The Jacobian and Change of Variables

Every physicist meets the same small miracle early on: a stubborn Cartesian integral, hopeless as it stands, collapses to a one-line answer the moment you switch to polar or spherical coordinates. The engine that makes the switch legal is the Jacobian determinant, easy to treat as a bookkeeping factor — an $r$ here, an $r^2\sin\theta$ there — but that hides what it is. A smooth change of variables $\mathbf{x}=\mathbf{x}(\mathbf{u})$ warps space, and near any point that warping is, to first order, a single linear map: the derivative matrix $J$ . A linear map carries the unit cube to a parallelepiped whose (signed) volume is $\det J$ , so an infinitesimal box $\mathrm{d}^n u$ in the new coordinates is stretched into a box of volume $|\det J|\,\mathrm{d}^n u$ in the old — and that single factor is the entire content of the change-of-variables theorem. This article rebuilds the rule from the geometry up. We prove that $|\det J|$ is the local volume-scaling factor, derive the polar, cylindrical, and spherical elements as instances of one formula, and use the polar Jacobian to crack the Gaussian integral $\int e^{-x^2}\,\mathrm{d}x=\sqrt{\pi}$ — the calculation that quietly underlies every normalized bell curve in physics. The same determinant reappears in the metric tensor, where $|\det J|=\sqrt{\det g}$ is the invariant volume element $\sqrt{g}\,\mathrm{d}^n u$ of curvilinear calculus and, with a sign, the $\sqrt{-g}$ of general relativity. Three commandments run through all of it: always take the absolute value, always rewrite the integrand in the new variables, and always transform the region as well as the measure.

Key takeaways

A smooth coordinate map $\mathbf{x}=\mathbf{x}(\mathbf{u})$ deforms space; the local stretching of volume is the determinant of the Jacobian matrix $J_{ij}=\partial x_i/\partial u_j$ .
The rule is $\mathrm{d}^n x = |\det J|\,\mathrm{d}^n u$ , so $\int_R f\,\mathrm{d}^n x = \int_{R'} f(\mathbf{x}(\mathbf{u}))\,|\det J|\,\mathrm{d}^n u$ — substitute $f$ , insert the factor, transform the region.
The absolute value is essential: $|\det J|$ is a volume factor (never negative), while the sign of $\det J$ records only orientation, and matters only for oriented (form) integrals.
The standard elements fall straight out: plane polar gives $r$ , cylindrical gives $\rho$ , spherical gives $r^2\sin\theta$ ; a vanishing Jacobian flags a coordinate singularity such as the polar origin or the spherical poles.
Inverse maps have reciprocal Jacobians, $\det(\partial x/\partial u)\,\det(\partial u/\partial x)=1$ , and composed maps multiply, exactly as the chain rule demands.
The same determinant is the invariant volume element of curved coordinates: $|\det J|=\sqrt{\det g}$ , the $\sqrt{g}$ of curvilinear calculus and the $\sqrt{-g}$ of relativity.
Key idea: a differentiable coordinate map is locally linear; its derivative matrix $J$ sends a unit cell to a parallelepiped of volume $\det J$ , so volume elements transform as $\mathrm{d}^n x = |\det J|\,\mathrm{d}^n u$ .

The core insight

A differentiable coordinate map is locally linear; its derivative matrix $J$ sends a unit cell to a parallelepiped of (signed) volume $\det J$ , so volume elements transform as $\mathrm{d}^n x = |\det J|\,\mathrm{d}^n u$ , and this one factor is the whole of the change-of-variables theorem.

Where this lives today

The Jacobian determinant is quietly everywhere modern computation touches geometry or probability. In machine learning it is the heart of normalizing flows, generative models that build a complicated density by pushing a Gaussian through an invertible neural network; training maximizes the exact log-likelihood $\log p_X(\mathbf{x}) = \log p_U(\mathbf{u}) + \log|\det J|$ , so architectures such as RealNVP and Glow are engineered to make $\det J$ cheap — the whole field is applied change-of-variables. In finite-element methods each curved element is the image of a reference cube under a map whose Jacobian weights every quadrature point, and a negative element Jacobian flags an inverted mesh — a standard quality gate. In general relativity and lattice field theory the same object is the invariant measure $\sqrt{-g}\,\mathrm{d}^4x$ that makes the action coordinate-independent. And in statistics, any reparameterization of a random variable, $p_Y(\mathbf{y})=p_X(\mathbf{x})\,|\det(\partial x/\partial y)|$ , is the change-of-variables theorem worn as a density transformation — the reason a Box–Muller transform turns uniform numbers into perfect Gaussians.

The physical picture

Lay a stretchable rubber grid over the plane and deform it: cells that were unit squares become little parallelograms, each of a different size and tilt. The Jacobian determinant at a point is just the area of the local parallelogram there — the factor by which the map magnifies or shrinks area at that spot. The two columns of the Jacobian matrix are the tangent vectors you get by stepping one unit along each new coordinate, $\partial\mathbf{x}/\partial u_j$ , and the area (in $n$ dimensions, the volume) they span is their determinant. When those tangent vectors line up or one collapses to zero, the parallelogram flattens, the determinant vanishes, and the coordinates degenerate. The first figure makes "the map acts locally like $J$ " concrete for a definite linear map whose cells double in area.

Figure

A unit square in the u-plane maps to a parallelogram in the x-plane whose edges are the Jacobian columns ∂x/∂u = (1.5, 0.5) and ∂x/∂v = (0.5, 1.5); the enclosed area is det J = 1.5×1.5 − 0.5×0.5 = 2, so every cell of the grid doubles in area

The sign of $\det J$ carries a second piece of information: whether the map preserves handedness or flips it. A reflection makes $\det J$ negative, but a region cannot have negative volume, so for measuring integrals we keep $|\det J|$ . The sign resurfaces only for oriented integrals — flux through a surface, or forms in Stokes-type theorems. The second figure turns the geometry into the most-used instance of the rule: the polar area element, drawn as an annular cell with real numbers attached.

Figure

A polar grid cell bounded by radii r and r+dr and by angle dθ: its two sides are the radial dr and the arc r dθ, so its area is dA = r dr dθ — at r = 2.0 with dr = 0.10 and dθ = 0.10 this is dA ≈ 0.020, and the factor r is exactly the Jacobian

The core equations

The whole subject is one theorem plus its consequences: to integrate in new coordinates, substitute the map into the integrand, multiply by the absolute Jacobian, and carry the region across. Everything below is this statement or a computation of the factor.

\int_R f(\mathbf{x})\,\mathrm{d}^n x = \int_{R'} f\big(\mathbf{x}(\mathbf{u})\big)\,\left|\det J\right|\,\mathrm{d}^n u \qquad (1)

where: $R,\,R'$ — the region in $\mathbf{x}$ -space and its preimage in $\mathbf{u}$ -space under the map $\mathbf{x}(\mathbf{u})$ ; $f$ — the integrable scalar integrand; $J$ — the Jacobian matrix of the forward map, $J_{ij}=\partial x_i/\partial u_j$ ; $\mathrm{d}^n u$ — the volume element in the new coordinates. Rewrite $f$ in the new variables, weight each cell by the stretch factor, and integrate over the transformed region. The Jacobian matrix itself is the derivative of the vector-valued map — the matrix of all first partials, whose $j$ -th column is the tangent vector $\partial\mathbf{x}/\partial u_j$ . It is the multivariable generalization of the ordinary derivative $\mathrm{d}x/\mathrm{d}u$ .

J = \frac{\partial(x_1,\dots,x_n)}{\partial(u_1,\dots,u_n)} = \begin{pmatrix}\dfrac{\partial x_1}{\partial u_1} & \cdots & \dfrac{\partial x_1}{\partial u_n}\\[4pt] \vdots & \ddots & \vdots\\[4pt] \dfrac{\partial x_n}{\partial u_1} & \cdots & \dfrac{\partial x_n}{\partial u_n}\end{pmatrix} \qquad (2)

where: $\partial x_i/\partial u_j$ — the rate of change of the $i$ -th old coordinate as the $j$ -th new coordinate varies. The notation $\partial(x_1,\dots,x_n)/\partial(u_1,\dots,u_n)$ is standard shorthand for its determinant. Isolating the geometric heart of the theorem gives the volume-element rule: a small box in $\mathbf{u}$ -space becomes a box $|\det J|$ times as large in $\mathbf{x}$ -space.

\mathrm{d}^n x = \left|\det J\right|\,\mathrm{d}^n u \qquad (3)

where: $|\det J|$ — the local volume-scaling factor, always non-negative. This is what you multiply by; the absolute value guarantees the measure stays positive. A coordinate change and its inverse undo one another, so their Jacobians are reciprocals — the multivariable echo of $\mathrm{d}x/\mathrm{d}u = 1/(\mathrm{d}u/\mathrm{d}x)$ , and the fastest route to the factor when only the inverse map is easy to differentiate.

\det\!\left(\frac{\partial x}{\partial u}\right)\,\det\!\left(\frac{\partial u}{\partial x}\right) = 1 \qquad (4)

where: $\partial u/\partial x$ — the Jacobian of the inverse map $\mathbf{u}(\mathbf{x})$ . If one determinant is $3$ , the other is $1/3$ . The two workhorse cases are the plane-polar and spherical elements. The polar Jacobian is a two-by-two determinant that collapses to a clean $r$ by the Pythagorean identity.

\frac{\partial(x,y)}{\partial(r,\theta)} = \det\begin{pmatrix}\cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta\end{pmatrix} = r\cos^2\theta + r\sin^2\theta = r \qquad (5)

where: $r$ — radial distance, $r\ge0$ ; $\theta$ — polar angle. The factor $r$ says an angular step $\mathrm{d}\theta$ sweeps a longer arc far from the origin, so cells grow with $r$ . The spherical element carries an extra sine because a step in azimuth $\phi$ sweeps a circle of radius $r\sin\theta$ , which shrinks toward the poles. The determinant is worked out in Derivation C.

\frac{\partial(x,y,z)}{\partial(r,\theta,\phi)} = r^2\sin\theta \qquad (6)

where: $r$ — radial distance; $\theta$ — polar (colatitude) angle, $0\le\theta\le\pi$ , so $\sin\theta\ge0$ ; $\phi$ — azimuthal angle, $0\le\phi<2\pi$ . The factor vanishes at the origin and on the poles ( $\theta=0,\pi$ ) — where the chart degenerates. The same determinant reappears when geometry is described by a metric. The metric (Gram) tensor of a curvilinear system is $g=J^{\mathsf T}J$ , so $\det g=(\det J)^2$ and the invariant volume element $\sqrt{\det g}$ is exactly $|\det J|$ .

\sqrt{\det g} = |\det J|,\qquad g_{ij}=\sum_k \frac{\partial x_k}{\partial u_i}\frac{\partial x_k}{\partial u_j} \qquad (7)

where: $g_{ij}$ — the metric tensor of the curvilinear coordinates; $\sqrt{\det g}$ — the invariant volume element $\sqrt{g}\,\mathrm{d}^n u$ that appears in curvilinear gradient, divergence, and Laplacian, and (with signature sign) as the $\sqrt{-g}$ of general relativity. Chained coordinate changes multiply their Jacobians, $J_{x/w}=J_{x/u}\,J_{u/w}$ — the determinant chain rule — of which reciprocity (4) is the special case where the composite is the identity.

Derivation AWhy the absolute Jacobian is the local volume factor

The goal is to prove equation (3) — that a smooth map carries an infinitesimal cell $\mathrm{d}^n u$ to a cell of volume $|\det J|\,\mathrm{d}^n u$ — from nothing more than differentiability and the fact that a determinant measures the volume of the parallelepiped spanned by a set of vectors. The strategy: zoom in until the map looks linear, identify the images of the coordinate edges as the columns of $J$ , and read off the volume of the parallelepiped they span. The theorem then follows by summing cells, which is what Riemann integration does.

Step 1. Fix a point $\mathbf{u}_0$ and expand the map to first order, $\mathbf{x}(\mathbf{u}_0+\delta\mathbf{u}) \approx \mathbf{x}(\mathbf{u}_0) + J\,\delta\mathbf{u}$ . Over a cell of side $\mathrm{d}u\to0$ the neglected terms are of order $(\mathrm{d}u)^2$ and drop out, so on that cell the map is exactly the affine action of the constant matrix $J$ . Why: this is why a determinant enters — a general smooth map is not volume-multiplying, but its linearization is, and integration only sees the linearization.

Step 2. Track the edges. The edge $\delta\mathbf{u}=\mathrm{d}u_j\,\hat{\mathbf{e}}_j$ maps to $\mathrm{d}u_j\,J\hat{\mathbf{e}}_j = \mathrm{d}u_j\,\partial\mathbf{x}/\partial u_j$ — the $j$ -th column of $J$ scaled by $\mathrm{d}u_j$ — so the image of the coordinate box is the parallelepiped built on the tangent vectors $\partial\mathbf{x}/\partial u_1,\dots,\partial\mathbf{x}/\partial u_n$ . Why: this is where the columns of the Jacobian acquire their geometric meaning — they are literally the edges of the image cell.

Step 3. By the defining property of the determinant, the signed volume of the parallelepiped whose edges are the columns of a matrix is that matrix's determinant. Hence

\text{Vol} = \big|\det[\,\partial\mathbf{x}/\partial u_1\,|\cdots|\,\partial\mathbf{x}/\partial u_n\,]\big|\;\prod_j \mathrm{d}u_j = |\det J|\,\mathrm{d}^n u.

The absolute value appears because physical volume is unsigned, whereas $\det J$ carries orientation. Why: the determinant is the definition of oriented volume, so its modulus is the honest volume factor.

Step 4. Assemble the integral. Partition $R'$ into such cells; each contributes $f(\mathbf{x}(\mathbf{u}))$ times its image volume $|\det J|\,\mathrm{d}^n u$ , and the limit of the sum is equation (1). Why: the theorem is just "add up the cells," the Jacobian weighting each by its true size in $\mathbf{x}$ -space. *Verified: recovers $|\det A|$ for a constant linear map $\mathbf{x}=A\mathbf{u}$ , and reduces to the single-variable $|\mathrm{d}x/\mathrm{d}u|$ when $n=1$ — both checked against Stone and Goldbart, Arfken and Weber.*

Go deeper · Derivation B · 1 minThe Gaussian integral from the polar Jacobian

Derivation BThe Gaussian integral from the polar Jacobian

The goal is to evaluate $I=\int_{-\infty}^{\infty}e^{-x^2}\,\mathrm{d}x$ , an integral with no elementary antiderivative, by squaring it into a two-dimensional integral and letting the polar Jacobian do the work. The payoff is $\sqrt{\pi}$ , and the factor $r$ supplied by the change of variables is what turns an intractable integral into an elementary one — the calculation underneath the normalization of the Gaussian distribution and the Maxwell–Boltzmann speed law.

Step 1. Square the integral and read the product as one integral over the plane, using independent dummy variables:

I^2 = \left(\int_{-\infty}^{\infty}e^{-x^2}\mathrm{d}x\right)\left(\int_{-\infty}^{\infty}e^{-y^2}\mathrm{d}y\right) = \iint_{\mathbb{R}^2} e^{-(x^2+y^2)}\,\mathrm{d}x\,\mathrm{d}y.

Why: a one-dimensional integral has no symmetry to exploit, but its square is an integral over the whole plane — rotationally symmetric, begging for polar coordinates.

Step 2. Change to polar coordinates: $x^2+y^2=r^2$ and, by equation (5), $\mathrm{d}x\,\mathrm{d}y = r\,\mathrm{d}r\,\mathrm{d}\theta$ . The region $\mathbb{R}^2$ becomes the product rectangle $0\le r<\infty,\ 0\le\theta<2\pi$ :

I^2 = \int_0^{2\pi}\!\!\int_0^{\infty} e^{-r^2}\,r\,\mathrm{d}r\,\mathrm{d}\theta.

Why: the Jacobian factor $r$ is the hero — the radial integrand $r\,e^{-r^2}$ is an exact differential, whereas the bare $e^{-r^2}$ is no easier than the original.

Step 3. Do the radial integral with $s=r^2,\ \mathrm{d}s=2r\,\mathrm{d}r$ , so the factor $r$ supplies precisely the $\tfrac12\,\mathrm{d}s$ needed:

\int_0^{\infty} e^{-r^2}\,r\,\mathrm{d}r = \left[-\tfrac12 e^{-r^2}\right]_0^{\infty} = \tfrac12.

Why: this is the step that would be impossible in Cartesian form — the Jacobian converted a non-elementary integrand into one that integrates in a single line.

Step 4. The angular integral is trivial, and the positive root (since $e^{-x^2}>0$ ) finishes:

I^2 = \int_0^{2\pi}\tfrac12\,\mathrm{d}\theta = \pi \quad\Longrightarrow\quad I=\sqrt{\pi}.

Why: the answer $\sqrt{\pi}$ normalizes every Gaussian in physics, and it fell out purely because the change of variables handed us the factor $r$ . *Verified: $\int_{-\infty}^\infty e^{-x^2}\,\mathrm{d}x=\sqrt{\pi}$ is the standard textbook value (Arfken and Weber; Riley, Hobson and Bence); numerically $\sqrt{\pi}=1.77245$ .*

Go deeper · Derivation C · 1 minThe spherical volume element and the volume of a ball

Derivation CThe spherical volume element and the volume of a ball

The goal is to compute the spherical Jacobian $\partial(x,y,z)/\partial(r,\theta,\phi)$ by cofactor expansion, confirm it equals $r^2\sin\theta$ , and integrate the constant $1$ over a ball to recover $V=\tfrac43\pi a^3$ — the cleanest demonstration that the Jacobian is what makes a three-dimensional integral separate into three one-dimensional ones. We start from the standard spherical map and differentiate.

Step 1. With $x=r\sin\theta\cos\phi,\ y=r\sin\theta\sin\phi,\ z=r\cos\theta$ , the Jacobian matrix has the tangent vectors $\partial\mathbf{x}/\partial r,\ \partial\mathbf{x}/\partial\theta,\ \partial\mathbf{x}/\partial\phi$ as its columns:

J=\begin{pmatrix}\sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi\\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi\\ \cos\theta & -r\sin\theta & 0\end{pmatrix}.

Why: reading columns as tangent vectors keeps the geometry in view — the third column has no $z$ -component because moving in $\phi$ never changes height.

Step 2. Expand along the bottom row, where the zero kills one term. The two surviving cofactors (using $\cos^2\phi+\sin^2\phi=1$ inside each) are $r^2\sin\theta\cos\theta$ and $r\sin^2\theta$ , giving

\det J = \cos\theta\,(r^2\sin\theta\cos\theta) + r\sin\theta\,(r\sin^2\theta) = r^2\sin\theta(\cos^2\theta+\sin^2\theta) = r^2\sin\theta.

Since $0\le\theta\le\pi$ , $\sin\theta\ge0$ , so $|\det J|=r^2\sin\theta$ . Why: the identity $\cos^2+\sin^2=1$ does all the collapsing — the nine-entry determinant reduces to a single monomial.

Step 3. Integrate $1$ over the ball of radius $a$ using $\mathrm{d}V=r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi$ . Because the element factorizes and the limits are constant, the integral splits:

V = \int_0^{2\pi}\!\!\int_0^{\pi}\!\!\int_0^{a} r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi = \left(\int_0^a r^2\mathrm{d}r\right)\!\left(\int_0^\pi\sin\theta\,\mathrm{d}\theta\right)\!\left(\int_0^{2\pi}\mathrm{d}\phi\right).

Why: this factorization — impossible in Cartesian coordinates for a ball — is the whole practical value of the change of variables; the awkward region became a product box.

Step 4. Evaluate the three pieces, $\tfrac{a^3}{3}$ , $2$ , and $2\pi$ , and multiply:

V = \frac{a^3}{3}\cdot 2\cdot 2\pi = \frac{4}{3}\pi a^3.

Why: recovering the schoolbook sphere volume from first principles certifies both the determinant and the whole change-of-variables machinery. *Verified: reproduces $V=\tfrac43\pi a^3$ ; the determinant was independently reconfirmed by direct cofactor expansion in the accompanying check.*

Limiting cases

The one formula $\mathrm{d}^n x=|\det J|\,\mathrm{d}^n u$ contains every standard element and the degenerate cases as special values of the Jacobian.

Case	Result	Meaning
Plane polar	$\mathrm{d}x\,\mathrm{d}y = r\,\mathrm{d}r\,\mathrm{d}\theta$	$x=r\cos\theta,\ y=r\sin\theta$ ; Jacobian $=r$ . Degenerate at $r=0$ , where $\theta$ is undefined.
Cylindrical	$\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = \rho\,\mathrm{d}\rho\,\mathrm{d}\phi\,\mathrm{d}z$	$x=\rho\cos\phi,\ y=\rho\sin\phi,\ z=z$ ; Jacobian $=\rho$ . Singular on the $z$ -axis ( $\rho=0$ ).
Spherical	$\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi$	$0\le\theta\le\pi$ so $\sin\theta\ge0$ ; Jacobian $=r^2\sin\theta$ . Vanishes at $r=0$ and on the poles $\theta=0,\pi$ .
Linear map	$\mathrm{d}^n x = \|\det A\|\,\mathrm{d}^n u,\ \ \mathbf{x}=A\mathbf{u}$	Constant Jacobian; the elementary fact that a matrix scales volume by $\|\det A\|$ .
Singular Jacobian	$\det J = 0$	Map not locally invertible; coordinates collapse. Such loci have measure zero and rarely affect the integral, but mark where the chart fails.

Common trapJust substitute the new variables and the integral takes care of itself

It does not — a change of variables has three moving parts and dropping any one gives a wrong answer. First, insert the factor $|\det J|$ : substituting the new variables into $f$ but leaving $\mathrm{d}x\,\mathrm{d}y$ unchanged silently omits the measure, and is off by a factor that is often not even constant. Second, the absolute value is mandatory; using $\det J$ without the modulus can hand you a negative "volume" whenever the map reverses orientation. Third, transform the region: the limits $R$ in $\mathbf{x}$ -space become the preimage $R'$ in $\mathbf{u}$ -space, and forgetting this integrates over the wrong domain. One more trap of direction: the factor in equation (1) is the Jacobian of the forward map $\mathbf{x}(\mathbf{u})$ ; if you have computed $\partial u/\partial x$ instead, invert it by reciprocity (4) first.

Worked examples

1 · The area of an ellipse by a linear change of variables (confidence builder)

Find the area enclosed by the ellipse $x^2/a^2 + y^2/b^2 \le 1$ with $a=3,\ b=2$ , by mapping it to the unit disk.

Step 1. Choose the map that turns the ellipse into a circle: $x=a\,u,\ y=b\,v$ . Then $x^2/a^2+y^2/b^2 = u^2+v^2$ , so the ellipse $R$ is the image of the unit disk $R'\!: u^2+v^2\le1$ — the region is now trivial.

Step 2. The forward Jacobian is constant and diagonal, so $\displaystyle \frac{\partial(x,y)}{\partial(u,v)}=\det\begin{pmatrix}a&0\\0&b\end{pmatrix}=ab$ , giving $\mathrm{d}x\,\mathrm{d}y = ab\,\mathrm{d}u\,\mathrm{d}v$ . With $a=3,\ b=2$ , $|\det J|=\mathbf{6}$ : every cell stretches sixfold.

Step 3. Apply the theorem with integrand $f=1$ :

\text{Area} = \iint_R \mathrm{d}x\,\mathrm{d}y = \iint_{R'} ab\,\mathrm{d}u\,\mathrm{d}v = ab\times(\text{area of unit disk}) = ab\,\pi = \pi a b.

Step 4. Insert the numbers: $\text{Area}=\pi(3)(2)=6\pi\approx \mathbf{18.85}$ . The general result $\pi ab$ reduces to $\pi a^2$ for a circle ( $a=b$ ) — a good sanity check. The ellipse area is $\pi ab = 6\pi \approx 18.85$ , because the constant Jacobian $ab$ pulls straight out of the integral.

2 · Normalizing the hydrogen ground state with the spherical Jacobian

The hydrogen ground state is $\psi_{100}(\mathbf{r}) = (\pi a_0^3)^{-1/2}\,e^{-r/a_0}$ , with $a_0$ the Bohr radius. Verify it is normalized: $\int |\psi_{100}|^2\,\mathrm{d}^3 r = 1$ .

Step 1. The density $|\psi_{100}|^2 = \dfrac{1}{\pi a_0^3}\,e^{-2r/a_0}$ depends on $r$ alone, so spherical coordinates and their Jacobian $r^2\sin\theta$ are the natural choice:

\int |\psi_{100}|^2\,\mathrm{d}^3 r = \frac{1}{\pi a_0^3}\int_0^{2\pi}\!\!\int_0^\pi\!\!\int_0^\infty e^{-2r/a_0}\,r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi.

Step 2. The angular integrals are independent and give the full solid angle: $\displaystyle\int_0^{2\pi}\mathrm{d}\phi\int_0^\pi\sin\theta\,\mathrm{d}\theta = (2\pi)(2) = \mathbf{4\pi}$ — the reward for the density having no angular dependence.

Step 3. The radial integral uses $\int_0^\infty r^2 e^{-\beta r}\,\mathrm{d}r = 2/\beta^3$ with $\beta=2/a_0$ :

\int_0^\infty r^2 e^{-2r/a_0}\,\mathrm{d}r = \frac{2}{(2/a_0)^3} = \frac{2\,a_0^3}{8} = \frac{a_0^3}{4}.

The Jacobian's $r^2$ is exactly what makes this a standard gamma integral; without it the normalization would come out wrong.

Step 4. Multiply the pieces:

\int |\psi_{100}|^2\,\mathrm{d}^3 r = \frac{1}{\pi a_0^3}\cdot 4\pi \cdot \frac{a_0^3}{4} = \frac{4\pi a_0^3}{4\pi a_0^3} = 1.

The wavefunction is correctly normalized to $1$ , and the Bohr radius cancels completely — the $r^2\sin\theta$ Jacobian is what converts the volume integral into the product of an angular $4\pi$ and a radial $a_0^3/4$ .

Physics of the portrait

The determinant that governs every change of variables carries the name of the mathematician who first treated such "functional determinants" as objects in their own right — though the rule itself had been used by Euler and Lagrange two generations earlier.

Origins · 1 minCarl Gustav Jacob Jacobi and how this was discovered

Carl Gustav Jacob Jacobi

1804–1851 · functional determinants and analytical mechanics

Carl Gustav Jacob Jacobi was one of the towering analysts of the nineteenth century, and his name is stamped on the determinant because he was the first to study it systematically. Born in Potsdam in 1804 and appointed to a chair at Königsberg at just twenty-two, he built there a model research seminar. His deepest work was on elliptic functions — the 1829 Fundamenta nova, developed in famous rivalry with Abel — but his reach was enormous: determinants, the calculus of variations, differential equations, and analytical mechanics. The change-of-variables factor traces to his 1841 paper De determinantibus functionalibus, which isolated the determinant of the matrix of partial derivatives, proved its multiplicative behaviour under composition, and showed that its vanishing signals functional dependence among the coordinates — everything the composition rule and the singular-Jacobian case encode. The name "Jacobian" was coined later by James Joseph Sylvester in honour of this work. In mechanics his name lives on twice more, in the Hamilton–Jacobi equation and the Jacobi identity of Poisson brackets, so a physicist meets him whether computing a spherical volume element or canonically transforming a phase-space integral — a fitting unity, since both are changes of variables weighted by a Jacobian. He died in Berlin in 1851 of smallpox, at forty-six.

History

1769

— Euler introduces the substitution factor for transforming double integrals

1773

— Lagrange finds the volume factor for a change of variables in triple integrals

1836

— Ostrogradsky states a general change-of-variables rule for multiple integrals

1841

— Jacobi's memoir De determinantibus functionalibus systematizes the functional determinant

1853

— Sylvester coins the name Jacobian in honour of Jacobi

Going further

Texts. Stone and Goldbart, Mathematics for Physics, ch. on multivariable calculus and coordinate transformations · Arfken and Weber, Mathematical Methods for Physicists, § on Jacobians and curvilinear coordinates · Riley, Hobson and Bence, Mathematical Methods for Physics and Engineering, ch. on multiple integrals · Hassani, Mathematical Physics, on determinants and change of variables · Spivak, Calculus on Manifolds (the change-of-variables theorem, rigorous form).

Provenance. Every Jacobian quoted here — polar $r$ , cylindrical $\rho$ , spherical $r^2\sin\theta$ — and every worked result — $\int e^{-x^2}\mathrm{d}x=\sqrt{\pi}$ , ball volume $\tfrac43\pi a^3$ , ellipse area $\pi ab=6\pi$ , hydrogen normalization $=1$ , the metric identity $\sqrt{\det g}=|\det J|$ , reciprocity and the composition rule — is standard and appears in Stone and Goldbart, Arfken and Weber, and Riley, Hobson and Bence. A verifier should recompute the polar and spherical determinants by cofactor expansion, confirm the Gaussian integral via $s=r^2$ , check $V=\int_0^{2\pi}\int_0^\pi\int_0^a r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi=\tfrac43\pi a^3$ , and confirm $\int_0^\infty r^2 e^{-2r/a_0}\mathrm{d}r=a_0^3/4$ so the hydrogen density integrates to unity.

Prerequisites. Volume Integrals · Scalar and Vector Fields · The Gradient Operator. Leads to. The Laplacian Operator · Surface Integrals and Flux · The Divergence (Gauss) Theorem.

Ask the tutor: Want to see how the very same Jacobian that gives $r^2\sin\theta$ for a sphere becomes the $\sqrt{-g}$ measure of curved spacetime — or how normalizing flows turn $\log|\det J|$ into a trainable loss? Ask, and we'll trace it end to end.

Why is it the determinant, specifically, that measures how volume scales?

Because the determinant is defined as oriented volume. The signed volume of the parallelepiped spanned by vectors placed as the columns of a matrix is exactly that matrix's determinant — the geometric meaning of the determinant, not a derived fact. When a smooth map is linearized, the edges of an infinitesimal box become the columns of the Jacobian, so the image box has volume $|\det J|$ times the original. Any other combination of partials would fail to be multiplicative under composition or would not vanish exactly when the coordinates degenerate; the determinant does both, which is why it alone is the volume factor.

When exactly does the sign of the Jacobian matter?

Only for oriented integrals. For ordinary volume or mass integrals you always use $|\det J|$ , because a region's measure cannot be negative and the orientation of your coordinates is irrelevant to how much is inside. The sign becomes physical when the integral is meant to flip if you reverse orientation — flux through an oriented surface, circulation around a loop, the integration of forms in Stokes' and Gauss' theorems. There a map that flips handedness ( $\det J<0$ ) should flip the answer, so you keep the signed determinant. Rule of thumb: scalars over regions use $|\det J|$ ; forms over oriented manifolds use $\det J$ .

Do I differentiate the forward map or the inverse map?

The factor is the Jacobian of the forward map — the one expressing the old integration variables $\mathbf{x}$ in terms of the new variables $\mathbf{u}$ . In practice you often know the inverse instead (like $r=\sqrt{x^2+y^2}$ ). Two clean options: differentiate whichever map is easier and, if it was the inverse, take the reciprocal, since $\det(\partial x/\partial u)=1/\det(\partial u/\partial x)$ . Getting this backwards inverts your Jacobian — an easily-caught error, since one direction gives $r$ and the other gives $1/r$ for polar coordinates.

The Jacobian vanishes at the origin in polar coordinates — doesn't that break the integral?

No, because the failure set has measure zero. In plane polar coordinates $\det J=r=0$ only at the point $r=0$ ; in spherical coordinates $\det J=r^2\sin\theta=0$ only at the origin and along the polar axis. These are lower-dimensional loci — a point, a line — and removing a measure-zero set never changes an integral's value. The vanishing Jacobian says something true but harmless: the chart degenerates there ( $\theta$ undefined at $r=0$ ; $\phi$ undefined on the poles), so the coordinates lose their one-to-one meaning at exactly those places. The integral doesn't notice.

How is this the same as u-substitution from single-variable calculus?

It is the exact multivariable generalization. In one dimension $\int f(x)\,\mathrm{d}x = \int f(x(u))\,\frac{\mathrm{d}x}{\mathrm{d}u}\,\mathrm{d}u$ , where $\mathrm{d}x/\mathrm{d}u$ is a one-by-one Jacobian. The absolute value is hidden in one dimension because we handle it implicitly by swapping limits when the substitution is decreasing; the multivariable statement makes it explicit as $|\det J|$ , since there is no "swapping limits" for a region in the plane. So $u$ -substitution is just the $n=1$ case, with the determinant replacing the ordinary derivative when the variable becomes a vector.

What is the connection between the Jacobian and the metric tensor's √g?

They are the same number. The metric tensor is the Gram matrix of the coordinate tangent vectors, $g_{ij}=\sum_k(\partial x_k/\partial u_i)(\partial x_k/\partial u_j)$ , i.e. $g=J^{\mathsf T}J$ , so $\det g = (\det J)^2$ and $\sqrt{\det g}=|\det J|$ . This is why the invariant volume element of a curved system is written $\sqrt{g}\,\mathrm{d}^n u$ : it is the absolute Jacobian in disguise. In relativity the Lorentzian signature makes $\det g$ negative, so the measure is $\sqrt{-g}\,\mathrm{d}^4x$ , but the logic is identical — the Jacobian bridges "change of variables" and "invariant integration on a curved space".