2-forms and dual 0-forms

Functions on a manifold cannot be integrated per se. This is also the reason for the notation

$\displaystyle \int_U f \, dx_1 \ldots dx_n$

if one wants to integrate a function $f$ over a region $U\subset \mathbb{R }^n$. If you express $f$ in different coordinates  (for example polar coordinates) the integral changes (unless you stick in some Jacobi determinant). Objects living on regions in $\mathbb{R }^n$ that can be integrated independent of the choice of coordinates are called $n$-forms. In the above formula $dx_1 \ldots dx_1$ is such an $n$-form and its integral over $U\subset \mathbb{R }^n$ always calculates the n-dimensional volume of $U$, no matter which coordinates you use. Conversely, if you know the integral of an n-form over any subregion of $U$ you know the $n$-form.

This gives us a clue how to define 2-forms on our discrete surfaces: A 2-form $\sigma$ assigns to each face $\varphi$ a real number $\sigma(\varphi)$. We also use the notation

$\displaystyle \sigma(\varphi) =: \int_\varphi \sigma$.

If $U \subset F$ is a union of faces we define

$\displaystyle \int_U \sigma := \sum_{\varphi \in U} \, \int_\varphi \sigma$.

Also we write

$\displaystyle \int_M \sigma := \displaystyle \int_F \sigma$.

In many situations it is possible to an area to each face. Such an assignment of areas to faces is a special case of a 2-form, usually referred to as the volume form $\sigma_\textrm{vol}$ of $M$. The total area of $M$ is then

$\displaystyle \textrm{vol}(M):=\int_M \sigma_\textrm{vol}$.

Even in the presence of a volume form discrete functions $f \in \Omega_0(M)$ cannot be integrated. This is because they represent function values at vertices and there is no canonical way to multiply such an $f$ with $\sigma_\textrm{vol}$ in order to get a 2-form that can be intergrated.

We use this as a motivation to introduce another kind of objects: dual 0-forms. We think of them as real-valued functions on the surface that are constant on the interior of all faces. Formally speaking they are also just defined as functions

$f: F \rightarrow \mathbb{R}$.

The above remarks should have made it clear though that it is not wise to confuse dual 0-forms with 2-forms: functions cannot be integrated while this is perfectly possible for 2-forms. Functions can however be multiplied with 2-forms, the result being another 2-form:

$\displaystyle \int_\varphi (f\sigma) := f(\varphi) \int_\varphi \sigma$.

Note that this seems to be a reasonable formula for functions $f$ that are constant on each face.

We denote the vector space of all functions by $\Omega_0(M^*)$. This space is indeed canonically isomorphic to the dual vector space of $\Omega_1(M)$: To verify this statement it enough to exhibit a nondegenerate bilinear pairing

$\langle \,,\rangle: \Omega_0(M^*) \times \Omega_2(M) \rightarrow \mathbb{R}$

$\displaystyle\langle f,\sigma\rangle := \int_M f \sigma$.

Finally: If we are given a volume form, we can integrate dual functions by defining

$\displaystyle \int_M f :=  \int_M f \, \sigma_\textrm{vol}$.

Integral of dual functions over unions $U$ of faces can then be defined in a similar way.

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