The Jacobian

Background

The matrix is named for Carl Gustav Jacob Jacobi, whose 1841 memoir on “functional determinants” studied the determinant of this matrix systematically. The idea underneath is older and simpler: a derivative is the best linear approximation to a function near a point. For a function of one variable that approximation is a number, the slope. For a map between spaces of several dimensions it has to be a matrix, because each output can respond to each input separately.

Reading the matrix

Row \(i\) is the gradient of the \(i\)-th component, \(\nabla F_i\), written as a row. It says how that one output changes as you move in any direction. Column \(j\) is \(\partial\vF/\partial x_j\): the velocity of the whole output as you slide the input along the \(x_j\) axis. Multiply the matrix by a direction and you get the directional derivative of the whole field:

\[J\,\vv=(\vv\cdot\nabla)\vF=\lim_{t\to0}\frac{\vF(\vr+t\vv)-\vF(\vr)}{t}\]

Three shapes are worth recognizing. A scalar function \(f:\mathbb R^n\to\mathbb R\) has a \(1\times n\) Jacobian, which is its gradient laid on its side. A curve \(\vr(t):\mathbb R\to\mathbb R^n\) has an \(n\times1\) Jacobian, its velocity. A vector field \(\mathbb R^3\to\mathbb R^3\) has a square \(3\times3\) Jacobian, and square matrices have a trace, a determinant and eigenvalues. That last case is the one Maxwell’s equations live in:

\[J_{\vF}=\begin{bmatrix}\partial_xF_x & \partial_yF_x & \partial_zF_x\\ \partial_xF_y & \partial_yF_y & \partial_zF_y\\ \partial_xF_z & \partial_yF_z & \partial_zF_z\end{bmatrix}\]

The chain rule is matrix multiplication

Compose two maps and their best linear approximations compose too:

\[J_{\mathbf G\circ\vF}(\vr)=J_{\mathbf G}\big(\vF(\vr)\big)\,J_{\vF}(\vr)\]

Every multivariable chain rule you have memorized, with its sums over intermediate variables, is one entry of this matrix product.

The linearization says the error in \(\vF(\vr+\vh)\approx\vF(\vr)+J\vh\) shrinks like \(|\vh|^2\), faster than \(\vh\) itself. Geometrically, a small square with sides \(h\,\mathbf e_1\) and \(h\,\mathbf e_2\) lands, after the map, very close to the parallelogram with sides \(h\,J\mathbf e_1\) and \(h\,J\mathbf e_2\), the scaled columns of \(J\). Its area is \(|\det J|\,h^2\). Halve the square and the curved edges straighten: the relative mismatch halves too.

For a vector field there is a second reading that connects straight to the main guide. Lab 0 there explores linear fields \(\vF=M\vr\), whose divergence is \(\operatorname{tr}M\) and whose curl is \(M_{21}-M_{12}\) everywhere. Near any point, a smooth field differs from its value at that point by approximately such a linear field, with \(M=J\). So the divergence and curl of any field at a point are the trace and the antisymmetric part of its Jacobian there.

Every square matrix splits in exactly one way into a symmetric part and an antisymmetric part. For a Jacobian, both parts have physical meaning.

S is the stretch. Because \(S\) is symmetric, the spectral theorem gives it real eigenvalues and perpendicular eigenvectors. Along each eigenvector the field changes at the rate given by the eigenvalue, without turning: these are the principal stretch axes, the dashed green lines in Lab J. Their rates add up to the trace, which is the divergence.

A is the rotation. \(A\) has three independent entries, and they are the components of the curl. For every vector \(\mathbf u\),

\[A\,\mathbf u=\tfrac12\,(\nabla\times\vF)\times\mathbf u\]

Acting on a small displacement, \(A\) turns it rigidly about the axis \(\nabla\times\vF\) at angular rate \(\tfrac12|\nabla\times\vF|\). This is the paddle wheel from the Primer in the main guide.

Take the trace out of \(S\) as well and the Jacobian falls into three pieces, with nothing left over:

\[J=\underbrace{\tfrac13(\nabla\cdot\vF)\,I}_{\text{expansion: 1 number}}+\underbrace{\Big(S-\tfrac13(\nabla\cdot\vF)\,I\Big)}_{\text{shear: 5 numbers}}+\underbrace{\tfrac12\,[\nabla\times\vF]_{\times}}_{\text{rotation: 3 numbers}}\]

Here \([\vw]_\times\) is the matrix that computes \(\vw\times\mathbf u\). The counts add to \(1+5+3=9\), the entries of a \(3\times3\) matrix. Maxwell’s equations specify the first piece and the last piece at every point, through the divergence and the curl. They say nothing locally about the five shear numbers, and they don’t need to: once the divergence and curl are known everywhere, and the field dies away at infinity, Helmholtz’s theorem fixes the whole field, shear included.

A small box of volume \(dV\) lands, after the map, on a region of volume \(|\det J|\,dV\). A negative determinant means the map also flips orientation, turning a left hand into a right hand. That is all the change-of-variables formula says: chop \(U\) into small boxes, map each one, and correct the integrand for how much each box grew.

The familiar volume elements are Jacobian determinants. Polar coordinates give \(dA=r\,dr\,d\theta\), cylindrical coordinates give \(dV=\rho\,d\rho\,d\phi\,dz\), and spherical coordinates give \(dV=r^2\sin\theta\,dr\,d\theta\,d\phi\). Example 5.1 works out the polar case.

Where \(\det J\neq0\), the inverse function theorem says the map can be undone near that point, and the Jacobian of the inverse is the inverse matrix, \(J_{\Phi^{-1}}=J_\Phi^{-1}\). Where \(\det J=0\), the map crushes some direction flat. Coordinates fail there, as spherical coordinates do on the polar axis, where \(\sin\theta=0\).

Divergence is the rate of change of volume

Let a field drive a flow, \(\dot{\vr}=\vF(\vr)\), and let \(\Phi_t\) be the Jacobian of the flow map: how the position at time \(t\) depends on the starting position. Differentiating \(\dot{\vr}=\vF(\vr)\) with respect to the starting point gives \(\dot\Phi_t=J_{\vF}\,\Phi_t\). Jacobi’s formula for the derivative of a determinant, \(\tfrac{d}{dt}\det\Phi=\det\Phi\,\operatorname{tr}\!\big(\Phi^{-1}\dot\Phi\big)\), then gives the equation in the box above:

\[\frac{d}{dt}\det\Phi_t=\det\Phi_t\,\operatorname{tr}\!\big(\Phi_t^{-1}J_{\vF}\Phi_t\big)=\det\Phi_t\,\operatorname{tr}J_{\vF}=(\nabla\cdot\vF)\,\det\Phi_t\]

A small blob carried by the flow grows in volume at the fractional rate \(\nabla\cdot\vF\). This is the flux definition of divergence from the main guide, seen from inside the flow. When \(\nabla\cdot\vF=0\) the flow preserves volume; Liouville’s theorem in Hamiltonian mechanics and incompressibility in fluid flow are both this statement.