Change of Variables and the Jacobian

Change of Variables & The Jacobian

A change of variables transforms a hard integral into a simpler coordinate system, generalising single-variable substitution $dx = g'(u)du$ to multiple dimensions.

If transformation $T: (u,v) \mapsto (x,y)$ maps region $S$ to $R$, the integral becomes:

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

Here, $J$ is the Jacobian determinant, which measures how the transformation locally scales area or volume.

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

Standard Jacobians & Pitfalls

Certain coordinate systems appear frequently. Memorise their scaling factors:

Common pitfall: Forgetting the absolute value $|J|$ or omitting the scaling factor entirely. If a disk integral returns $2\pi R$ instead of $\pi R^2$, a missing polar $r$ is always the culprit.