Jacobian matrix and change of variables

The formula

The Jacobian matrix of a vector function f that takes n inputs and returns m outputs is the m × n grid of every first-order partial derivative:

        [ ∂f₁/∂x₁   ∂f₁/∂x₂   ...   ∂f₁/∂xₙ ]
J(x) =  [ ∂f₂/∂x₁   ∂f₂/∂x₂   ...   ∂f₂/∂xₙ ]
        [   ...                              ]
        [ ∂fₘ/∂x₁   ∂fₘ/∂x₂   ...   ∂fₘ/∂xₙ ]

Reading: row i is the gradient of output i; column j says how every output reacts when you nudge input j.

f(x + h) ≈ f(x) + J(x)·h

Reading: near a point, the messy nonlinear map behaves like one matrix multiply. J is the best linear approximation of f.

det J   (only when m = n, a square map)

Reading: the determinant of the square Jacobian is the factor by which the map stretches (or shrinks) a tiny area or volume at that point. A det J of 3 means a small box comes out three times bigger.

∬_R f(x,y) dx dy = ∬_S f(x(u,v), y(u,v)) · |det J| du dv

Reading: the change-of-variables rule. Swap coordinates, and you must multiply by the absolute value of the Jacobian determinant or the integral is simply wrong.

p_Y(y) = p_X(x) · |det J⁻¹|   ,  where  y = f(x)

Reading: push a probability density through a transform and it re-weights by the inverse Jacobian, because probability mass has to stay conserved.

The two textbook cases every engineer has met (both verified by direct computation):

polar:      x = r·cos θ ,  y = r·sin θ        →   det J = r          →   dA = r dr dθ
spherical:  standard ρ, φ, θ convention        →   det J = ρ²·sin φ   →   dV = ρ² sin φ dρ dφ dθ

Where you meet it

Robot arm on the integration bench. The manipulator Jacobian maps joint speeds to tool-tip speed: ẋ = J(q)·q̇, where q is the vector of joint angles. When you command a straight-line move and the arm suddenly demands huge joint rates or refuses to move a direction, you have hit a singularity — a configuration where det J → 0 and the matrix loses rank. Manipulability, √det(J·Jᵀ), is the scalar people watch to stay clear of it.
Monte Carlo and error budgets at the review board. When a measured quantity feeds a nonlinear function — RF power from a voltage, range from a time-of-flight — the output distribution is the input distribution re-weighted by |det J⁻¹|. Linearized uncertainty propagation (Σ_y = J·Σ_x·Jᵀ) is the same Jacobian doing the same job on a covariance matrix. It is why a small input spread can blow up near a steep part of the transfer curve.
Any integral that changes coordinate systems. Computing a moment of inertia over a circular cross-section, a radiation pattern over a sphere, or a flux through a cylinder, you move to polar, cylindrical, or spherical coordinates. The r, the ρ²·sin φ, the r in r dr dθ dz — those are Jacobian determinants, not decoration. Drop them and the area is wrong.
Navigation and coordinate transforms in the test range software. Converting between lat/lon/alt, ECEF, and local level frames, or between rectangular and polar radar returns, every conversion carries a Jacobian. Filter code (Kalman and friends) linearizes those transforms with the Jacobian at each step.

How it works

The whole idea is local linearization. A nonlinear map does something different at every point, but zoom in far enough and it looks like a single linear transformation — stretch, rotate, shear. The Jacobian is that local linear transformation written as a matrix. Everything else follows from that one fact.

The determinant is the volume story. A linear map with determinant d scales every volume by |d|. So when you carve an integration region into tiny cells and reparametrize, each cell's volume changes by |det J|, and you multiply the integrand by exactly that to keep the total honest. The polar r is not magic: a wedge of a disk gets wider as r grows, so a cell at r = 5 covers five times the area of a cell at r = 1. det J = r says precisely that.

Sign versus magnitude is the first gotcha. In the integral you take |det J| — absolute value. The bare determinant can go negative, which just means the map flipped orientation (a mirror). Area does not care about handedness; you take the magnitude. In physics and orientation problems the sign carries information, so do not throw it away out of habit either.

The big limit of validity is invertibility. The inverse function theorem says f has a smooth local inverse exactly where det J ≠ 0. Where the determinant hits zero, the map crushes some direction flat — a whole line of inputs lands on a single output — and you cannot uniquely go backward. That is the robot singularity, the caustic in optics, the fold in a coordinate grid. The change-of-variables formula quietly assumes you stay away from those points; the transform has to be one-to-one on the region.

The common mistake is inverting the wrong way. When you transform a probability density you need the Jacobian of the map from the new variable back to the old one, or equivalently 1/|det J| of the forward map. Get the direction backward and your density integrates to something other than 1, and you will not notice until a downstream result is off by a factor. Verify it the cheap way: the transformed density must still integrate to 1. Jacobi himself proved the structural check that catches a bad setup — if your n functions of n variables are actually dependent (redundant), the Jacobian is identically zero; if they are genuinely independent, it cannot be.

Last practical note: for a big map, computing the full symbolic Jacobian is expensive, so real code uses automatic differentiation or finite differences to build it numerically at the operating point. The matrix is only ever needed at a point anyway.

History

Carl Gustav Jacob Jacobi was born in Potsdam in 1804 and died in Berlin in 1851 ^[1]. He was not the first to write down the functional determinant that now carries his name — it turns up in a paper by Augustin-Louis Cauchy back in 1815 ^[1]^[2]. What Jacobi did was make it a tool.

In 1841 he published a long memoir, De determinantibus functionalibus — "on functional determinants" — devoted entirely to this object: the determinant built from the n² partial derivatives of n functions in n variables ^[1]^[2]^[3]. Among the results he proved is the one engineers still lean on as a sanity check: if the functions are functionally related, the determinant is identically zero, and if they are independent it cannot be ^[1]. The object of that memoir is the same determinant that today does the bookkeeping when you change variables in a multiple integral ^[2], which is why the r and the ρ²·sin φ you write without thinking carry his name.

The name itself came a decade later: James Joseph Sylvester coined the term "Jacobian" in 1852, and in an 1853 paper spelled out why — "the differential determinant, so called by Jacobi, but which for greater brevity I call the Jacobian" ^[3]. The label is Sylvester's; the substance is Jacobi's 1841 contribution ^[1]^[3].

Related tools

/tools/convert-angle
/tools/moment-of-inertia-shapes
/tools/doppler-shift
/tools/section-modulus-rect

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →