# 1 Theory¶

## 1.0 Introduction¶

The package `DifferentialForms`

(in file `dform.spad`

) builds on the
domain `DeRhamComplex`

. In the following section we give a brief overview
of the functions that are going to be implemented. The focus is on precise
definitions of the notions, since those may be varying in the literature.
In section (2) we will describe the exported functions and how they work,
in section (3) some short implementation notes will be given and finally
the last section is devoted to some examples.

## 1.1 Definitions¶

Let \(\mathcal{M}\) be a n-dimensional manifold (sufficiently smooth and orientable). To each point \(P \in \mathcal{M}\) there is a neighborhood which can be diffeomorphically mapped to some region in \(\mathbb{R}^n\), with coordinates

for all \(P' \in \mathcal{U} (P) \subset \mathcal{M}\). The tangent space \(T_{P'} (\mathcal{M})\) at the point \(P'\) is a vector space, that is spanned by the basis

which also is often denoted by

A tangent vector \(v\) has the form

The cotangent space \(T_{P'}^{} (\mathcal{M})^{\star}\) is the vector space of linear functionals

spanned by the basis \(e^1 (P'), \ldots, e^n (P')\) which (corresponding to the basis \(\partial_j\)) is also denoted by

The latter notation indicates the dependency on the moving point \(P'\). The dual basis is by definition comprised of those linear functionals such that

Therefore we have

where \(\alpha = \sum_{j = 1}^n \alpha_j e^j\).

### 1.1.1 Inner product of differential forms (**dot**)¶

Let \(g_x\) be a symmetric \(n \times n\) matrix which is nondegenerate (i.e. \(\det (g_x) \neq 0\)). The index x indicates that this matrix depends on the coordinates \(x_1 (P), \ldots, x_n (P)\) and may be varying from point to point. If this dependency is smooth (enough) we speak of a (pseudo-) Riemannian metric (locally). This way we get an isomorphism between tangent vectors and 1-forms (= covectors):

Clearly, \(\sum_k g^{j k} g_{k l} = \delta^j_l\), in other words
\((g^{j k})\) is the inverse of g. The metric g defines an *inner product*
of vectors

and by duality also on 1-forms:

Now, this inner product is extended to arbitrary p-forms by

and linearity.

### 1.1.2. The volume form \(\eta\) (**volumeForm**)¶

The Riemannian *volume form* \(\eta\) is (by definition) given by the
n-form

This definition makes sense because by a (orientation preserving) change of coordinates \(\sqrt{\mathrm{det} g}\) transforms like the component of a n-form.

### 1.1.3. Hodge dual (**hodgeStar**)¶

The *Hodge dual* of a differential p-form \(\beta\) is the (n - p)-form
\(\star \beta\) such that

holds, for all p-forms \(\alpha\). The linear operator \((\star)\) is
called the *Hodge star operator*. By the *Riesz representation theorem* the
Hodge dual is uniquely defined by the expression above.

**Warning**

Flanders [4] defines the Hodge dual by the equality

where \(\lambda\) is a p-form and \(\mu\) a (n - p)-form. This can result in different signs (actually \(\star_F = s(g)\star\), where \(s(g)\) is the sign of the determinant of \(g\)).

The generally adopted definition (2016) is the one given at the beginning of this subsection.

The components of \(\star \beta\) are

what is equal to

### 1.1.4 Interior product (**interiorProduct**)¶

The *interior product* of a vectorfield \(v\) and a p-form \(\alpha\)
is a (p -1)-form \(i_v (\alpha)\) such that

holds, for all vectorfields \(v_1, \ldots, v_{p - 1}\). Therefore, the components of \(i_v (\alpha)\) are

One can express the interior product by using the \(\star\)-operator. Let \(\alpha\) be the 1-form defined by the equation

That means in components: \(\alpha_j = g_{j k} v^k\), thus we have

Clearly, the interior product is independent of any metric, whereas the
Hodge operator is **not**! So, usually one should not use the Hodge
operator to compute the interior product.

We will use the fact that the interior product is an *antiderivation*,
which allows a recursive implementation.

### 1.1.5 The Lie derivative (**lieDerivative**)¶

The *Lie derivative* with respect to a vector field \(v\) can be
calculated (and defined) using Cartan’s formula:

There are other ways to define \(\mathcal{L}_v \alpha\), however, it is convenient to compute it this way when \(d\) and \(i_v\) are already at hand.

### 1.1.6 The CoDifferential \(\delta\) (**codifferential**)¶

The *codifferential* \(\delta\) is defined on a p-form as follows:

where \(g\) is the metric and \(s(g)\) is related to the
**signature** of \(s(g)\) as described next.

### 1.1.7 The sign of a metric \(s(g)\) (**s**)¶

The signature of a metric \(g\) is defined as the difference of the number of positive (p) and negative (q) eigenvalues, i.e:

and the **sign** functions **s** is defined as:

Since, as we always assume, \(g\) is non-degenerate, we have \(p+q=n\), and consequently:

### 1.1.8 The inverse Hodge star \(\star^{-1}\) (**invHodgeStar**)¶

Applying the Hodge star operator twice on a p-form twice we get the identity up to sign:

Therefore

### 1.1.9 The Hodge-Laplacian \(\Delta_g\) (**hodgeLaplacian**)¶

The **Hodge-Laplacian** also known as *Laplace-de Rham operator* is
defined on any manifold equipped with a (pseudo-) Riemannian
metric \(g\) and is given by

Note that in the Euclidean case \(\Delta_g = - \Delta\), where latter is the ordinary Laplacian.

## Bibliography¶

References

[1] | Ralph Abraham, Jerrold E.Marsden and Tudor Ratiu.Manifolds, Tensor Analysis, and Applications. Springer, Auflage: 2nd Corrected ed. 1988. Corr. 2nd printing 1993 edition. |

[2] | Henri Cartan. Di erential Forms. Dover Pubn Inc, Au age: Tra edition. |

[3] | Herbert Federer. Geometric Measure Theory. Springer, Au age: Reprint of the 1st ed. Berlin, Heidelberg, New York 1969 edition. |

[4] | Harley Flanders and Mathematics. Differential Forms with Applications to the Physical Sciences. Dover Pubn Inc, Auflage: Revised. edition. |

[5] | L. A. Lambe and D. E. Radford. Introduction to the Quantum Yang-Baxter Equation and Quantum Groups:An Algebraic Approach. Springer, Auflage: 1997 edition. |

[6] | Walter Rudin and RudinWalter. Principles ofMathematicalAnalysis.Mcgraw Hill Book Co, Au age: Revised. edition. |

[7] | Hassler Whitney. Geometric Integration Theory: Princeton Mathematical Series, No. 21. Literary Licensing, LLC. |