Deriving the Faraday 2–form from the 1865 Maxwell’s equations

The connection to rotations

by Stéphane Haussler

This page presents the geometric differential formulation of electromagnetism, emphasizing the direct connection between Maxwell’s equations and rotations expressed with differential form. By calculating the exterior derivative these rotations in spacetime and straightforward identification, we will see that the equations governing electromagnetism are, in fact, spacetime rotations with the exterior derivative applied. In layman’s terms, we can think of electromagnetism as a torque over all possible 6 planes of rotations in 4–dimensional spacetime.

I hope my approach will feel more intuitive than the conventional method where the electromagnetic 2-form as well as the Maxwell’s equations in geometric differential form are given upfront, and equivalence to the vector formulation proven backward. For an example of the standard proof, see a nonetheless great video by Michael Penn. The mathematical expressions obtained in this section correspond to the well-recognized differential geometric formulation of Maxwell’s equations, where \(\mathbf{F}\) is the field 2–form and \(\mathbf{J}\) the current 3–form.

\[\begin{split}d\:\mathbf{F} = 0 \\ d⋆ \mathbf{F} = \mathbf{J}\end{split}\]

Unless explicitly stated otherwise, I explicitely indicate the nature of the mathematical objects involved using musical notation. The electromagnetic field 2–form \(\mathbf{F}\) is written with a double musical flat \(F^{♭♭}\), the current 3–form with a triple musical flat \(J^{♭♭♭}\), and the Hodge dual current 1–form with a single musical flat \(J^♭\).

This argumentation in this page proceeds as follows: First, I reordered Maxwell’s equations to align with their original formulation, using modern notation. This reorered form will serve the foundation for expressing the electromagnetic field equations in terms of geometric differential forms. Next, I represent a generic rotation in spacetime to which I apply the exterior derivative. Finally, I identified Maxwell’s equations with the exterior derivative of this spacetime rotation.

The discussion assumes a solid understanding of the exterior derivative \(d\), Élie Caretan’s differential forms, as well as Hodge duality. This page is self-contained and can be read independently of other content on this site.

Vector formulation of Mr. Heaviside

Mr. Maxwell’s groundbreaking work in 1865, A Dynamical Theory of the Electromagnetic Field (pdf), utilized differential expressions rather than the modern vector formulation proposed by Mr. Heaviside. Indeed in 1865, the concept of vectors had not yet been introduced.

Mr. Heaviside proposed both the concept of vectors as well as the vector form of the Maxwell equations. This is the most widespread formulation today. I therefore start from there and unpack into a form closer in spirit to the 1865 equation. The equations consist of two inhomogeneous (Gauss’s law and Ampère’circuital law) and two homogeneous vector equations (Gauss’s law for magnetism and Faraday’s law of induction), which is how I organize the equations.

While these equations are undoubtedly intriguing, within the context of this derivation, a deep comprehension of the physics behind them is unnecessary. I simply present them as the initial basis for the derivation. With the exception of Maxwell’s modification to Ampère’s circuital law, these equations represent the mathematical expression of empirical observations. Therefore, they can be regarded as established experimental facts.

Gauss’s Law

\[\overrightarrow{∇} \cdot \overrightarrow{E} = ρ / ε_0\]

Ampère’s Circuital Law

\[\overrightarrow{∇} \times \overrightarrow{B} = μ_0 \overrightarrow{J} + \frac{1}{c^2} \frac{∂}{∂t} \overrightarrow{E}\]

Faraday’s law of induction

\[\overrightarrow{∇} ⨯ \overrightarrow{E} = -\frac{∂}{∂t} \overrightarrow{B}\]

Gauss’s Law for Magnetism

\[\overrightarrow{∇} \cdot \overrightarrow{B} = 0\]

With the electric field \(\overrightarrow{E} = \begin{bmatrix} E^x \\ E^y \\ E^z \end{bmatrix}\), magnetic field \(\overrightarrow{B} = \begin{bmatrix} B^x \\ B^y \\ B^z \end{bmatrix}\), and operator \(\overrightarrow{∇} = \begin{bmatrix} \frac{∂}{∂x} \\ \frac{∂}{∂y} \\ \frac{∂}{∂z} \end{bmatrix}\)

The equations of Mr. Maxwell

Unpacking the vector equations into their component form, we obtain in spirit the 1865 Maxwell formulation, albeit with modern notation and conventions.

Gauss’s Law

\[\frac{∂}{∂x} E^x + \frac{∂}{∂y} E^y + \frac{∂}{∂z} E^z = ρ / ε_0\]

Ampère’s Circuital Law

\[\begin{split}\frac{∂}{∂y} B^z - \frac{∂}{∂z} B^y = μ_0 J^x + \frac{1}{c^2} \frac{∂}{∂t} E^x \\ \frac{∂}{∂z} B^x - \frac{∂}{∂x} B^z = μ_0 J^y + \frac{1}{c^2} \frac{∂}{∂t} E^y \\ \frac{∂}{∂x} B^y - \frac{∂}{∂y} B^x = μ_0 J^z + \frac{1}{c^2} \frac{∂}{∂t} E^z \\\end{split}\]

Gauss’s Law for Magnetism

\[\frac{∂}{∂x} B^x + \frac{∂}{∂y} B^y + \frac{∂}{∂z} B^z = 0\]

Faraday’s Law of Induction

\[\begin{split}\frac{∂}{∂y} E^z - \frac{∂}{∂z} E^y = - \frac{∂}{∂t} B^x \\ \frac{∂}{∂z} E^x - \frac{∂}{∂x} E^z = - \frac{∂}{∂t} B^y \\ \frac{∂}{∂x} E^y - \frac{∂}{∂y} E^x = - \frac{∂}{∂t} B^z \\\end{split}\]

Underlying structure

Gathering and reordering the terms, a clear structures becomes apparent:

Inhomogenous equations: Gauss’s law and Ampère’s circuital law

\[\begin{split}\begin{alignedat}{4} & + \frac{∂E^x}{∂x} & + \frac{∂E^y}{∂y} & + \frac{∂E^z}{∂z} & = + ρ/ε_0 \\ + \frac{1}{c^2} \frac{∂E^x}{∂t} & & - \frac{∂B^z}{∂y} & + \frac{∂B^y}{∂z} & = - μ_0 J^x \\ + \frac{1}{c^2} \frac{∂E^y}{∂t} & + \frac{∂B^z}{∂x} & & - \frac{∂B^x}{∂z} & = - μ_0 J^y \\ + \frac{1}{c^2} \frac{∂E^z}{∂t} & - \frac{∂B^y}{∂x} & + \frac{∂B^x}{∂y} & & = - μ_0 J^z \\ \end{alignedat}\end{split}\]

Homogenous equations: Guauss’s law and Faraday’s law of induction

\[\begin{split}\begin{alignedat}{4} & + \frac{∂B^x}{∂x} & + \frac{∂B^y}{∂y} & + \frac{∂B^z}{∂z} &= 0 \\ + \frac{∂B^x}{∂t} & & + \frac{∂E^z}{y∂} & - \frac{∂E^y}{∂z} &= 0 \\ + \frac{∂B^y}{∂t} & - \frac{∂E^z}{∂x} & & + \frac{∂E^x}{∂z} &= 0 \\ + \frac{∂B^z}{∂t} & + \frac{∂E^y}{∂x} & - \frac{∂E^x}{∂y} & &= 0 \\ \end{alignedat}\end{split}\]

Ordered equations

Recognizing the emerging structure, we slightly modify the expressions. These modifications are not intricate. The objective is merely to present a compact and symmetrical form, where all terms are aligned.

To eliminate the factor \(1/c\), we introduce \(\tilde{E}^x = E^x / c\), \(\tilde{E}^y = E^y / c\), and \(\tilde{E}^z = E^z / c\). Additionally, we define for the time dimension \(∂_t = \frac{∂}{∂(ct)}\), and for the spatial dimensions \(∂_x = \frac{∂}{∂ x}\), \(∂_y = \frac{∂}{∂y}\), as well as \(∂_z = \frac{∂}{∂z}\). The equations are now:

Inhomogenous equations

\[\begin{split}\begin{alignedat}{4} & + ∂_x \E^x & + ∂_y \E^y & + ∂_z \E^z & = + μ_0 c ρ \\ + ∂_t \E^x & & - ∂_y B^z & + ∂_z B^y & = - μ_0 J^x \\ + ∂_t \E^y & + ∂_x B^z & & - ∂_z B^x & = - μ_0 J^y \\ + ∂_t \E^z & - ∂_x B^y & + ∂_y B^x & & = - μ_0 J^z \\ \end{alignedat}\end{split}\]

Homogenous equations

\[\begin{split}\begin{alignedat}{4} & + ∂_x B^x & + ∂_y B^y & + ∂_z B^z & = 0 \\ + ∂_t B^x & & + ∂_y \E^z & - ∂_z \E^y & = 0 \\ + ∂_t B^y & - ∂_x \E^z & & + ∂_z \E^x & = 0 \\ + ∂_t B^z & + ∂_x \E^y & - ∂_y \E^x & & = 0 \\ \end{alignedat}\end{split}\]

For readers well-versed in the tensor formulation of electromagnetism, the presence and nature of the Faraday tensor and its dual are likely evident. Moreover, for those acquainted with matrix multiplication principles, it should be apparent that we can employ matrices operations.

Note

Although beyond our current discussion’s scope, defining \(∂_μ\) unifies all dimensions to a unit of inverse distance. \(\frac{1}{c}\frac{∂}{∂t} =\frac{∂}{∂(ct)}\) has the units of an inverse distance, exactly like the partial derivative with respect to the spatial dimensions \(\frac{∂}{∂x}\), \(\frac{∂}{∂y}\), and \(\frac{∂}{∂z}\).

Note

The experimental relation between the speed of light \(c\), the permitivity of free space \(ε_0\), and and the permeability of free space \(μ_0\) is used:

\[c = \frac{1}{\sqrt{ε_0 μ_0}}\]

We now link the ordered Maxwell equations to the exterior derivative of rotations in spacetime. For this, we first express a generic rotation as a linear combination of bivectors, flatten to a 2–form, and calculate the exterior derivative.

Rotations in differential forms

General rotations can be expressed as linear combinations of rotations in each planes. For Minkowski space with 4 directions \(∂_t\), \(∂_x\), \(∂_y\) and \(∂_t\), this result in 6 planes of rotations. Each plance is expressed as the exterior product \(∧\) of basis vectors as \(∂_μ ∧ ∂_ν\). Rotations in spacetime are expresssed as linear combination of basis rotations in the six available planes of rotations.

Rotations in Minkowski space

\[\begin{split}R^{♯♯} = \begin{bmatrix} Q^x \; ∂_t ∧ ∂_x \\ Q^y \; ∂_t ∧ ∂_y \\ Q^z \; ∂_t ∧ ∂_z \\ R^x \; ∂_y ∧ ∂_z \\ R^y \; ∂_z ∧ ∂_x \\ R^z \; ∂_x ∧ ∂_y \\ \end{bmatrix}\end{split}\]

Using the Minkowski metric \(η\), we flatten a basis vector with the flat operator \(♭\):

\[(∂_μ)^♭ = η_{μν} dx^ν\]

Likewise any index of the doubly contravariant exterior product can be flattened:

\[\begin{split}\begin{matrix} (∂_μ ∧ ∂_ν)^{♭♯} &= η_{γμ} dx^γ ∧ ∂_ν \\ (∂_μ ∧ ∂_ν)^{♯♭} &= η_{γν} ∂_μ ∧ dx^γ \\ (∂_μ ∧ ∂_ν)^{♭♭} &= η_{δμ} η_{γν} dx^δ ∧ dx^γ \\ \end{matrix}\end{split}\]

To obtain the doubly covariant representation of rotations in spacetime, we apply the flat operators \(♭♭\) to each components \(R^{♭♭} = (R^{♯♯})^{♭♭}\):

Doubly covariant representation of rotations in Minkowski space

\[\begin{split}R^{♭♭} = \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right]\end{split}\]

Calculations

Apply the flat operators

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; ∂_t ∧ ∂_x \\ Q^y \; ∂_t ∧ ∂_y \\ Q^z \; ∂_t ∧ ∂_z \\ R^x \; ∂_y ∧ ∂_z \\ R^y \; ∂_z ∧ ∂_x \\ R^z \; ∂_x ∧ ∂_y \\ \end{bmatrix}^{♭♭}\end{split}\]

Distribute the musical operators

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; ∂_t^♭ ∧ ∂_x^♭ \\ Q^y \; ∂_t^♭ ∧ ∂_y^♭ \\ Q^z \; ∂_t^♭ ∧ ∂_z^♭ \\ R^x \; ∂_y^♭ ∧ ∂_z^♭ \\ R^y \; ∂_z^♭ ∧ ∂_x^♭ \\ R^z \; ∂_x^♭ ∧ ∂_y^♭ \\ \end{bmatrix}\end{split}\]

Expand

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; η_{tμ} d^xμ ∧ η_{xμ} dx^μ \\ Q^y \; η_{tμ} d^xμ ∧ η_{yμ} dx^μ \\ Q^z \; η_{tμ} d^xμ ∧ η_{zμ} dx^μ \\ R^x \; η_{yμ} d^xμ ∧ η_{zμ} dx^μ \\ R^y \; η_{zμ} d^xμ ∧ η_{xμ} dx^μ \\ R^z \; η_{xμ} d^xμ ∧ η_{yμ} dx^μ \\ \end{bmatrix}\end{split}\]

Identify non-zero terms

\[\begin{split}R^{♭♭} = \begin{bmatrix} Q^x \; η_{tt} dt ∧ η_{xx} dx \\ Q^y \; η_{tt} dt ∧ η_{yy} dy \\ Q^z \; η_{tt} dt ∧ η_{zz} dz \\ R^x \; η_{yy} dy ∧ η_{zz} dz \\ R^y \; η_{zz} dz ∧ η_{xx} dx \\ R^z \; η_{xx} dx ∧ η_{yy} dy \\ \end{bmatrix}\end{split}\]

Apply numerical values

\[\begin{split}R^{♭♭} = \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right]\end{split}\]

The exterior derivative of rotations

In the article Differential operators expressed as exterior derivatives, I systematically compute the exterior derivative of all possible k–forms in both 3D Euclidean space and 4D spacetime with the Minkowski metric. The motivation for these systematic calculations is that all differential operators acting on fields (such as the gradient and the laplacian) or vector fields (such as the divergence and the curl) can be expressed using combinations of the Hodge star operator \(⋆\) and the exterior derivative \(d\).

Since the combinations of these two operators are finite, this approach allows for the calculation of all conceivable differential operators, which are generalized within the framework of exterior calculus. Specifically, we derive expressions for the exterior derivative of rotations and the exterior derivative of their Hodge duals the following:

Hodge dual of rotations

\[\begin{split}⋆ R^{♭♭} = \left[ \begin{aligned} & Q^x \; dy ∧ dz \\ & Q^y \; dz ∧ dx \\ & Q^z \; dx ∧ dy \\ & R^x \; dt ∧ dx \\ & R^y \; dt ∧ dy \\ & R^z \; dt ∧ dz \\ \end{aligned} \right]\end{split}\]

Calculations

Apply the Hodge star by linearity

\[\begin{split}⋆ R^{♭♭} = ⋆ \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right] = \left[ \begin{aligned} - & Q^x \; ⋆ dt ∧ dx \\ - & Q^y \; ⋆ dt ∧ dy \\ - & Q^z \; ⋆ dt ∧ dz \\ & R^x \; ⋆ dy ∧ dz \\ & R^y \; ⋆ dz ∧ dx \\ & R^z \; ⋆ dx ∧ dy \\ \end{aligned} \right]\end{split}\]

Apply the Hodge star to the basis 2-Forms

Using the tables for the Hodge dual:

\[\begin{split}⋆ R^{♭♭} = \left[ \begin{aligned} & Q^x \; dy ∧ dz \\ & Q^y \; dz ∧ dx \\ & Q^z \; dx ∧ dy \\ & R^x \; dt ∧ dx \\ & R^y \; dt ∧ dy \\ & R^z \; dt ∧ dz \\ \end{aligned} \right]\end{split}\]

Exterior derivative of the Hodge dual of rotations

\[\begin{split}d⋆R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Calculations

Take the exterior derivative

\[\begin{split}d⋆R^{♭♭} = d \begin{bmatrix} Q^x \; dy ∧ dz \\ Q^y \; dz ∧ dx \\ Q^z \; dx ∧ dy \\ R^x \; dt ∧ dx \\ R^y \; dt ∧ dy \\ R^z \; dt ∧ dz \\ \end{bmatrix}\end{split}\]

Distribute the exterior derivative

\[\begin{split}d⋆R^{♭♭} = \begin{bmatrix} d( Q^x \; dy ∧ dz) \\ d( Q^y \; dz ∧ dx) \\ d( Q^z \; dx ∧ dy) \\ d( R^x \; dt ∧ dx) \\ d( R^y \; dt ∧ dy) \\ d( R^z \; dt ∧ dz) \\ \end{bmatrix}\end{split}\]

Apply

\[\begin{split}d⋆R^{♭♭})= \left[ \begin{alignedat}{5} ∂_t (+ Q^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ Q^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^y ) \; & dy ∧ dz ∧ dx \\ ∂_t (+ Q^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^z ) \; & dz ∧ dx ∧ dy \\ ∂_y (+ R^x ) \; & dy ∧ dt ∧ dx & + & ∂_z (+ R^x ) \; & dz ∧ dt ∧ dx \\ ∂_x (+ R^y ) \; & dx ∧ dt ∧ dy & + & ∂_z (+ R^y ) \; & dz ∧ dt ∧ dy \\ ∂_x (+ R^z ) \; & dx ∧ dt ∧ dz & + & ∂_y (+ R^z ) \; & dy ∧ dt ∧ dz \\ \end{alignedat} \right]\end{split}\]

Reorder

\[\begin{split}d⋆R^{♭♭} = \left[ \begin{alignedat}{5} ∂_t (+ Q^x )(+1) \; & dt ∧ dy ∧ dz & + & ∂_x (+ Q^x )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^y )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^y )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^z )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^z )(+1) \; & dx ∧ dy ∧ dz \\ ∂_y (+ R^x )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^x )(-1) \; & dt ∧ dz ∧ dx \\ ∂_x (+ R^y )(-1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^y )(+1) \; & dt ∧ dy ∧ dz \\ ∂_x (+ R^z )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^z )(-1) \; & dt ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Apply values

\[\begin{split}d(⋆R^{♭♭}) = \left[ \begin{alignedat}{5} ∂_t (+ Q^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ Q^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^y ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ Q^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^z ) \; & dx ∧ dy ∧ dz \\ ∂_y (+ R^x ) \; & dt ∧ dx ∧ dy & + & ∂_z (- R^x ) \; & dt ∧ dz ∧ dx \\ ∂_x (- R^y ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^y ) \; & dt ∧ dy ∧ dz \\ ∂_x (+ R^z ) \; & dt ∧ dz ∧ dx & + & ∂_y (- R^z ) \; & dt ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}d ⋆ R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z & \: ) \; & dx ∧ dy ∧ dz \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y & \: ) \; & dt ∧ dy ∧ dz \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x & \: ) \; & dt ∧ dz ∧ dx \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & & \: ) \; & dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Exterior derivative of rotations

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Calculations

Distribute the exterior derivative

\[\begin{split}dR^{♭♭} = \begin{bmatrix} d( - Q^x \; dt ∧ dx ) \\ d( - Q^y \; dt ∧ dy ) \\ d( - Q^z \; dt ∧ dz ) \\ d( + R^x \; dy ∧ dz ) \\ d( + R^y \; dz ∧ dx ) \\ d( + R^z \; dx ∧ dy ) \\ \end{bmatrix}\end{split}\]

Apply the exterior derivative

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{3} ∂_y (- Q^x ) \; & dy ∧ dt ∧ dx & + & ∂_z (- Q^x ) \; & dz ∧ dt ∧ dx \\ ∂_x (- Q^y ) \; & dx ∧ dt ∧ dy & + & ∂_z (- Q^y ) \; & dz ∧ dt ∧ dy \\ ∂_x (- Q^z ) \; & dx ∧ dt ∧ dz & + & ∂_y (- Q^z ) \; & dy ∧ dt ∧ dz \\ ∂_t (+ R^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ R^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^y ) \; & dy ∧ dz ∧ dx \\ ∂_t (+ R^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^z ) \; & dz ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Reorder the exterior products

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{3} ∂_y (- Q^x )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (- Q^x )(-1) \; & dt ∧ dz ∧ dx \\ ∂_x (- Q^y )(-1) \; & dt ∧ dx ∧ dy & + & ∂_z (- Q^y )(+1) \; & dt ∧ dy ∧ dz \\ ∂_x (- Q^z )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (- Q^z )(-1) \; & dt ∧ dy ∧ dz \\ ∂_t (+ R^x )(+1) \; & dt ∧ dy ∧ dz & + & ∂_x (+ R^x )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^y )(+1) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^y )(+1) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^z )(+1) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^z )(+1) \; & dx ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Simplify

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{3} ∂_y (- Q^x ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ Q^x ) \; & dt ∧ dz ∧ dx \\ ∂_x (+ Q^y ) \; & dt ∧ dx ∧ dy & + & ∂_z (- Q^y ) \; & dt ∧ dy ∧ dz \\ ∂_x (- Q^z ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ Q^z ) \; & dt ∧ dy ∧ dz \\ ∂_t (+ R^x ) \; & dt ∧ dy ∧ dz & + & ∂_x (+ R^x ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^y ) \; & dt ∧ dz ∧ dx & + & ∂_y (+ R^y ) \; & dx ∧ dy ∧ dz \\ ∂_t (+ R^z ) \; & dt ∧ dx ∧ dy & + & ∂_z (+ R^z ) \; & dx ∧ dy ∧ dz \\ \end{alignedat} \right]\end{split}\]

Rearange

\[\begin{split}dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Hodge dual of the exterior derivative of rotations

\[\begin{split}⋆ dR^{♭♭} = \left[ \begin{alignedat}{5} (& & - ∂_x R^x & - ∂_y R^y & - ∂_z R^z &\:) \; dt \\ (& - ∂_t R^x & & - ∂_y Q^z & + ∂_z Q^y &\:) \; dx \\ (& - ∂_t R^y & + ∂_x Q^z & & - ∂_z Q^x &\:) \; dy \\ (& - ∂_t R^z & - ∂_x Q^y & + ∂_y Q^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

Calculations

Apply the Hodge star

Apply the Hodge star to \(dR^{♭♭}\):

\[\begin{split}⋆dR^{♭♭} = ⋆\left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Distribute the Hodge star

\[\begin{split}⋆dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; ⋆ dx^x ∧ dx^y ∧ dx^z \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; ⋆ dx^t ∧ dx^y ∧ dx^z \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; ⋆ dx^t ∧ dx^z ∧ dx^x \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; ⋆ dx^t ∧ dx^x ∧ dx^y \\ \end{alignedat} \right]\end{split}\]

Apply the Hodge star to the basis 1-forms

Using the tables for the Hodge dual:

\[\begin{split}⋆dR^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x R^x & + ∂_y R^y & + ∂_z R^z &\:) \; (-dt) \\ (& + ∂_t R^x & & + ∂_y Q^z & - ∂_z Q^y &\:) \; (-dx) \\ (& + ∂_t R^y & - ∂_x Q^z & & + ∂_z Q^x &\:) \; (-dy) \\ (& + ∂_t R^z & + ∂_x Q^y & - ∂_y Q^x & &\:) \; (-dz) \\ \end{alignedat} \right]\end{split}\]

Conclude

\[\begin{split}⋆dR^{♭♭} = \left[ \begin{alignedat}{5} (& & - ∂_x R^x & - ∂_y R^y & - ∂_z R^z &\:) \; dt \\ (& - ∂_t R^x & & - ∂_y Q^z & + ∂_z Q^y &\:) \; dx \\ (& - ∂_t R^y & + ∂_x Q^z & & - ∂_z Q^x &\:) \; dy \\ (& - ∂_t R^z & - ∂_x Q^y & + ∂_y Q^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

Identifying the Faraday 2–form

To summarize the results and observations thus far, an arbitrary rotation in Minkowski space is represented by a 2–form:

By applying the exterior derivative \(d\) to this rotation or to its Hodge dual (effectively calculating a torque), we obtain:

\[\begin{split}d⋆R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z &\:) \; dx ∧ dy ∧ dz \\ (& + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y &\:) \; dt ∧ dy ∧ dz \\ (& + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x &\:) \; dt ∧ dz ∧ dx \\ (& + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & &\:) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

\[\begin{split}⋆ dR^{♭♭} = \left[ \begin{alignedat}{5} (& & - ∂_x R^x & - ∂_y R^y & - ∂_z R^z &\:) \; dt \\ (& - ∂_t R^x & & - ∂_y Q^z & + ∂_z Q^y &\:) \; dx \\ (& - ∂_t R^y & + ∂_x Q^z & & - ∂_z Q^x &\:) \; dy \\ (& - ∂_t R^z & - ∂_x Q^y & + ∂_y Q^x & &\:) \; dz \\ \end{alignedat} \right]\end{split}\]

We have also reordered the 1865 Maxwell equations to:

Inhomogenous equations

\[\begin{split}\begin{alignedat}{4} & + ∂_x \E^x & + ∂_y \E^y & + ∂_z \E^z & = + μ_0 c ρ \\ + ∂_t \E^x & & - ∂_y B^z & + ∂_z B^y & = - μ_0 J^x \\ + ∂_t \E^y & + ∂_x B^z & & - ∂_z B^x & = - μ_0 J^y \\ + ∂_t \E^z & - ∂_x B^y & + ∂_y B^x & & = - μ_0 J^z \\ \end{alignedat}\end{split}\]

Homogenous equations

\[\begin{split}\begin{alignedat}{4} & + ∂_x B^x & + ∂_y B^y & + ∂_z B^z & = 0 \\ + ∂_t B^x & & + ∂_y \E^z & - ∂_z \E^y & = 0 \\ + ∂_t B^y & - ∂_x \E^z & & + ∂_z \E^x & = 0 \\ + ∂_t B^z & + ∂_x \E^y & - ∂_y \E^x & & = 0 \\ \end{alignedat}\end{split}\]

Identification

Identifiying the components of the electric field \(\tilde{E}^i=E^i/c\) and the magnetic field \(B^i\) is straightforward:

\[\begin{split}\begin{matrix} \E^x = R^x \\ \E^y = R^y \\ \E^z = R^z \\ B^x = Q^x \\ B^y = Q^y \\ B^z = Q^z \\ \end{matrix}\end{split}\]

The Faraday 2–form \(F^{♭♭}\) is a rotation in Minkowski spacetime.

\[\begin{split}F^{♭♭} = \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right]\end{split}\]

Maxwell’s equations are obtained by applying the exterior derivative, both directly \(dF^{♭♭}\) and to its Hodge dual \(d⋆F^{♭♭}\):

Inhomogenous Maxwell equations

\[\begin{split}d\:⋆ \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right] = \begin{bmatrix} + μ_0 c ρ \; dx ∧ dy ∧ dz\\ - μ_0 J^x \; dt ∧ dy ∧ dz\\ - μ_0 J^y \; dt ∧ dz ∧ dx\\ - μ_0 J^z \; dt ∧ dx ∧ dy\\ \end{bmatrix}\end{split}\]

Homogenous Maxwell equations

\[\begin{split}d \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right] = 0\end{split}\]

We have flipped the sign of the ordered homegenous Maxwell equations to our conveniance to ensure the correct overall sign. With our explicit notation, using a double musical flat ♭♭ to identify 2–forms, and a single musical flat ♭ to identify 1–forms, we have:

\[\begin{split}d\:F^{♭♭} &= 0 \\ d⋆ F^{♭♭} &= J^♭ \\\end{split}\]

As advertised in the introduction and using the implicit notation of the wikipedia article, we fall back to the well-recognized differential geometric formulation of Maxwell’s equations, where \(\mathbf{F}\) is the field 2–form and \(\mathbf{J}\) is the current 3–form.

\[\begin{split}d\:\mathbf{F} &= 0 \\ d⋆ \mathbf{F} &= \mathbf{J} \\\end{split}\]

A single equation

Adding a further Hodge star to the Homogenous equations is equally valid and we can equally write \(d\:F^{♭♭} = 0\) or \(⋆d\:F^{♭♭}=0\). With our explicit notation using double or single musical flats to idenfity 1–forms (♭) and 2–forms (♭♭), and since \(⋆\:d\:F\) is a 1–form and \(d⋆F\) a 3–form, we notice that we can unambiguously merge inhomogeneous and homogeneous equations into a single equation [note2]:

\[\begin{split}d ⋆ \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right] + ⋆\:d \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right] = \begin{bmatrix} + μ_0 c ρ \; dx ∧ dy ∧ dz\\ - μ_0 J^x \; dt ∧ dy ∧ dz\\ - μ_0 J^y \; dt ∧ dz ∧ dx\\ - μ_0 J^z \; dt ∧ dx ∧ dy\\ \end{bmatrix}\end{split}\]

Maxwell’s equations in differential form

\[\begin{split}(d ⋆ + ⋆ d ) \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right] = \left[ \begin{aligned} + μ_0 c ρ \; dx ∧ dy ∧ dz\\ - μ_0 J^x \; dt ∧ dy ∧ dz\\ - μ_0 J^y \; dt ∧ dz ∧ dx\\ - μ_0 J^z \; dt ∧ dx ∧ dy\\ \end{aligned} \right]\end{split}\]

With a shorthand \(F^{♭♭}\) for the electromagnetic field 2-form and \(J^{♭♭♭}\) for the current 3-form, we conclude with the compact form [note3]:

Maxwell’s equations

\[(d ⋆ + ⋆ d) \; F^{♭♭} = J^{♭♭♭}\]

Maxwell’s equations are interpreted as a twist in spacetime, with a strength proportional to the 4-current.

Reducing to a single equation using the wikipedia notation

Using the wikipedia notation, we arrive at the same result. However the lack of an explicit distinction between 1–forms and 2–forms is unfortunate as it makes it harder to see the reduction to a single equation.

\[⋆ \: d \: \mathbf{F} + d⋆\mathbf{F} = \mathbf{J}\]

Since \(d\:\mathbf{F}\) is a 3–form, its Hodge dual \(⋆\:d\:\mathbf{F}\) is a 1–form. We can thus rewrite the homogenous equation as acting on 1–forms, and the inhomogeneous equations as acting on 3–forms:

\[\begin{split}\begin{alignedat}{2} ⋆ \: d \: \mathbf{F} &= 0 & \qquad\text{1-forms} \\ d⋆\mathbf{F} &= \mathbf{J} & \qquad\text{3-forms} \\ \end{alignedat}\end{split}\]

Expanding the right hand side of both equations for the argument, we have:

\[\begin{split}⋆ \: d \: \mathbf{F} =& \: 0\:dt + 0\:dx + 0\:dy + 0\:dz \\ d⋆\mathbf{F} =& \: \mu_0 ρ dx∧dy∧dz - μ_0 J^x dt∧dy∧dz - μ_0 J^y dt∧dz∧dx - μ_0 J^z dt∧dx∧dy \\\end{split}\]

Therefore, the differential geometric formulation can be unambiguously reduced to a single equation involving both 3–forms and 1–forms:

\[⋆ \: d \: \mathbf{F} + d⋆\mathbf{F} = \mathbf{J}\]

Notes

[note2]

An equation containing 3-forms and 2-forms indeed cannot be reduced. For example, the following equation: \(a \; dx ∧ dy + b \; dx ∧ dy ∧ dz = c \; dx ∧ dy\) cannot be simplified. Surface 2-forms and volume 3-forms are distinct objects, but they can be represented in the same equation using the \(+\) symbol, even though they cannot actually be added together. Similar examples include combining the real and imaginary parts of complex numbers, or adding bivectors and trivectors in Clifford algebra. With the exemplary equation above, we thus necessarily have \(a = c\) as well as \(b = 0\). This is how we can write the Maxwell equations via differential forms into a single equation.

[note3]

Flipping the sign of \(⋆ d\) is equally valid.