The Jacobian
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 5.7: “Change of Variables in Multiple Integrals” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/5-7-change-of-variables-in-multiple-integrals |
| Textbook used in class | Stewart, Calculus, Section 15.10: “Change of Variables in Multiple Integrals” (Examples 1, 2) |
Quick Reference
Jacobian of the transformation $x = g(u,v)$, $y = h(u,v)$: $$\frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix}\partial x/\partial u & \partial x/\partial v \\ \partial y/\partial u & \partial y/\partial v\end{vmatrix} = \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}.$$
Change of area element: $$dA = \left|\frac{\partial(x,y)}{\partial(u,v)}\right|\,du\,dv.$$
For polar: $x = r\cos\theta$, $y = r\sin\theta$ gives Jacobian $= r$, so $dA = r\,dr\,d\theta$.
Motivation
Substitution in a single integral changes the variable: $\int f(x)\,dx = \int f(g(u))g'(u)\,du$. The factor $g'(u)$ adjusts for how the transformation stretches or compresses the interval. In two (or three) dimensions, the Jacobian determinant plays the same role: it measures how much the transformation scales areas (or volumes) locally. Without this correction factor, a change of variables would give the wrong answer.
Key Concepts
1. What the Jacobian Measures
The Jacobian $\partial(x,y)/\partial(u,v)$ is the (signed) ratio of area element $dA_{xy}$ to $dA_{uv}$. If the Jacobian is 2 at a point, then a tiny region in $(u,v)$-space maps to a region twice as large in $(x,y)$-space.
Taking the absolute value accounts for orientation: the transformation might reverse orientation (negative determinant), but areas are always positive.
2. Jacobian for 3D Transformations
For $x = x(u,v,w)$, $y = y(u,v,w)$, $z = z(u,v,w)$, the Jacobian is the $3\times 3$ determinant: $$\frac{\partial(x,y,z)}{\partial(u,v,w)} = \begin{vmatrix}x_u & x_v & x_w \\ y_u & y_v & y_w \\ z_u & z_v & z_w\end{vmatrix}.$$
For spherical coordinates: $\partial(x,y,z)/\partial(\rho,\phi,\theta) = \rho^2\sin\phi$, which gives $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$.
Worked Example
Compute the Jacobian for the polar transformation $x = r\cos\theta$, $y = r\sin\theta$. (Stewart 15.10, Example 2.)
$$\frac{\partial(x,y)}{\partial(r,\theta)} = \begin{vmatrix}\partial x/\partial r & \partial x/\partial\theta \\ \partial y/\partial r & \partial y/\partial\theta\end{vmatrix} = \begin{vmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{vmatrix}.$$
$$= \cos\theta(r\cos\theta) - (-r\sin\theta)(\sin\theta) = r\cos^2\theta + r\sin^2\theta = r.$$
So $|J| = r$, confirming $dA = r\,dr\,d\theta$. The Jacobian for polar coordinates is simply $r$.
the Jacobian is always 1 for “simple” transformations. The Jacobian is 1 only for transformations that preserve area (like rotations and reflections). Any transformation that stretches, compresses, or shears the plane has a Jacobian different from 1 in magnitude. Even a simple scaling $x = au$, $y = bv$ has Jacobian $ab \neq 1$ (unless $ab = 1$).
Leveled Practice
Problem 1. Find the Jacobian of $x = u^2 - v^2$, $y = 2uv$.
Show answer
$$J = \begin{vmatrix}2u & -2v \\ 2v & 2u\end{vmatrix} = 4u^2 + 4v^2 = 4(u^2+v^2).$$
$dA = 4(u^2+v^2)\,du\,dv$.