The Jacobian determinant
An integral adds up little pieces. Change the coordinates you measure them in and every piece gets resized — and in two dimensions the resize factor is a single number: the determinant of the Jacobian. Deform a grid, drag a probe, and watch the determinant be the local area scale that makes substitution exact.
The one-dimensional ancestor
Substitution is a local length scale
Substitution in one variable is the whole idea in miniature. If $x = g(u)$ reparameterises the line, then a small step $du$ becomes a step $g'(u)\,du$ in $x$: it stretches by the local derivative. Writing the integral in the new variable, that stretch factor appears exactly once,
and it is what makes the two integrals count the same quantity. The factor $g'(u)$ is a local length scale — the ratio of a tiny new length to a tiny old one.
Two dimensions are the same story with a bigger bookkeeping problem. Our running example is a two-link arm: its joint angles $(\theta_1,\theta_2)$ live in one plane, the hand position $(x,y)$ lives in another, and the forward-kinematics map $T$ carries one to the other. A tiny square of joint angles lands as a tiny curved parallelogram in the workspace. Comparing their areas needs not one number but a matrix of derivatives — and the area ratio turns out to be that matrix's determinant. Pick a map below and slide the morph; the faint square is the original grid, the coloured grid is its image.
The same $(u,v)$ grid under four maps. The faint dashed grid is the identity; the coloured grid is the morph between identity and the chosen map.
The Jacobian is the local area scale
A matrix where the 1D case had a number
Near a point, every smooth map is well approximated by its derivative. Linearise $T(u,v) = (x(u,v), y(u,v))$ at $(u,v)$:
The image of a little square of sides $du$ and $dv$ is a parallelogram whose edge vectors are the columns of $J$ scaled by $du$ and $dv$. Its area is $|\det J|\,du\,dv$. So $\det J$ is exactly the $g'(u)$ of two variables: the ratio of a mapped cell's area to the original cell's area, at that point.
Shade the small grid below by $|\det J|$, then move the probe. The readout shows the Jacobian matrix, its determinant, and the actual area of the mapped cell next to the area of the source cell — they agree.
A coarse $(u,v)$ grid mapped by the chosen map, cells shaded by $|\det J|$ (dark = large area scale). The outlined cell contains the probe.
Change of variables in two dimensions
Making substitution exact
Because every small cell is resized by $|\det J|$, integrating in the new coordinates just means carrying that factor along:
where $S$ is the region in the $(u,v)$-plane that $T$ sends onto $D$. The absolute value keeps areas positive; the determinant's sign is handled separately in Step 5.
The demo integrates the same function two ways: directly over the parallelogram $D$ in the $x$–$y$ plane, and over the source square in $(u,v)$ with the $|\det J|$ weight. Raise the resolution and watch the two sums converge to the same number — that agreement is the formula.
The source square mapped by a shear into the parallelogram $D$. The sampled quads are the pieces the transformed sum adds up.
Polar: where the $r$ comes from
$dA = r\,dr\,d\theta$
For polar coordinates $T(r,\theta) = (r\cos\theta,\; r\sin\theta)$ the Jacobian is
The factor $r$ is not decoration. A thin annulus of width $dr$ at radius $r$, spanning an angle $d\theta$, has arc length $r\,d\theta$ and therefore area $r\,dr\,d\theta$. The same angular slice sweeps a longer arc the farther out it sits, so the area element itself grows with $r$. Move the probe radius and read the factor; then compare a disc's area computed with the $r$ and without it.
Concentric circles and radial spokes. The shaded ring is the annulus at the probe radius; its area is $2\pi r\,dr$.
Sign and orientation
Negative determinant is a mirror
A determinant can be negative, and the sign is information: it records whether the map preserves or reverses orientation. When $\det J < 0$, the corners of a cell are traversed in the opposite rotational sense — the picture has been reflected, like a page turned over. Areas themselves stay positive, which is precisely why the change-of-variables formula uses $|\det J|$: the magnitude is the area scale, the sign only says which way round the image sits.
The map $(u,v)\mapsto(a u,\,v)$. The highlighted cell's corners are numbered in order; when $a<0$ the traversal reverses.
Where this shows up
The determinant as the price of changing coordinates
Every time a problem is easier in different coordinates, this factor is the toll. In Multi-view geometry, camera and world frames are related by coordinate transforms, and a 3D density (or a bundle of rays) must be rescaled by the Jacobian determinant when it is pushed between them — the same $|\det J|$ that makes the integral invariant. The factor returns for higher-dimensional regions in Part 6: Multiple integrals & coordinates, where cylindrical and spherical volume elements are exactly $\det J$. And when a random variable is pushed through a map, its probability density is divided by $|\det J|$ — the change-of-variables rule for densities, developed in Part 11: Calculus of probability.
Notation to carry forward
| Notation | Reads as | Where it comes up |
|---|---|---|
g'(u) du | The 1D length scale: a small new step stretched into the old variable | Substitution in Volume I |
T(u,v) | The coordinate map — joints to workspace, or any reparameterisation | This part; forward kinematics |
J = ∂(x,y)/∂(u,v) | The Jacobian matrix of first partial derivatives of the map | This part; Parts 8, 13 |
det J | The local area scale, signed: magnitude resizes, sign orients | Cell-area ratio; densities |
|det J| du dv | The mapped area element — the exact substitution factor | The 2D change-of-variables formula |
r dr dθ | The polar area element; the r is $\det J$ for polar coordinates | Part 6; any radial integral |
det J < 0 | Orientation reversed — a reflection, with areas still counted positive | Sign and orientation |
Further reading
- 3Blue1Brown, The Jacobian matrix and determinant — the same local area story, animated.
- MIT 18.02SC, Multivariable Calculus — change of variables and the Jacobian in full rigour.
- Khan Academy, Integrating multivariable functions — extra practice on double integrals and substitution.