From components to vectors

On Physical Lines of Force → Modern derivations → Components to vectors

What Maxwell had

Maxwell wrote this paper in 1861 and 1862. The notation a modern reader takes for granted — boldface for a directed quantity, $\mathrm{\nabla }\cdot$ for a divergence, $\mathrm{\nabla }\times$ for a curl — did not exist. Heaviside and Gibbs built it independently in the 1880s, two decades later, and one of the things they built it out of was this paper and its successors. Vector analysis is in part a distillation of Maxwell.

So there are no vectors in On Physical Lines of Force. Every relation is written three times, once for each coordinate axis, and the reader is expected to hold the three lines together. This page is about learning to see the single vector statement hiding behind each triple. Once you can do it, the paper becomes perhaps three times shorter.

Maxwell's letters and their modern counterparts

The concordance in brief

Maxwell	Modern	What it is
α, β, γ	$𝐇$	the magnetic force
μα, μβ, μγ	$𝐁=\mu 𝐇$	the magnetic induction
p, q, r	$𝐉$	the electric current per unit area
P, Q, R	$𝐄$	the electromotive force
F, G, H	$𝐀$	the electrotonic state — the vector potential
f, g, h	$𝐃/4\pi$	the electric displacement (see the sign note below)
ψ	$\phi$	the electric tension, or potential
E	$\sqrt{1/\epsilon }$	a coefficient of the dielectric, not the electric field
e	$\rho$	the density of free electricity

The full table, with Maxwell's own glosses, is on the notation page.

Two collisions to watch for. Maxwell's E is a dielectric coefficient — an elasticity — and not the electric field; the electric field is his P, Q, R. His H is the third component of the vector potential and not the magnetic force; the magnetic force is his α, β, γ. Both letters were reassigned by later writers, and both traps catch readers who are skimming.

Three patterns worth recognising

Nearly every displayed formula in the paper is one of three shapes.

A divergence

Three derivatives, each of a component with respect to its own axis, added:

$$\frac{d\alpha }{dx}+\frac{d\beta }{dy}+\frac{d\gamma }{dz}⟶\mathrm{\nabla }\cdot 𝐇$$

The pattern is x-with-x, y-with-y, z-with-z, all with plus signs. This measures how much of the field is flowing out of a small region — whether the point is a source. It is a single number at each point, not a direction, which is why a divergence appears in the paper as one equation rather than three.

A curl

Three equations, each with two terms, each term a derivative of one component with respect to a different axis, and the two terms subtracted:

$$\frac{d\gamma }{dy}-\frac{d\beta }{dz}⟶(\mathrm{\nabla }\times 𝐇{)}_x$$

The pattern is antisymmetric: swap the two indices and the term changes sign. This measures the circulation of the field round a small loop — how much the field curls. It has a direction (the axis of the circulation), which is why a curl appears in the paper as three equations.

Circulation round a small loop: the curl

A gradient

Three derivatives of a single scalar, one along each axis:

$$\frac{d\psi }{dx},\frac{d\psi }{dy},\frac{d\psi }{dz}⟶\mathrm{\nabla }\psi$$

This points in the direction in which $\psi$ increases fastest, with a length equal to that rate. Wherever a lone letter like ψ appears differentiated three ways, you are looking at a gradient.

The cyclic rule

Here is the practical reading aid. Maxwell writes the x-equation of a triple in full, and the other two follow from it by the substitution

$$x\to y\to z\to x$$

carried out on everything at once: the coordinates, and the component letters that go with them. So α → β → γ → α, and p → q → r → p, and P → Q → R → P, and F → G → H → F.

Apply it to $4\pi p=\frac{d\gamma }{dy}-\frac{d\beta }{dz}$. Replace $p$ by $q$; replace $\gamma$ by $\alpha$ and $y$ by $z$; replace $\beta$ by $\gamma$ and $z$ by $x$. The result is $4\pi q=\frac{d\alpha }{dz}-\frac{d\gamma }{dx}$, which is exactly the second equation Maxwell prints. Do it once more and you get the third.

This is worth practising, because it means you only ever have to read one line of any triple carefully. It is also a check on a transcription: if the second and third lines are not the cyclic images of the first, either the reconstruction is wrong or the relation is not of this simple kind, and either way it is worth stopping to find out which.

A worked example

Take the equations of electric currents, Maxwell's (9), the relation between a current and the magnetic force circulating round it:

$$\begin{array}{cc}4\pi p& =\frac{d\gamma }{dy}-\frac{d\beta }{dz}\\ 4\pi q& =\frac{d\alpha }{dz}-\frac{d\gamma }{dx}\\ 4\pi r& =\frac{d\beta }{dx}-\frac{d\alpha }{dy}\end{array}\mathrm{(}\mathrm{9}\mathrm{)}$$

Provenance: the equation number is attested; the body is a reconstruction in Maxwell's notation. See Sources.

Now proceed in four steps.

1. Collect the triples. $(\alpha ,\beta ,\gamma )$ are the three components of one directed quantity, the magnetic force, which we name $𝐇$. $(p,q,r)$ are the three components of another, the current density, which we name $𝐉$. Grouping three numbers into a vector is a claim, not just a shorthand: it asserts that the three transform together when you turn the axes, so that the object survives a change of coordinates. Both triples do.
2. Identify the pattern. Every right-hand side has the antisymmetric two-term shape of a curl, and the three of them are cyclic images of one another. That is the signature of $\mathrm{\nabla }\times$.
3. Write out the modern curl and compare, remembering that $H_x=\alpha$, $H_y=\beta$, $H_z=\gamma$.
4. Assemble. The left sides are $4\pi$ times the components of $𝐉$, the right sides the components of $\mathrm{\nabla }\times 𝐇$, and three equations become one.

The third step, written out in full:

$$\begin{array}{cc}(\mathrm{\nabla }\times 𝐇{)}_x& =\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}\\ (\mathrm{\nabla }\times 𝐇{)}_y& =\frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}\\ (\mathrm{\nabla }\times 𝐇{)}_z& =\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}\end{array}$$

Line by line, these are Maxwell's right-hand sides. So the fourth step gives

$$\mathrm{\nabla }\times 𝐇=4\pi 𝐉$$

which is Ampère's circuital law, in the form Maxwell had before Prop. XIV added the displacement current to it.

What it says today

The whole of the paper's field theory reduces to four vector statements, each of which Maxwell writes as three (or, for the divergences, one) component equations. The other pages in this series take them one at a time. What the notation buys is not new physics — every one of these results is in the 1861 text — but the ability to see at a glance which of the three patterns you are looking at, and therefore what the equation is about.

Where the conventions differ

	Maxwell 1861	Modern
form	three scalar equations per vector relation	one vector equation
derivative	$\frac{d\gamma }{dy}$ — total-derivative notation throughout	$\frac{\partial \gamma }{\partial y}$
time derivative	$d/dt$ for both the rate at a fixed point and the rate following a moving element; the prose carries the distinction	$\partial /\partial t$ and $D/Dt$ are different symbols
the factor $4\pi$	explicit in the field equations, Gaussian-style	absorbed into ${\epsilon }_0,{\mu }_0$ in SI; still explicit in Gaussian
E	a dielectric coefficient, $\sqrt{1/\epsilon }$	the electric field
H	the third component of the vector potential	the magnetic field strength
ρ	the density of the cell substance (Part I)	charge density

Two of these deserve a further word.

The $4\pi$. Maxwell works in a Gaussian-style system in which the factor is written out rather than hidden in a constant. It is not decoration: it is the surface area of a unit sphere, and it is there because his unit of current is fixed by the magnetic effect of a closed circuit. Tracking it is most of the labour of converting his results, and it is the single commonest source of error.

The two meanings of $d/dt$. Maxwell uses one symbol for a rate of change measured at a fixed point in space and for a rate of change measured while drifting along with a moving piece of the medium. These are different quantities whenever the medium moves, and the difference is the whole content of the motional term in Prop. XI — the term we would now call $𝐯\times 𝐁$. Reading Prop. XI while assuming $d/dt$ means $\partial /\partial t$ will make it incomprehensible.

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See also: Notation and concordance · Index of numbered equations