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

Raphael Ducatez

► To cite this version:

Raphael Ducatez. For the use of exterior form in daily physics, an introduction without coordinate frame. 2026. hal-05475765

HAL Id: hal-05475765

https://hal.science/hal-05475765v1

Preprint submitted on 25 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
For the use of exterior form in daily physics, an
introduction without coordinate frame.

Raphael Ducatez

January 25, 2026

This is a short introduction of the exterior form formalism focus on its applications in physics and then mostly aimed to physics students. If exterior forms are more than a century old they are unfortunately still seen (and taught) as a high level mathematics object and then little used outside theoretical physics. We then focus here on simple examples which occur in daily physics. There exists already a lot of very good mathematical textbooks and courses on the subject but the originality of these notes, the physical applications aside, is that we keep a completely geometrical approach. 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. To say differently we will always prefer the following

Geometric definition > Coordinate definition

Geometric proof > Computational proof.

A large part of these notes are just “notations rewriting” of well known
physical objects but this should not be underestimate as it gives short and
elegant expressions that are useful both for insights and computations. Appart
from the *game* explained above most the material presented here is very well
known [Fra11, Fla63, BBB85] with exception maybe of Corollary 26 for which we
have surprisingly not found its statement in the litterature and the discussion
around Corollaries 20, 21 which we believe deserves more publicity. Hopefully
the approach presented here could be generalized in a very abstract setting.

# 1 Exterior forms

## 1.1 Submanifolds as elementary objects

We denote $\Omega$ as the “physical universe” and we will always think $\Omega$ as a 3 or 4 dimensional manifold ($\mathbb{R}^3$ or $\mathbb{R}^4$ for example). Our basic objects will be submanifolds of $\Omega$

$\nu \subset \Omega$
<figure><img src="image_2.png" /><figcaption>Figure 1: What can we do if we just have submanifolds and no coordinate frame ?</figcaption></figure>

In any textbook the next step is to attach to Ω a set of local coordinate. This is
the step we will try to avoid here, or at least we will try to write down as much as
we can without any choice of local coordinate. For example if one consider the
church “Notre Dame”, one could say that it is at 48° North-2° East on the surface
of the Earth but one could also say that the church is just a volume embedded
in the universe V<sub>Notre Dame</sub> ⊂ Ω and this is a more fundamental mathematical
representation. In these notes we ask the following.

**Question.** *If we restrict ourself to work only with submanifold without local coordinate, what could we do from a mathematical point of view?*

Here we will use exterior forms but will try to avoid Grassmann algebra as
well.

1.2 Integration on a submanifold

Usually, introduction about exterior form start at the coordinate level giving
the formal definition of an exterior algebra [Pau07, God70, LCC$^+$09]. However,
it is not so obvious at first sight why such an object is usefull and interesting in
physics. So we would instead propose the following definition to go immediatly
to the main point [Fla63].

**Definition 1.** A $k$-form is *what* to be integrated on a $k$-dimension submanifold.

This definition is made vague and unformal on purpose. Because it much
closer of what any physics would want as a mathematical object. That is to
choose an submanifold<fnref n="1" />: a path, a surface, a volume or a time interval × a volume and have physical quantity to integrate on it. This should really be though as the “meaningful physical quantity”, as it doesn’t depend on the choice of coordinate, not even the space-time metric $g$. If one think of a glass of water the quantity of water in the volume defined by the glass (a 3 dimension submanifold) should exist and be independant on the methods we measure space or time. From the lowest mathematical level this is just an application $\alpha$ that for any $k$-dimension submanifold $\mathcal{V}$ associate a real number $\alpha(\mathcal{V}) \in \mathbb{R}$.

$$ \alpha : k\text{-dimension submanifold} \rightarrow \mathbb{R} \\ \mathcal{V} \rightarrow \alpha(\mathcal{V}) $$

We also denote

$$ \alpha(\mathcal{V}) = \int_{\mathcal{V}} \alpha $$

but this notation is best to be seen as the definiton of the r-h-s than of the l-h-s.
Here are some examples in the case $\Omega$ is 3 dimensional.

<table><thead><tr><th>dimension</th><th>basis</th><th>quantity</th><th>unit</th></tr></thead><tbody><tr><td rowspan="4">0-form</td><td rowspan="4">1</td><td>Temperature</td><td>K</td></tr><tr><td>Pressure</td><td>Pa</td></tr><tr><td>Potential</td><td>V</td></tr><tr><td>Reflective index</td><td>1</td></tr><tr><td rowspan="5">1-form</td><td rowspan="5">dx, dy, dz</td><td>Temperature gradient</td><td>K.m<sup>-1</sup></td></tr><tr><td>Pressure gradient</td><td>Pa.m<sup>-1</sup></td></tr><tr><td>Electric field</td><td>V.m<sup>-1</sup></td></tr><tr><td>Vector potential</td><td>Vs.m<sup>-1</sup></td></tr><tr><td>Magnetizing field</td><td>Cs<sup>-1</sup>.m<sup>-1</sup></td></tr><tr><td rowspan="5">2-form</td><td rowspan="5">dx &wedge; dy, <br>dx &wedge; dz, <br>dy &wedge; dz</td><td>mass flow</td><td>kg.s<sup>-1</sup>.m<sup>-2</sup></td></tr><tr><td>charge flow</td><td>C.s<sup>-1</sup>.m<sup>-2</sup></td></tr><tr><td>radiation</td><td>W.m<sup>-2</sup></td></tr><tr><td>magnetic field</td><td>Vs.m<sup>-2</sup></td></tr><tr><td>Electric displacement field</td><td>C.m<sup>-2</sup></td></tr><tr><td rowspan="4">3-form</td><td rowspan="4">dx &wedge; dy &wedge; dz</td><td>density of mass</td><td>kg.m<sup>-3</sup></td></tr><tr><td>density of charge</td><td>C.m<sup>-3</sup></td></tr><tr><td>density of energy</td><td>J.m<sup>-3</sup></td></tr><tr><td>heat capacity</td><td>JK<sup>-1</sup>.m<sup>-3</sup></td></tr></tbody></table>

In $\mathbb{R}^3$ the dimension of the $k$-form $\Lambda^k(\mathbb{R}^3)$ are respectively 1, 3, 3 and 1. The main message here is that exterior form are natural to describe daily physics quantities and are not more complicated to use than the standard scalars of vector fields. We still stress a few differences. What are usual called “scalsars” are here distinguished between 0-form and 3-form in a very similar way as we

<footnote n="1">¹More precisely an “oriented submanifold”. The integration on path from A to B or from B to A have a different sign. Similarly the integration of a flow through a surface going inward or outward have a different sign.</footnote>
distinguish intensive and extensive properties. And what are usual called “vector fields” are distinguished between 1-form and 2-form. It is still interesting to make such a difference at the mathematical level since they would not behave the same way with a change of coordinate. A fact that stills appears in the system of units.

In $\mathbb{R}^4$ the dimension of the $k$-form $\Lambda^k(\mathbb{R}^3)$ are respectively 1, 4, 6, 4, and 1. The 0-forms and 4-forms are the usual scalars while 1-forms and 3-forms are the usual 4-vectors. The 2-form are less commun but correspond to anti-symmetric 2-tensors. Generally speaking anti-symmetric properties of tensor are related to integration so have some geometric meaning. Here are some examples of physical quantities written as exterior forms in 4-dimensional.

<table><thead><tr><th>dimension</th><th>basis</th><th>quantity</th><th>unit</th></tr></thead><tbody><tr><td>0-form</td><td>1</td><td>(same as in R<sup>3</sup>)</td><td>*</td></tr><tr><td>1-form</td><td>dx, dy, dz,<br>dt</td><td>4-gradient<br>(∂<sub>t</sub>f, ∇f)</td><td>*.m<sup>-1</sup> or<br>*.s<sup>-1</sup></td></tr><tr><td>2-form</td><td>dx ∧ dy, ...<br>dt ∧ dx, ...</td><td>Electromagnetic Field<br>F = (E, B)</td><td>*.m<sup>-2</sup> or<br>*.m<sup>-1</sup>s<sup>-1</sup></td></tr><tr><td>3-form</td><td>dx ∧ dy ∧ dz,...<br>dt ∧ dx ∧ dy,...</td><td>Density and flow<br>J = (ρ, j)</td><td>*.m<sup>-3</sup> or<br>*.m<sup>-2</sup>s<sup>-1</sup></td></tr><tr><td>4-form</td><td>dt ∧ dx ∧ dy ∧ dz</td><td>Field Lagrangian</td><td>*.m<sup>-3</sup>s<sup>-1</sup></td></tr></tbody></table>

For example integrating $J = j dt \wedge dx \wedge dy$ would give the amount of matter that has been through a (horizontal) surface within a time interval.

## 1.3 Tangent vector fields

“Vector field” is also usually used to design another mathematical object that to avoid ambiguity we will call here *tangent vector fields*. If exterior forms are associated with integration, one instead should think of tangent vector fields [LCC+09] informally as *what* describes a flow, ie a transport.

**Definition 2.** A flow is a family of application on $\Omega$

$$ \phi_t : \Omega \to \Omega \quad t \in \mathbb{R}_+ $$

We should stress here that $\phi_t$ is well defined at a geometrical level. And even if it looks abstract, it is very simple and natural because it is nothing but the description of how the system has evolved at different times. Moreover if we assume that is a semi-group

$$ \phi_{t_1+t_2} = \phi_{t_1} \circ \phi_{t_2} \quad t_1, t_2 \in \mathbb{R}_+ $$

one can introduce a tangent vector field such the flow is the solution of a differential equation.

**Definition 3.** For a flow $\phi_t$ the associated tangent vector field $X$ is such that for any point $x_0 \in \Omega$,

$$ x(t) = \phi_t(x_0) \Leftrightarrow \begin{cases} x(0) = x_0 \\ \partial_t x(t) = X(x(t)). \end{cases} \qquad (1) $$
One could see the tangent vector field as the derivative of the flow "X = $\frac{d}{dt}\phi_t$". So Equation (1) can be thought as the usual way to measure a velocity flow which is to observe the system at different time and "differentiate" this evolution. The main message here is that a tangent vector field it is a completely different mathematical object than the exterior forms. It is used to describe different physical notions and has different basis notation and system of units as for example :

<table><thead><tr><th>object</th><th>basis</th><th>quantity</th><th>unit</th></tr></thead><tbody><tr><td>tangent vector</td><td>$\partial_x, \partial_y, \partial_z$</td><td>velocity of a fluid</td><td>$m.s^{-1}$</td></tr></tbody></table>

As illustrated by the system of units tangent vector fields are usually refered as covariant tensors while exterior form are contravariant tensors. In $\mathbb{R}^4$, we also consider transport in the time direction and use the basis element $\partial_t$. For example one can consider a motionless object but still think of it as transported from the past toward the future. In general we see the evolution from a point $(t_0, x_0, y_0, z_0)$ to another point $(t_1, x_1, y_1, z_1)$ as a transport both in time and space.

## 1.4 Transported and enlarged submanifold

Tangent vector fields allow us to define a few opérations on a submanifold $V \subset \Omega$ which are, we recall, our basic object of interest.

**Definition 4.** With $X$ a tangent vector field associated to a semi-group $\phi_t$ and $V$ a submanifold, we define

* An transported submanifold

$$V(t) = \phi_t(V) = \{x(t), x_0 \in V\}$$

with $x(t)$ solution of (1).

* An enlarged submanifold

$$I_t(V) = \bigcup_{s \in [0,t]} \phi_s(V) = \{x(s), x_0 \in V, s \in [0,t]\}$$

that is a submanifold which is one dimensional larger than $V$.

For example if one think of $X$ describing the velocity field of flow of water, then a transported submanifold would be used to describe anything that is dived in the water and then carried away by the flow. For the enlarged submanifold one can see it as keeping track of the successive positions of transported manifold along the flow as in a numerical simulation. For example in the case of a point $V = \{x_0\}$, this is just the line traced by the trajectory $x(t)$. In the case of the leave it is the volume in the water that has been covered by the leave on its way.

Both $V(t)$ and $I_t(V)$ are natural objects on which integrate an exterior form. This is will be presented in the next section.
<figure><img src="image_3.png" /><figcaption>Figure 2: The transported submanifold $\mathcal{V}(t)$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: The enlarged submanifold $\mathcal{V}(t)$.</figcaption></figure>
## 1.5 Pullback transport, Lie derivative and interior product

We now present some basic operations to manipulate exterior form as appears in math textbooks [JJ08, LCC+09, Fra11]. All these operations also have a simple definition in a coordinate system but the goal of these notes is to stay at a geometric point of view and to give an motivation to introduce these objects. Also it makes it clear that they are independent of the choice of the coordinate and therefore should have some physical meaning.

**Definition 5.** With $X$ a tangent vector field associated to a semi-group $\phi_t$ and $\alpha$ a $k$ form we introduce the following quantities

* The pullback transport $\phi_t^*(\alpha)$ defined as a $k$ form such that for any $k$ dimensional submanifold $\mathcal{V}$

$$[\phi_t^*(\alpha)](\mathcal{V}) = \alpha(\mathcal{V}(t)).$$

* The Lie derivative $L_X\alpha$ defined as

$$L_X\alpha = \frac{d}{dt}\Big|_{t=0}\phi_t^*(\alpha)$$

Remark that the Lie derivative is then defined as the $k$ form such that for any $k$ dimensional submanifold $\mathcal{V}$

$$[L_X\alpha](\mathcal{V}) = \lim_{t \to 0} \frac{\alpha(\mathcal{V}(t)) - \alpha(\mathcal{V}(0))}{t}.$$

**Definition 6.** With $X$ a tangent vector field associated to a semi-group $\phi_t$ and $\alpha$ a $k$ form we introduce the following quantities

* A $k-1$ form $I_t\alpha$ defined such that for any $k-1$ dimensional submanifold $\mathcal{V}$

$$[I_t\alpha](\mathcal{V}) = \alpha(\mathcal{I}_t(\mathcal{V}))$$

with $\mathcal{I}_t(\mathcal{V})$ the enlarged submanifold.

* The interior product $i_X\alpha$ defined as

$$i_X\alpha = \frac{d}{dt}\Big|_{t=0}I_t\alpha$$

That is the $k-1$ form such that for any $k-1$ dimensional submanifold $\mathcal{V}$

$$[i_X\alpha](\mathcal{V}) = \lim_{t \to 0} \frac{\alpha(\mathcal{I}_t(\mathcal{V}))}{t}.$$

This has a natural physical meaning: for example with $\rho$ a density of matter (a 3-form) and $X$ a velocity field, $i_X\rho$ (a 2-form) describes the density flow (For any surface $S$, $i_X\rho(S)$ is the flow of matter that is going through $S$).
The approach presented here is actually very general : any operation on submanifolds can be translated into an operation on forms. Indeed considering an application that from a k submanifold V gives another submanifold V → ˜V . We can define an application that from a k form α gives another form α → ˜α such that

$$
\tilde{\alpha}(\tilde{V}) = \alpha(V) \tag{2}
$$

Equation (2) can also be used to have “dual” application ˜α → α that from a form ˜α define a k form α .

**Question.** If we restrict ourself to the use of this operators, what physics equa- tions could we write ?

We will work on that question later but first we have to introduce the exterior derivative.

2 Exterior Derivative

2.1 Integration on the boundary

The usual formal definition of the exterior derivative use the derivation at the coordinate level [JJ08, LCC$^{+}$09, Fra11] but as before it is not very clear at first sight why such an objet would be interesting in physics. So here again is another definition that is informal but that looks very natural and capture the main interest of the object.

**Definition 7.** For a $k$-form $\alpha$, the exterior derivative $d\alpha$ is defined as the ($k+1$)-form such that for any ($k+1$) dimensional submanifold $V$

$$
d\alpha(\nu) = \alpha(\partial\nu).
$$

This is of course the very famous Stokes Theorem.

$$
\int_V d\alpha = \int_{\partial V} \alpha
$$

but as in Definition 1 such an equation could also be seen as a definition of an application on the ( k + 1) − submanifolds, that is a ( k + 1) − form and such a definition does not depend on a choice coordinate.

Also boundaries of submanifold are very commun objects : the extremal points of a path, the circle surrounding a disc, the surface around a volume. One can also think of the initial and final configurations of a system as the 3 dimension boundaries of the evolving system in time and space seen as a 4 dimension submanifold.
<figure><img src="image_5.png" /><figcaption>Figure 4: Boundaries of the submanifolds in Figure 1.</figcaption></figure>

In a 3 dimension space whether the form $\alpha$ is a 0, 1 or 2-form, the exterior derivative $d\alpha$ is called gradient, curl and divergence<fnref n="2" />.

$$ \Lambda^0(\mathbb{R}^3) \xrightarrow{\text{grad}} \Lambda^1(\mathbb{R}^3) \xrightarrow{\text{curl}} \Lambda^2(\mathbb{R}^3) \xrightarrow{\text{div}} \Lambda^3(\mathbb{R}^3). $$

where $\Lambda^k(\mathbb{R}^3)$ denote the space of $k$-form.

## 2.2 Conserved quantity

With the previous section in $\mathbb{R}^3$ a divergence-free vector field is then a 2-form $\alpha$ such that $d\alpha = 0$. In $\mathbb{R}^4$ we can propose a similar definition of a conserved quantity. If we go back to our example of evolving system in time and space. For conservation we ask that the integration on the initial and final configurations gives the same result. This is guaranty if $d\alpha = 0$ so we can propose the following definition.

**Definition 8.** A conserved quantity is a 3-form<fnref n="3" /> $J$ such that $dJ = 0$.

Writting the 3-form in a coordinate frame $J = (\rho, j_x, j_y, j_z)$, the condition $dJ = 0$ is just the Continuous Equation [Fla63]

$$ \partial_t \rho + \text{div}(j) = 0. $$

An advantage of the above formulation is that it does not depends on the coordinate frame or the metric. Equivalently for any $V$ we have $J(\partial V) = 0$. In word: “What is going into $V$ is the same as what is going out of $V$” which is also a very natural non mathematical definition of what a conserved quantity is.

<footnote n="2">²The most common convention use the metric tensor to transform these forms into vector field. See Section 3.2.</footnote>

<footnote n="3">³And more generally a $(n-1)$-form if $\Omega$ is $n$ dimensional</footnote>
## 2.3 Gauge Invariance

We also state the following important observation: «the boundary of a manifold has no boundary» $\partial(\partial V) = \emptyset$. Therefore for any form $\alpha$, we have

$$ (d \circ d\alpha)(V) = \alpha(\partial(\partial V)) = 0 $$

and we write the following proposition.

**Proposition 9.** $d \circ d = 0$

For example in $\mathbb{R}^3$ this is the well known curl $\circ$ grad = 0 and div $\circ$ curl = 0.

**Corollary 10.** (Gauge invariance) For a $k$-form $\alpha$ and any $(k-1)$ form $\beta$ we have

$$ d(\alpha + d\beta) = d\alpha. $$

If the physical quantity of interest is $d\alpha$, then the $k$-form $\alpha$ is not a unique and any $\alpha + d\beta$ works as well. Setting a particular $\beta$ is to make a Gauge choice. For example for the magnetic field $B = \mathrm{curl}(A + \mathrm{grad}(f))$.

## 2.4 Cartan's magic formula

We go back to the math textbooks [JJ08, LCC+09, Fra11]. Here are some remarks related to the boundaries (see for example Figure 2): The boundary of the transported submanifold is just the transport of the boundary of the initial submanifold:

$$ \partial[V(t)] = [\partial V](t). $$

As a consequence we have this Proposition.

**Proposition 11.** With $X$ a tangent vector field associated to a flow $\phi_t$ and $\alpha$ a $k$ form we have

$$ \phi_t^*(d\alpha) = d\phi_t^*(\alpha) \quad \text{and} \quad dL_X\alpha = L_X(d\alpha). $$

We also remark following. The boundary of $\mathcal{I}_t(V)$ has three terms: $V$, $V(t)$ and the union of $\partial V(s)$ for $s \in [0, t]$. The latter also corresponds to $\mathcal{I}_t(\partial V)$. Therefore we have$^4$

$$ [I_t d\beta](V) = d\beta(\mathcal{I}_t(V)) = \beta(V(t)) - \beta(V(0)) - \beta(\mathcal{I}_t(\partial V)) $$

and then obtain the relation known as Cartan's magic formula.

**Proposition 12.** (Cartan's magic formula) With $X$ a tangent vector field associated to a flow $\phi_t$ and $\alpha$ a $k$ form we have

$$ I_t d\alpha = \phi_t^*(\alpha) - \alpha - dI_t\alpha \quad \text{and} \quad L_X = d \circ i_X + i_X \circ d. \qquad (3) $$

<footnote>$^4$The orientation of the boundary parts is a bit tricky.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 5: A “Geometric proof” of Cartan’s formula $i_X \circ d = L_X - d \circ i_X$ seen as a consequence of $\partial[I_t(V)] = V(t) \cup V(0) \cup I_t(\partial V) : i_X \circ d$ 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.</figcaption></figure>

## 2.5 Lie Derivative and Hydrodynamics

In the following small second sections, we add two applications of the Lie Derivative. First in the context of fluid hydrodynamics, it is natural to consider exteriors forms $\alpha_t$ that evolve in time, such that for transported submanifolds $V(t)$, the quantity $\alpha_t(V(t))$ stay constant. We would then have $\alpha_t = \phi_{-t}^*(\alpha_0)$. And if we add some modifications of $\alpha_t$ along the way, we obtain the following equation.

**Definition 13.** (Euler Equation) $\partial_t \alpha + L_X \alpha = $ ”additional terms”$^5$.

Here again, our main message is that one need to be careful whether $\alpha$ is a 0,1,2,3 form to compute $L_X \alpha$ in a coordinate system. We can write

$$L_X \alpha = \sum_{i=1}^{3} X_i L_{e_i} \alpha + \sum_{i=1}^{3} dX_i \wedge i_{e_i} \alpha.$$

with $X = \sum_{i=1}^3 X_i e_i$ and the first term is the usual $\vec{X}$.grad that appears in the Lagrangian derivative. We could call it a “translation term”. For a general tangent vector field $X$, the submanifolds are deformed along the flow (rotation, dilatation,...) and we need to add what we could call a “deformation term” for which we can use the following table.

<table><thead><tr><td>dimension</td><td>deformation term</td><td>examples of equations</td></tr></thead><tbody><tr><td>0-form f</td><td>Ø</td><td>Lagrangian derivative</td></tr><tr><td>1-form A</td><td>(∇X)A</td><td>Deformation gradient</td></tr><tr><td>2-form B</td><td>(B.∇)X - (∇.X)B</td><td>Cauchy momentum equation, the vorticity equation</td></tr><tr><td>3-form ρ</td><td>ρ(∇.X)</td><td>Continuity equation</td></tr></tbody></table>

Where we use the notations $\nabla X = (\partial_i X_j)_{1\le i,j \le 3}$ for the Jacobian, $\nabla . X = \sum_{i=1}^3 \partial_i X_i$ for the divergence and $B.\nabla = \sum B_i \partial_i$ for the Lagrangian derivative.

<footnote n="5">⁵Such as exterior forces, pressure, viscosity, source...</footnote>
## 2.6 Lie Derivative and forces applied at the boundary

To consider the total force applied on a submanifold $V$ one would simply write $\overrightarrow{F} = \int_V \overrightarrow{F}$. Unfortunately if $V \subset \Omega$ a general manifold ($\neq \mathbb{R}^d$) one can't just sum vectors that are defined at different points and the last equation doesn't make any sense. To overcome this difficulty we will consider a displacement described by a tangent vector field $X$ and consider the work generated by the force as the displacement is made. We call the quantity $F_X = \int_V \overrightarrow{F} \cdot \overrightarrow{X}$ the force in $X$-direction. For example in the case $\Omega = \mathbb{R}^3$ and a constant vector field $\overrightarrow{X} = \overrightarrow{e_x}, \overrightarrow{e_y}$ or $\overrightarrow{e_z}$ (that describe a translation) one recover the standard definition $\overrightarrow{F} = (F_x, F_y, F_z)$ where $F_x = \int_V \overrightarrow{F} \cdot \overrightarrow{e_x}$. But here $F_X$ is a very general notion because $X$ can then be any tangent vector field. For example in the case of $X$ are the rotations it gives the moment of force. And more generally it could be the force associated to any deformation as for example a dilatation or a contraction. This is the quantity that can be generalized in the exterior form formalism.

**Definition 14.** If $\alpha$ describes the energy of a system and $X$ is a tangent vector field then $L_X \alpha$ is the associated force in the $X$-direction.

We then have a 0,1,2 or 3 form whether it is a ponctual force, a linear force, a surface force or a volume force.

Here is the motivation for such a definition. If $V$ is submanifold that describe a physical object and $\alpha(V)$ is the energy of the system. The tangent vector field $X$ defined a displacement of the object as the transported manifold $V(s)$. Then the variation of the energy is given by

$$ -\frac{d}{ds}\alpha(V(s)) = \frac{d}{ds}[\phi_s^*\alpha](V) = (L_X\alpha)(V). $$

**Remark 15.** If $d\alpha = 0$ then $^{6}(L_X\alpha)(V) = (i_X\alpha)(\partial V)$.

We mention here a few examples.

* Archimedes principle: For the pressure which is a 3-form $P$ that describes the energy per volume. Then the force on a volume $V$ is $L_X P(V)$ which may correspond to the weight of the fluid and is also given by $(i_X P)(\partial V)$ which is pressure force integrated at the surface.

* Magnetic force: For the magnetic field $B$ is a 2-form and a surface $S$ for which $\partial S$ is a closed loop with a circulating electric courant $i_0$, the energy is given by $i_0 B(S)$ and then the force is $i_0 L_X B(S)$ which is the variation of the magnetic flux. But it is also given by $i_0(i_X B)(\partial S)$ which in a coordinate system reads $i_0 \oint_{\partial S} \overrightarrow{X} \cdot \overrightarrow{B} \times \overrightarrow{d\ell}$ (Lorentz force).

<footnote n="6">⁶Cartan formula</footnote>
# 3 The metric and Maxwell equations

## 3.1 The volume form

Surprisingly so far we only have a manifold $\Omega$ and didn't attached any metric on it<fnref n="7" />. But of course it is of fundamental importance in physics. Formally a metric symmetric 2-tensor $g_{ij}$ so it is neither a form nor a tangent vector. The first use of the metric for differential form is the definition of a “volume form” $\nu$.

**Definition 16.** A metric $g$ locally gives a orthonormal basis $dx_1, dx_2, \cdots, dx_n$ and we define

$$ \nu = dx_1 \wedge dx_2 \wedge \cdots \wedge dx_n. $$

For example once you have defined the meter and what orthogonal means then you can construct the meter-cube. And then have a 3-form $\nu$ such that for any 3 dimensional submanifold $\mathcal{V}$

$$ \nu(\mathcal{V}) = \text{"Volume of } \mathcal{V} \text{ in } m^3\text{"}. $$

## 3.2 The ⋆ operator

From a purely mathematical point of view of the exterior forms the metric also appears in a quite indirect way. Remark that the space of $k$-form has the same dimension as the space of $(n-k)$ form. They are therefore isomorphic. The $g$ metric gives a method to construct such an isomorphism that is called the Hodge ⋆ operator. Again a metric $g$ locally gives a orthonormal basis $dx_1, dx_2, \cdots, dx_n$ and to give a very simplified definition.

**Definition 17.** That ⋆-operator associates a $k$-form to a $(n-k)$-form that is the “complement” in the basis<fnref n="8" />:

$$ \star \left( \bigwedge_{i \in J} dx_i \right) = \pm \bigwedge_{i \notin J} dx_i \quad \text{for } J \subset \{1, \cdots, n\} \text{ with k element} $$

For example : $\star(dx) = dy \wedge dz \wedge dt$ and $\star(dx \wedge dt) = dy \wedge dz$. Notice that in physics it happens that we use the same name for objects that may be different at a formal mathematical level. For example one usually doesn't make a difference between a density $\rho = \rho(x)\nu$ that is a 3-form, and the function $\rho(x)$ is a 0-form. Or a “vector field $X$” may refer to the tangent vector field, or to $n-1$-form $i_X \nu$ induced by this tangent vector field, or to the 1-form $\alpha = \star i_X \nu$.

$$ \begin{array}{ccc} \text{1-form} & & (n-1)\text{-form} \\ \star\alpha & \leftrightarrow & \alpha = i_X\nu \quad \leftrightarrow \quad X \end{array} $$

<footnote n="7">⁷For mathematicians we use differential geometry but not Riemannian geometry</footnote>

<footnote n="8">⁸The formula is just wrong because we do not precise the sign which depend on the order of the $dx_i$ and on the signature of $g$. But from the beginning we didn't precise the orientation of the submanifold neither... We refer to the previously mentioned textbook for a correct definition.</footnote>
In Newton second law for instance, we identify a acceleration which is a tangent vector to a force which is a 1 form (ma = F = grad(f)). In all these transformations, to be rigorous one should add the metric g and/or the factor $\sqrt{\det g}$. If g is the identity matrix, this is of course never written down, but it has to be done for example in relativity when "moving up and down indices": $J^\mu \to J_\mu$ or $F^{\mu\nu} \to F_{\mu\nu}$.

## 3.3 The Laplace-De Rham Operator

We now define the Laplace-De Rham Operator.

**Definition 18.** (Laplace-De Rham)

$$ \Delta = \partial \circ d + d \circ \partial $$

where $\partial = \star \circ d \circ \star$. It is an operator that from a k-form gives an k-form.

In $\mathbb{R}^3$ and for a 0-form $f$ (or 3-form), because $\partial f = 0$ this is the usual formula

$$ \Delta f = \operatorname{div}(\operatorname{grad}(f)) $$

and for a 1-form $u$ (or 2-form), we have the well known<fnref n="9" />

$$ \Delta u = \operatorname{grad}(\operatorname{div}(u)) - \operatorname{curl}(\operatorname{curl}(u)). $$

For a general manifold $\Omega$ with metric $g$ and a 0-form $f$, $\Delta f = (\star d \star d) f$ which reads

$$ \Delta f = \frac{1}{\sqrt{\det g}} \sum_{ij} \partial_j (\sqrt{\det g} g^{ij} \partial_i f). $$

In the Minkovski space $\mathbb{R}^{1,3}$, because $g$ is not positive we obtain d'Alembert operator and denote $\square = \partial \circ d + d \circ \partial$ instead. In the case $d\alpha = 0$ (or $\partial\alpha = 0$) we may also denote $\square\alpha = \star d \star \alpha$ (or $\square\alpha = d \star d\alpha$).

## 3.4 Maxwell equations

One of the most beautiful application of the exterior form formalism is that it gives a clean and unified picture of the classical theory of electromagnetism [Fla63, Fra11]. Here are the Maxwell equations written with differential forms where we drop the constant $\epsilon_0, \mu_0, c$. Maxwell Theory is given by the wonderful table

<table><tr><td>Λ<sup>0</sup>(ℝ<sup>4</sup>)</td><td>d/∂</td><td>Λ<sup>1</sup>(ℝ<sup>4</sup>)</td><td>d/∂</td><td>Λ<sup>2</sup>(ℝ<sup>4</sup>)</td><td>d/∂</td><td>Λ<sup>3</sup>(ℝ<sup>4</sup>)</td><td>d/∂</td><td>Λ<sup>4</sup>(ℝ<sup>4</sup>)</td></tr><tr><td>f</td><td>(1)<br>(2)</td><td>(V, A)</td><td>(3)<br>→</td><td>(E, B)</td><td>(4)<br>→</td><td>0</td><td></td><td></td></tr><tr><td colspan="4"></td><td>★(E, B)</td><td>(5)<br>→</td><td>(ρ, j)</td><td>(6)<br>→</td><td>0</td></tr></table>

<footnote n="9">⁹But seen here as the definition of $\Delta u$.</footnote>
Notice that the dimension of $\Lambda^1(\mathbb{R}^4)$, $\Lambda^2(\mathbb{R}^4)$ and $\Lambda^3(\mathbb{R}^4)$ are 4, 6 and 4. We denote here

* $A^\mu = (V, A)$: the potential and potential vector (usually a 4-vector). This is a 1-form as it is linked to the phase of the wave function of the electron (as for example in the Aharonov-Bohm effect).

* $F = (E, B)$: the electromagnetic field (usually already a antisymmetric 2-tensor). This is a 2-form, one would want to integrate the magnetic field on a surface for example.

* $J^\mu = (\rho, j)$: charge density and current (usually a 4-vector). This is a 3-form, the density is integrate on a volume and the current on a surface times a time intervale.

A great advantage here that everything is at the “geometric” level. There is no choice of parametrisation of the space. No worry about the change of referencial.

Here every arrow of the diagramm is just the exterior derivative. Below are their usual meaning:

(1) This is a Gauge invariance, one can change

$$V \to V - \frac{\partial f}{\partial t} \quad \text{et} \quad A \to A + \text{grad}(f)$$

without modifying electromagnetic field, indeed $(3) \circ (1) = 0$.

(2) A particular choice of Gauge called Lorentz's Gauge (2) = 0 that is

$$\frac{\partial V}{\partial t} + \text{div}(A) = 0.$$

(3) Electromagnetic fields are expressed with the potential and potential vector:

$$E = -\frac{\partial A}{\partial t} - \text{grad}(V) \quad \text{et} \quad B = \text{curl}(A).$$

(4) Here are the Maxwell-Faraday et Maxwell-Thomson equation

$$\frac{\partial B}{\partial t} + \text{curl}(E) = 0 \quad \text{et} \quad \text{div}(B) = 0$$

This follows of course from $(4) \circ (3) = 0$.

(5) Here are now Maxwell-Gauss and Maxwell-Ampere equations

$$\text{div}(E) = \rho \quad \text{et} \quad -\frac{\partial E}{\partial t} + \text{curl}(B) = j$$

(6) This is the conservation law of the charge

$$\frac{\partial \rho}{\partial t} + \text{div}(j) = 0$$

Again that is $(6) \circ (5) = 0$.
We can finish by a small historical remark, in 1865 the fantastic idea of Maxwell were to notice that $(6)\circ(5) \neq 0$ with Ampere equation as stated at that time. He modified the equation adding the term $\frac{\partial E}{\partial t}$ to obtain a coherent theory. Therefore it is indeed a differential geometry approach that gave nowday's classical electromagnetic theory.

# 4 De Rham (Trivial) Cohomologie

We have seen in Section 2 that if a form $\alpha$ can be written as $\alpha = d\beta$ it satisfies $d\alpha = 0$. A natural question would be to asked whether the converse is always true. The answer is yes for $\mathbb{R}^n$.

**Proposition 19.** (De Rham (trivial) Cohomology) In $\mathbb{R}^n$, for any form $\alpha$ that is not a constant 0-form

$$d\alpha = 0 \Leftrightarrow \text{ there exists } \beta \text{ such that } \alpha = d\beta.$$

This proposition is not true for more complicated topological space. It gives rise to a whole domain of study called De Rham Cohomology which happens to be one of the most powerful tools to study and characterize topological object in differential geometry and algebraic topology<fnref n="10" /> [God70]. But in $\mathbb{R}^3$ all this is very well known by any undergraduate student:

* $\alpha$ is a 0-form: $\mathrm{grad}(\alpha) = 0$ iff there exists $c \in \mathbb{R}$ constant such that $\alpha = c$,

* $\alpha$ is a 1-form: $\mathrm{curl}(\alpha) = 0$ iff there exists a 0-form $\beta$ such that $\alpha = \mathrm{grad}(\beta)$,

* $\alpha$ is a 2-form: $\mathrm{div}(\alpha) = 0$ iff there exists a 1-form $\beta$ such that $\alpha = \mathrm{curl}(\beta)$,

* $\alpha$ is a 3-form: The equation $\mathrm{div}(\beta) = \alpha$ always has a solution.

Notice that it is possible to construct $\beta$ explicitly. For example, if $\alpha$ is a 1-form, define the 0 form $f(x) = \int_{x_0}^x \alpha$ with $x_0$ fixed and then $\alpha = \mathrm{grad}(f)$. Or if $\rho_0$ is a 3-form seen as the density of mass that has been transported from far away, we have with the continuous equation

$$\rho_0 = - \int_{-\infty}^{0} \mathrm{div}(\vec{v}_t \rho_t) dt = \mathrm{div} \left( - \int_{-\infty}^{0} \vec{v}_t \rho_t dt \right).$$

In higher dimension, it also implies important properties.

**Corollary 20.** If $J$ is a conserved quantity, then there exists $\beta$ such that $d\beta = J$.

<footnote n="10">¹⁰There, the formula $\phi_t^*(\alpha) = \alpha + dI_t\alpha$ is still very useful since it implies that $\alpha$ and $\phi_t^*\alpha$ belongs to the same “class”.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 6: A “Geometric proof” of $d\alpha = 0 \Rightarrow \alpha = d\beta$. Choose a tangent vector field which induces a contraction of the space to a single point. Then in Cartan’s magic formula, the terms $\alpha(\mathcal{V}(t))$ and $\alpha(\partial(I_t \mathcal{V}))$ vanish and we are left with $0 = -\alpha(\mathcal{V}) - \alpha(I_t(\partial\mathcal{V})).$ Therefore $\alpha = d(-I_t\alpha)$.</figcaption></figure>

In particular in $\mathbb{R} \times \mathbb{R}^3$ for any $(\rho, j)$ that satisfies the continuous equation there exists a field $\beta = (\mathcal{E}, \mathcal{B})$ for which Maxwell-Gauss and Maxwell-Ampere equations are valid

$$ \mathrm{div}(\mathcal{E}) = \rho \quad \text{and} \quad -\frac{\partial \mathcal{E}}{\partial t} + \mathrm{curl}(\mathcal{B}) = j. $$

This is a purely mathematical construction. This field should be seen as an equivalent to the potential flow that describes an irrotational velocity field as a gradient in hydrodynamics.

Also such a $\beta$ is not unique and one can make a choice of Gauge. In the case $d(\star\beta) = 0$ we have the following

**Corollary 21.** If $J$ is a conserved quantity, then there exists $\alpha$ such that

$$ d \star d\alpha = J. $$

In a coordinate system it is the usual propagation equation with a source

$$ \Box \alpha = J \qquad (4) $$

We should stress that this is true for any conserved quantity with no more information about the physics of the system. However there are important examples where $\alpha$ appears in theoretical model or is indeed a real physical quantity. For example:

* The electric charge is conserved. We have the electromagnetic field: (4) with $J = (\rho, j)$ the charge density and current and $\alpha = (V, A)$ potential and vector potential.

So we can ask the following.

**Question 22.** What is the associated $\alpha$ for the conservation of energy, momentum, moment of inertia, weak charge, ...?
# 5 Euler Lagrange Equations

This last section is more “theoretical physics” than “daily physics” as we will focus on the Lagrangian approach for a classical field theory and its very nice consequences such as Noether Theorem. Again we will express everything using exterior forms.

## 5.1 Euler Lagrange Equations

As a most simple model the Lagrangian $\mathcal{L}$ is a $n$-form<fnref n="11" /> and we assume the following.

**Assumption 23.** $\mathcal{L} = \mathcal{L}(\alpha, d\alpha)$ only depends only on a $k$-form $\alpha$ and its exterior derivative $d\alpha$.

For $k \ge 1$, the last hypothesis is more restrictive than considering all the derivatives $(\frac{\partial \alpha}{\partial x_i})_{1 \le i \le n}$ but is also reasonable if we believe that $d\alpha$ has more geometric or physical meaning<fnref n="12" />. Also a nice aspect of this formalism is that it gives a justification why the second order derivatives $\frac{\partial^2 \alpha}{\partial x_i \partial x_j}$ are not used in the Lagrangian : indeed $d \circ d\alpha = 0$.

We write $\mathcal{L}_{\alpha} := \partial_1 \mathcal{L}(\alpha, d\alpha)$ that is an $(n-k)$-form and $\mathcal{L}_{d\alpha} := \partial_2 \mathcal{L}(\alpha, d\alpha)$ that is a $(n-k-1)$-form such that for a small perturbation $\delta\alpha$ at first order we have

$$ \mathcal{L}(\alpha + \delta\alpha, d\alpha + d\delta\alpha) \approx \mathcal{L}(\alpha, d\alpha) + \delta\alpha \wedge \mathcal{L}_{\alpha} + (d\delta\alpha) \wedge \mathcal{L}_{d\alpha} $$

With this formulation we also assume that $\mathcal{L}$ depends only locally on $\alpha$, in the sense that if $\delta\alpha$ is supported on a region $\Omega' \subset \mathbb{R}^n$ then $\mathcal{L}$ is not modified outside of $\Omega'$. We now write down the Euler-Lagrange Equation. Remark that we have

$$ \mathcal{L}(\alpha + \delta\alpha, d\alpha + d\delta\alpha) \approx \mathcal{L}(\alpha, d\alpha) + \delta\alpha \wedge (\mathcal{L}_{\alpha} - (-1)^k d\mathcal{L}_{d\alpha}) + d(\delta\alpha \wedge \mathcal{L}_{d\alpha}). $$

For a submanifold $\mathcal{V}$, in order to maximise $\int_{\mathcal{V}} \mathcal{L}(\alpha, d\alpha)$ with fixed boundary conditions on $\partial\mathcal{V}$ for $\alpha$ we obtain the following.

**Definition 24.** (Euler Lagrange Equation)

$$ \mathcal{L}_{\alpha} - (-1)^k d\mathcal{L}_{d\alpha} = 0 $$

For $\alpha$ a 0-form this is the usual Euler Lagrange Equation in Field Theory

$$ \frac{\partial \mathcal{L}}{\partial \alpha} = \partial_{\mu} \left( \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \alpha)} \right). $$

A nice example with $\alpha$ a 1-form is of course the electromagnetic field with $\alpha = (V, A)$ and $J = (\rho, j)$ we have

<footnote n="11">¹¹Most commun conventions call the Lagrangian $\star\mathcal{L}$ which is then a 0-form (a scalar).</footnote>

<footnote n="12">¹²If $\alpha$ is a 0-form then considering $d\alpha$ is similar to considering all the derivatives $(\frac{\partial \alpha}{\partial x_i})_{1 \le i \le n}$.</footnote>
**Definition 25.** (Electromagnetism Lagrangian)

$$ \mathcal{L}^{(\text{EM})} = \alpha \wedge J - \frac{1}{2}(d\alpha \wedge \star d\alpha). $$

Here $\mathcal{L}_{\alpha}^{(\text{EM})} = J$, $\mathcal{L}_{d\alpha}^{(\text{EM})} = \star d\alpha$ and the Euler Lagrange Equations gives again the Maxwell equations :

$$ d \star d\alpha = J $$

## 5.2 A conserved quantity in the 1-Form case

Remark that Euler Lagrange Equation directly implies that $d\mathcal{L}_{\alpha} = 0$. So in particular we have the following.

**Corollary 26.** If $\alpha$ is a 1-form then $\mathcal{L}_{\alpha}$ is a conserved quantity and $\mathcal{L}_{d\alpha}$ is an associated field<fnref n="13" />.

One way to understand Corollary 26 is that assumption 23 add a Gauge symmetry to the system. For example with a perturbation of the form $\delta\alpha = d(\delta f)$ we have $d(\delta\alpha) = 0$ and then

$$ \mathcal{L}(\alpha + d\delta f, d\alpha) \approx \mathcal{L}(\alpha, d\alpha) - \delta f \wedge d\mathcal{L}_{\alpha} + d(\delta f \wedge \mathcal{L}_{\alpha}). $$

and finally obtain $d\mathcal{L}_{\alpha} = 0$ as for the Euler Lagrange Equation. Again we have a nice example with the electromagnetic field where $\mathcal{L}_{\alpha}^{(\text{EM})} = J$ is a conserved quantity and $d\mathcal{L}_{d\alpha}^{(\text{EM})} = J$.

## 5.3 Noether Theorem

We now mention the famous Noether Theorem. Remark that Euler Lagrange equation implies that

$$ \mathcal{L}(\alpha + \delta\alpha, d\alpha + d\delta\alpha) - \mathcal{L}(\alpha, d\alpha) \approx d(\delta\alpha \wedge \mathcal{L}_{d\alpha}) $$

We denote $\xi$ an infinitesimal transformation $\alpha \to \alpha + \delta\alpha^{\xi}$ and $\mathcal{L} \to \mathcal{L} + \delta\mathcal{L}^{\xi}$. If the system is invariant by this transformation $\delta\mathcal{L}^{\xi} = 0$ then $d(\delta\alpha^{\xi} \wedge \mathcal{L}_{d\alpha}) = 0$ so $\delta\alpha^{\xi} \wedge \mathcal{L}_{d\alpha}$ is a conserved quantity. We can state a more general result [Olv93].

**Theorem 27.** (Noether Theorem) If $\delta\mathcal{L}^{\xi} = d(\delta\Lambda^{\xi})$ then $\delta\alpha^{\xi} \wedge \mathcal{L}_{d\alpha} - \delta\Lambda^{\xi}$ is a conserved quantity.

A very nice application of Noether Theorem is of course the conservation of energy and momentum.

<footnote n="13">¹³As in Corollary 20</footnote>
## 5.4 The stress-energy tensor

We consider translations of the system and more generally the transport along a flow given by vector field $X$. Notice that we have $L_X \mathcal{L} = d(i_X \mathcal{L})$ (Cartan's magic formula) so we can apply Noether Theorem. We compute

$$L_X \mathcal{L} = L_X \alpha \wedge \mathcal{L}_\alpha + (dL_X \alpha) \wedge \mathcal{L}_{d\alpha} + L_X \mathcal{L}|_{\alpha,d\alpha}$$

where the last term is the derivative of $\mathcal{L}$ assuming $\alpha, d\alpha$ fixed and then we obtain

$$d(L_X \alpha \wedge \mathcal{L}_{d\alpha} - i_X \mathcal{L}) = -L_X \mathcal{L}|_{\alpha,d\alpha}.$$

For example in the case of the electromagnetism Lagrangian and $X = \partial_t$ for translation in time we obtain Poynting's theorem

$$\partial_t(\frac{1}{2}(|E|^2 + |B|^2) + \vec{A} \cdot \vec{j}) + \text{div}(E \times B) = \vec{A} \cdot \partial_t \vec{j}.$$

For more general Lagrangian $\mathcal{L}$ we can define the following.

**Definition 28.** We call $L_X \alpha \wedge \mathcal{L}_{d\alpha} - i_X \mathcal{L}$ the stress-energy tensor if $X$ is a translation.

In a coordinate system and with the translations $X = \partial_\nu$, this is the usual formula for the stress-energy tensor defined from a Lagrangian

$$T_{\mu\nu} = \partial_{\nu}\alpha^{\eta}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\alpha)}\right)_{\eta} - \eta_{\mu\nu}\mathcal{L}.$$

And we also state the conservation of energy and momentum in a general setting.

**Corollary 29.** If $L_X \mathcal{L}|_{\alpha,d\alpha} = 0$, ie. the Lagrangian is invariant by the transformation induced by the vector field $X$, then $(L_X \alpha) \wedge \mathcal{L}_{d\alpha} - i_X \mathcal{L}$ is a conserved quantity.

## 5.5 Hamilton Equation

The energy tensor

$$\mathcal{H} = L_X \alpha \wedge \mathcal{L}_{d\alpha} - i_X \mathcal{L}.$$

corresponds to the definition of the Hamiltonian in the case $X = \partial_t$. With some computation we obtain for small perturbation

$$\delta\mathcal{H} = L_X \alpha \wedge \delta\mathcal{L}_{d\alpha} - \delta\alpha \wedge L_X \mathcal{L}_{d\alpha} - di_X(\delta\alpha \wedge \mathcal{L}_{d\alpha}) \\ - i_X(\delta\alpha \wedge (\mathcal{L}_\alpha - (-1)^k d\mathcal{L}_{d\alpha}))$$

There if we assume that $\mathcal{H} = \mathcal{H}(\alpha, \mathcal{L}_{d\alpha})$ and that the small perturbation is given by

$$\delta\mathcal{H} = \delta\alpha \wedge H_1 + \delta\mathcal{L}_{d\alpha} \wedge H_2$$

we obtain the Hamilton Equations.
**Proposition 30.** (Hamilton Equations) Euler Lagrange Equations are equivalent to

$$H_1 = -L_X \mathcal{L}_{d\alpha} \quad \text{and} \quad H_2 = (-1)^{k+1} L_X \alpha.$$

*Remark 31.* The stationary form (for which $L_X \alpha = 0$ and then $L_X \mathcal{L}_2 = 0$) which that are critical point of the Hamiltonian ($\delta H = 0$ for any $\delta \alpha$) solve Euler Lagrange equation. In particular if $\alpha$ is such that it minimizes the energy.

## 5.6 Gravitational waves (?)

We finish by giving a partial answer for Question 22. One can find in a book on General Relativity this equation<fnref n="14" /> used to describe gravitational waves

$$\Box \tilde{h} = T$$

where $\tilde{h}$ is constructed with the perturbation of the metric $g$ around the flat Minkowski metric and $T$ is the $4 \times 4$ Stress-Energy tensor. Here one can think of each line of $T$ as a 3-form which corresponds to the conservation of energy (first line) and the conservation of momentum (the three others lines) and then the line of $\tilde{h}$ play the role of the associated $\alpha$ in Corollary 21 and the associated $\beta$ in Corollary 20 can be interpreted as the classical gravitational field.

Unfortunately general relativity is much more complicated and the above statement is true only at first order.

## 6 Summary table

<table><tr><td>$m^{-2}s^{-1}$</td><td>$dx \wedge dy \wedge dt$</td></tr><tr><td>gradient, curl, divergence</td><td>$d$</td></tr><tr><td>irrotational, divengence free, conserved quantity</td><td>$d\alpha = 0$</td></tr><tr><td>Gauge invariance</td><td>$d(\alpha + d\beta) = d\alpha$</td></tr><tr><td>De Rham cohomology</td><td>$d\alpha = 0 \Rightarrow \alpha = d\beta$ ?</td></tr><tr><td>Hydrodynamics</td><td>$\partial_t \alpha + L_V \alpha = ...$</td></tr><tr><td>Maxwell equations</td><td>$dF = 0, \quad d \star F = 0$</td></tr><tr><td>Propagation wave with source</td><td>$d \star d\alpha = J$</td></tr><tr><td>Euler Lagrange equation</td><td>$\mathcal{L}_{\alpha} - (-1)^k d\mathcal{L}_{d\alpha} = 0$</td></tr><tr><td>Stress-energy tensor</td><td>$L_X \alpha \wedge \mathcal{L}_{d\alpha} - i_X \mathcal{L}$</td></tr></table>

## References

[BBB85] William L Burke, William L Burke, and William L Burke. *Applied differential geometry*. Cambridge University Press, 1985.

[Fla63] Harley Flanders. *Differential forms with applications to the physical sciences*, volume 11. Courier Corporation, 1963.

<footnote n="14">¹⁴Only true in the weak gravitation regime where Einstein equation can be linearized.</footnote>
[Fra11] Theodore Frankel. *The geometry of physics: an introduction*. Cambridge university press, 2011.

[God70] Claude Godbillon. *Element de topologie algebrique*. Hermann, 1970.

[JJ08] Jürgen Jost and Jeurgen Jost. *Riemannian geometry and geometric analysis*, volume 42005. Springer, 2008.

[LCC+09] Jeffrey M Lee, Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, et al. Manifolds and differential geometry. *Topology*, 643:658, 2009.

[Olv93] Peter J Olver. *Applications of Lie groups to differential equations*, volume 107. Springer Science & Business Media, 1993.

[Pau07] Frédéric Paulin. Géométrie différentielle élémentaire. *Notes de cours de première année de mastère, Ecole Normale Supérieure*, 2007.