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For the use of exterior form in daily physics, an introduction without coordinate frame

Raphael Ducatez

TL;DR

The paper presents a coordinate-free introduction to exterior differential forms, framing physics in purely geometric terms by treating submanifolds as fundamental objects and defining forms as quantities to integrate over them. It builds a cohesive toolkit—flows, pullbacks, Lie derivatives, interior products, the exterior derivative, and the Hodge star—then shows how Maxwell theory, conservation laws, and Noetherian structures arise naturally within this language. The work emphasizes gauge invariance, de Rham cohomology, and the stress-energy construction, linking Lagrangian and Hamiltonian formalisms to physically meaningful conserved quantities and dynamics. Its coordinate-free perspective aims to clarify physical meaning and provide elegant, compact derivations that illuminate classical field theories and their geometric underpinnings.

Abstract

This is a short introduction of the exterior form formalism focus on its applications in physics and then mostly aimed to physics students. As a rule of a game played here we never use a coordinate frame neither in the definitions nor in the proofs but only at the end in order to recover the classical physics equations. This approach is unusual but we think is helpful for the understanding and very valuable to grab the physical meanings of the mathematical object.

For the use of exterior form in daily physics, an introduction without coordinate frame

TL;DR

The paper presents a coordinate-free introduction to exterior differential forms, framing physics in purely geometric terms by treating submanifolds as fundamental objects and defining forms as quantities to integrate over them. It builds a cohesive toolkit—flows, pullbacks, Lie derivatives, interior products, the exterior derivative, and the Hodge star—then shows how Maxwell theory, conservation laws, and Noetherian structures arise naturally within this language. The work emphasizes gauge invariance, de Rham cohomology, and the stress-energy construction, linking Lagrangian and Hamiltonian formalisms to physically meaningful conserved quantities and dynamics. Its coordinate-free perspective aims to clarify physical meaning and provide elegant, compact derivations that illuminate classical field theories and their geometric underpinnings.

Abstract

This is a short introduction of the exterior form formalism focus on its applications in physics and then mostly aimed to physics students. As a rule of a game played here we never use a coordinate frame neither in the definitions nor in the proofs but only at the end in order to recover the classical physics equations. This approach is unusual but we think is helpful for the understanding and very valuable to grab the physical meanings of the mathematical object.
Paper Structure (27 sections, 11 theorems, 54 equations, 6 figures)

This paper contains 27 sections, 11 theorems, 54 equations, 6 figures.

Table of Contents

  1. Exterior forms
  2. Submanifolds as elementary objects
  3. Integration on a submanifold
  4. Tangent vector fields
  5. Transported and enlarged submanifold
  6. Pullback transport, Lie derivative and interior product
  7. Exterior Derivative
  8. Integration on the boundary
  9. Conserved quantity
  10. Gauge Invariance
  11. Cartan's magic formula
  12. Lie Derivative and Hydrodynamics
  13. Lie Derivative and forces applied at the boundary
  14. The metric and Maxwell equations
  15. The volume form
  16. ...and 12 more sections

Key Result

Proposition 9

$d\circ d=0$

Figures (6)

  • Figure 1: What can we do if we just have submanifolds and no coordinate frame ?
  • Figure 2: The transported submanifold ${\cal V}(t)$.
  • Figure 3: The enlarged submanifold ${\cal V}(t)$.
  • Figure 4: Boundaries of the submanifolds in Figure \ref{['manifold']}.
  • Figure 5: A "Geometric proof" of Cartan's formula $i_{X}\circ d=L_{X}-d\circ i_{X}$ seen as a consequence of $\partial[{\cal I}_{t}({\cal V})]={\cal V}(t)\cup{\cal V}(0)\cup{\cal I}_{t}(\partial{\cal V})$ : $i_{X}\circ d$ is associated to the figure on the left, $L_{X}$ to the figure in the center and $d\circ i_{X}$ to the figure on the right.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 9
  • Corollary 10
  • ...and 19 more