Change of Variables in Double Integrals
From single-variable substitution to the Jacobian determinant — the art of re-coordinatizing the plane
1. The Single-Variable Ancestor: Substitution Recalled
1.1 The One-Variable Rule
You already know the substitution rule for a definite integral. If $x = g(u)$ is a differentiable, one-to-one function mapping $[c,d]$ onto $[a,b]$, then
$$ \int_a^b f(x)\,dx \;=\; \int_c^d f\!\bigl(g(u)\bigr)\,\bigl|g'(u)\bigr|\,du. $$Two things happen simultaneously:
- The limits transform — $a$ and $b$ are replaced by their pre-images $c$ and $d$.
- The derivative factor is paid — the factor $|g'(u)|$ compensates for the stretching or compressing that $g$ does to the $u$-axis.
1.2 The Answer: The Jacobian Determinant
The one-dimensional derivative $g'(u)$ measures how much a tiny interval $du$ is stretched into $dx$. In two dimensions, a mapping $T(u,v) = \bigl(x(u,v),\, y(u,v)\bigr)$ stretches tiny rectangles into tiny parallelograms. The factor that measures this local area magnification is the absolute value of the Jacobian determinant:
$$ \left|\frac{\partial(x,y)}{\partial(u,v)}\right| \;=\; \left|\det\begin{pmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\[6pt] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{pmatrix}\right| \;=\; \left|\frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\right|. $$This is the Jacobian — the machinery we built when we studied local invertibility and area magnification, now installed in the integral formula.