ellipses
Elliptic (a, b axes)	x = a r cosθ, y = b r sinθ	ab r	Ellipses (x/a)² + (y/b)² ≤ 1
Sum/difference	u = x+y, v = x−y	1/2	Symmetric integrands in x±y
Hyperbolic	x = u v, y = u/v	2u/v	xy = const, y = const · x

Worked example: a linear change of variables

To compute the area of an ellipse (x/a)² + (y/b)² ≤ 1, set x = au, y = bv. The map is linear with matrix diag(a,b), so |J| = ab. The new region is the unit disk, area π. So

Area = ∬_{disk} ab du dv = ab · π = πab.

For a more general linear map x = Au + b, the Jacobian is just |det A| — the determinant measures how much the linear map scales volume, exactly as it does for any matrix.

When to use the inverse direction

Sometimes the integrand is naturally written in (u,v) but the region is defined in (x,y) — or vice versa. In that case you need the Jacobian of the inverse map. Two facts make this easy:

• The Jacobian of the inverse is the reciprocal: ∂(u,v)/∂(x,y) = 1 / [∂(x,y)/∂(u,v)].
• You can compute either one and use whichever is convenient.

For example, if u = x + y, v = x − y, then ∂(u,v)/∂(x,y) = det[[1,1],[1,−1]] = −2, so |∂(u,v)/∂(x,y)| = 2 and |∂(x,y)/∂(u,v)| = 1/2.

Where the Jacobian shows up

• Probability transforms. If Y = g(X) and X has density f_X, then f_Y(y) = f_X(g⁻¹(y)) |J| where J is the Jacobian of g⁻¹. The same formula governs the change-of-variables in joint distributions.
• Statistical mechanics. Phase-space integrals over (q, p) coordinates use Jacobians under canonical transformations — Liouville's theorem says |J| = 1 for them.
• Computer graphics. Texture mapping and surface parametrizations use Jacobians to compute how a unit area on the texture stretches across the model.
• General relativity. The volume element in curved space is √|g| d⁴x where g is the metric determinant — a generalised Jacobian.
• Optimization. Newton's method and other root finders for vector-valued equations rely on the Jacobian matrix at each step.

Common mistakes

• Forgetting the absolute value. A negative determinant just means the map flips orientation. Volume is positive, so we always use |J|. Without the absolute value, you can get negative areas and wrong answers.
• Computing the wrong direction. If you wrote x and y in terms of u and v, then J = ∂(x,y)/∂(u,v) and the formula is dx dy = |J| du dv. If you wrote u and v in terms of x and y, take the reciprocal.
• Skipping the Jacobian in spherical or cylindrical. "I'll convert to spherical, dV = dρ dφ dθ" is a fast way to wrong answers. The factor ρ² sinφ (or r) is essential.
• Forgetting to transform the region. A disk in (x,y) becomes a rectangle in (r,θ) — the bounds change with the variables. Always map the region too.
• Using the wrong sign convention. Some textbooks define the Jacobian without an absolute value, then handle sign through orientation. For ordinary (unsigned) integrals, always take |J|.