Differential operators expressed as exterior derivatives

Differential operators expressed as exterior derivatives#

by Stéphane Haussler

Table of Contents:

The following pages systematically explore the application of the exterior derivative in both Euclidean space and Minkowski spacetime. Traditional differential operators from vector calculus are reinterpreted using the language of differential forms, musical operators, and the Hodge dual. These operations are then generalized and systematically applied in Minkowski spacetime. A central focus lies in the analysis of 2-forms, as the primary objective of this work is to highlight the profound connection between electromagnetism and rotations.

Exterior derivative in 3D Euclidean space#

All standard differential operators commonly encountered in vector calculus are expressed in the framework of differential forms, musicality, and Hodge duality:

Gradiant

\[(df)^♯ = \mathbf{∇} f\]

Divergence

\[⋆ d ⋆ F^♭ = \mathbf{∇} \cdot \mathbf{F}\]

Curl

\[(⋆(dF^♭))^♯ = ∇^♯ ⨯ F^♯\]

Laplacian

\[⋆ d ⋆ d f = \mathbf{∇}^2 f\]

Exterior derivative in Minkowski space#

At the heart of this work lies the exploration of the exterior derivative applied to rotations expressed in differential form, employing the Cartan-Hodge formalism, and within the context of Minkowski spacetime. I identify in a further article Of Maxwell Equations and Rotations that a twist in spacetime leads to the equations governing electromagnetism. Readers well versed in the formulation of electromagnetism will readily recognize the Faraday tensor, its dual, and the Maxwell equations.

The most generic possible form in Minkowski space is:

\[\begin{split}f + \left[ \begin{alignedat}{2} & A^t & \: dt \\ & A^x & \: dx \\ & A^y & \: dy \\ & A^z & \: dz \\ \end{alignedat} \right] + \left[ \begin{alignedat}{2} & Q^x & \: dt ∧ dx \\ & Q^y & \: dt ∧ dy \\ & Q^z & \: dt ∧ dz \\ & R^x & \: dy ∧ dz \\ & R^y & \: dz ∧ dx \\ & R^z & \: dx ∧ dy \\ \end{alignedat} \right] + \left[ \begin{alignedat}{2} & D^t & \: dx ∧ dy ∧ dz \\ & D^x & \: dt ∧ dy ∧ dz \\ & D^y & \: dt ∧ dz ∧ dx \\ & D^z & \: dt ∧ dy ∧ dz \\ \end{alignedat} \right] + G \: dt ∧ dx ∧ dy ∧ dz\end{split}\]

The differential operator available is the exterior derivative \(d\). Since \(d d F = 0\), to avoid obtaining zero, the Hodge star operator \(⋆\) must be applied before any further application of the exterior derivative. Consequently, all differential operators are formed as combinations of the exterior derivative and the Hodge star operator.

Observe that the number of distinct differential operators is inherently limited. The non-zero combinations are at most \(d ⋆ d ⋆\) and \(⋆ d ⋆ d\). With these, we can systematically compute all differential operators for the most general form in Minkowski space.

0–forms — Functions#

\[⋆ f = - \; f \; dt ∧ dx ∧ dy ∧ dz\]
\[d f = ∂_t f dt + ∂_x f dx + ∂_y f dy + ∂_z f dz\]
\[d ⋆ f = 0\]
\[\begin{split}⋆ d f = \left[ \begin{aligned} ∂_t f dx ∧ dy ∧ dz \\ ∂_x f dt ∧ dy ∧ dz \\ ∂_y f dt ∧ dz ∧ dx \\ ∂_z f dt ∧ dx ∧ dy \\ \end{aligned} \right]\end{split}\]
\[⋆ d ⋆ f = 0\]
\[d ⋆ d f = (∂_t^2 - ∂_x^2 - ∂_y^2 - ∂_z^2) \; f \; dz ∧ dt ∧ dx ∧ dy\]

1–forms — Translations#

\[\begin{split}⋆ \left[ \begin{alignedat}{2} & A^t & dt \\ - & A^x & dx \\ - & A^y & dy \\ - & A^z & dz \\ \end{alignedat} \right] = \left[ \begin{alignedat}{2} & A^t & dx ∧ dy ∧ dz \\ - & A^x & dt ∧ dy ∧ dz \\ - & A^y & dt ∧ dz ∧ dx \\ - & A^z & dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]
\[\begin{split}d \left[ \begin{alignedat}{2} & A^t & dt \\ - & A^x & dx \\ - & A^y & dy \\ - & A^z & dz \\ \end{alignedat} \right] = \left[ \begin{alignedat}{1} ( & - & ∂_t A^x & - & ∂_x A^t) & \; dt ∧ dx \\ ( & - & ∂_t A^y & - & ∂_y A^t) & \; dt ∧ dy \\ ( & - & ∂_t A^z & - & ∂_z A^t) & \; dt ∧ dz \\ ( & - & ∂_y A^z & + & ∂_z A^y) & \; dy ∧ dz \\ ( & - & ∂_z A^x & + & ∂_x A^z) & \; dz ∧ dx \\ ( & - & ∂_x A^y & + & ∂_y A^x) & \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]
\[\begin{split}d⋆ \left[ \begin{alignedat}{2} & A^t & dt \\ - & A^x & dx \\ - & A^y & dy \\ - & A^z & dz \\ \end{alignedat} \right] = \left( ∂_t A^t + ∂_x A^x + ∂_y A^y + ∂_z A^z \right) \; dt ∧ dx ∧ dy ∧ dz\end{split}\]
\[\begin{split}⋆d \left[ \begin{alignedat}{2} & A^t & dt \\ - & A^x & dx \\ - & A^y & dy \\ - & A^z & dz \\ \end{alignedat} \right] = \left[ \begin{alignedat}{4} (+ & ∂_z A^y & - & ∂_y A^z & ) & \; dt ∧ dx \\ (+ & ∂_x A^z & - & ∂_z A^x & ) & \; dt ∧ dy \\ (+ & ∂_y A^x & - & ∂_x A^y & ) & \; dt ∧ dz \\ (+ & ∂_x A^t & + & ∂_t A^x & ) & \; dy ∧ dz \\ (+ & ∂_y A^t & + & ∂_t A^y & ) & \; dz ∧ dx \\ (+ & ∂_z A^t & + & ∂_t A^z & ) & \; dx ∧ dy \\ \end{alignedat} \right]\end{split}\]
\[\begin{split} d ⋆ d⋆ - ⋆ d ⋆ d \left[ \begin{alignedat}{2} & A^t & dt \\ - & A^x & dx \\ - & A^y & dy \\ - & A^z & dz \\ \end{alignedat} \right] = \left[ \begin{alignedat}{7} ( & & - ∂_x^2 A^t & - ∂_y^2 A^t & - ∂_z^2 A^t & ) \; dt \\ ( & - ∂_t^2 A^x & & + ∂_y^2 A^x & + ∂_z^2 A^x & ) \; dx \\ ( & - ∂_t^2 A^y & + ∂_x^2 A^y & & + ∂_z^2 A^y & ) \; dy \\ ( & - ∂_t^2 A^z & + ∂_x^2 A^z & + ∂_y^2 A^z & & ) \; dz \\ \end{alignedat} \right]\end{split}\]

2–forms — Rotations#

\[\begin{split}d ⋆ \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right] = \left[ \begin{alignedat}{5} & ( & + ∂_x Q^x & + ∂_y Q^y & + ∂_z Q^z & ) \; dx ∧ dy ∧ dz \\ & ( + ∂_t Q^x & & - ∂_y R^z & + ∂_z R^y & ) \; dt ∧ dy ∧ dz \\ & ( + ∂_t Q^y & + ∂_x R^z & & - ∂_z R^x & ) \; dt ∧ dz ∧ dx \\ & ( + ∂_t Q^z & - ∂_x R^y & + ∂_y R^x & & ) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]
\[\begin{split}⋆ d \left[ \begin{aligned} - & Q^x\; dt ∧ dx \\ - & Q^y\; dt ∧ dy \\ - & Q^z\; dt ∧ dz \\ & R^x\; dy ∧ dz \\ & R^y\; dz ∧ dx \\ & R^z\; dx ∧ dy \\ \end{aligned} \right] = \left[ \begin{alignedat}{5} & ( & - ∂_x R^x & - ∂_y R^y & - ∂_z R^z & ) \; dt \\ & ( - ∂_t R^x & & - ∂_y Q^z & + ∂_z Q^y & ) \; dx \\ & ( - ∂_t R^y & + ∂_x Q^z & & - ∂_z Q^x & ) \; dy \\ & ( - ∂_t R^z & - ∂_x Q^y & + ∂_y Q^x & & ) \; dz \\ \end{alignedat} \right]\end{split}\]
\[\begin{split}( d ⋆ d ⋆ - ⋆ d ⋆ d ) \left[ \begin{aligned} - & Q^x \; dt ∧ dx \\ - & Q^y \; dt ∧ dy \\ - & Q^z \; dt ∧ dz \\ & R^x \; dy ∧ dz \\ & R^y \; dz ∧ dx \\ & R^z \; dx ∧ dy \\ \end{aligned} \right] &= \left[ \begin{alignedat}{5} & ( + ∂_t^2 Q^x & - ∂_x^2 Q^x & - ∂_y^2 Q^x & - ∂_z^2 Q^x & \: ) \; dt∧dx \\ & ( + ∂_t^2 Q^y & - ∂_x^2 Q^y & - ∂_y^2 Q^y & - ∂_z^2 Q^y & \: ) \; dt∧dy \\ & ( + ∂_t^2 Q^z & - ∂_x^2 Q^z & - ∂_y^2 Q^z & - ∂_z^2 Q^z & \: ) \; dt∧dz \\ & ( - ∂_t^2 R^x & + ∂_x^2 R^x & + ∂_y^2 R^x & + ∂_z^2 R^x & \: ) \; dy∧dz \\ & ( - ∂_t^2 R^y & + ∂_x^2 R^y & + ∂_y^2 R^y & + ∂_z^2 R^y & \: ) \; dz∧dx \\ & ( - ∂_t^2 R^z & + ∂_x^2 R^z & + ∂_y^2 R^z & + ∂_z^2 R^z & \: ) \; dx∧dy \\ \end{alignedat} \right]\end{split}\]

3–forms — Volumes#

4–forms — Hyper–volumes#