Change of Variables and the Jacobian

A change of variables transforms an integral into a more convenient coordinate system.

General formula — If $T : (u,v) \mapsto (x,y)$ maps $S$ to $R$:

$$\iint_R f(x,y)\,dA = \iint_S f(x(u,v),y(u,v))\;|J|\;du\,dv$$

where $J$ is the Jacobian determinant:

$$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$$

Common Jacobians

• Polar $(r,\theta)$: $|J| = r$
• Cylindrical $(r,\theta,z)$: $|J| = r$
• Spherical $(\rho,\phi,\theta)$: $|J| = \rho^{2}\sin\phi$

The Jacobian measures how the transformation scales area (or volume).

Key insight: The Jacobian generalises the substitution rule $dx = g'(u)\,du$ to multiple variables.

Common pitfall: Changing variables without the Jacobian silently rescales all areas and volumes: polar needs $r$, spherical needs $\rho^{2}\sin\varphi$. If your disk integral came out as $2\pi R$ instead of $\pi R^{2}$, the missing $r$ is the culprit.