Computational Geometry Workshop

These notes are a quite terse introduction to differential \(k\)-forms on \(\mathbb{R}^n\). I will fill in missing details and answer questions during the lecture. For more information, see one of the differential topology references on the References page. Specifically, Phil Parker's book-in-progress is a good place to start.

Last week we developed the theory of vector fields and 1-forms on a chart \(\left(U,(x^1,x^2,...,x^n)\right)\).

We now wish to consider the differential \(k\)-forms on \(U\), \(\Omega^k\). A differential \(k\)-form is a multi-linear alternating map \(\mathfrak{X}^k \to \mathfrak{F}\). To properly define these, we need the exterior algebra \(\wedge\, T^*U\); see [PP] and/or [MB], for example.

We will briefly review the ideas of exterior algebras in the seminar, but for the sake of these notes the most important piece of the definition is the word alternating.

In \(\wedge^2\, T^*U\), alternating means that \(dx^i \wedge dx^i = 0\) for all \(i = 1,...,n\). It can be shown that this implies skew-symmetry: \(dx^i \wedge dx^j = -dx^j \wedge dx^i\) for all \(i \neq j\). (You should show this!) Since we are working in real vector spaces, skew-symmetry also implies alternating. In \(\wedge^r\, T^*U\) for \(r > 2\), alternating means that transposing any two adjacent terms results in a factor of \(-1\).

Thus the space \(\wedge^k\, T^*_p U\), \(k \leq n\) is an \((_nC_k)\)-dimensional vector space spanned by elements of the form \(dx^{i_1} \wedge dx^{i_2} \wedge ... \wedge dx^{i_k}\) with \(i_1 < i_2 < \cdots < i_k.\) The \(k\)-forms are then \(\mathfrak{F}\)-linear combinations of these basis vectors in \(\wedge^k\, T^*U\).

The exterior derivative on forms is a generalizatoin of the differential of a function in \(\mathfrak{F}\), \(d: \Omega^k \to \Omega^{k+1}\). If \(\alpha \in \Omega^k\), then \(\alpha\) may be written in coordinates as \[ \alpha = \sum_{i_1 < \cdots < i_k}\alpha_{i_1i_2...i_k} dx^{i_1} \wedge dx^{i_2} \wedge ... \wedge dx^{i_k}. \] Then \(d\alpha\) is defined to be the \((k+1)\)-form \[ d\alpha = \sum_{i_1 < \cdots < i_k}d\alpha_{i_1i_2...i_k} \wedge dx^{i_1} \wedge dx^{i_2} \wedge ... \wedge dx^{i_k}, \] where \(d\alpha_{i_1i_2...i_k}\) is the usual differential of \(\alpha_{i_1i_2...i_k}\) in \(\mathfrak{F}\).

Finally, we notice that \(\wedge^0\,T^*U\) and \(\wedge^n\, T^*U\) are both 1-dimensional vector spaces (over \(\mathfrak{F}\)). Indeed, \(\Omega^0 = \mathfrak{F},\) while the basis element of \(\Omega^n\) is the (Euclidean) volume form on \(U\). That is, \[ dx^1 \wedge ... \wedge dx^n (X_1,...,X_n) \] gives the volume of the \(n\)-dimensional polyhedron determined by the vectors \(\left\{X_1,...,X_n\right\}\) in \(\mathbb{R}^n\).