1.0 Introduction
Differential forms occur in various fields of mathematics and physics providing
a method to formulate many objects in a way which is independent of a particular
coordinate system. Although primary usage certainly is in a theoretical context,
there are situations where actual computations have to be done. Due to the
antisymmetric nature and especially in the context of manifolds the calculations
may be quite involved and tedious, therefore, it is convenient to have tool box
for the most elaborate transformations. The price one has to pay for the
conciseness and beauty of the presentation is that concrete evaluations are
slightly more challenging than conventional methods like vector calulus for
instance.
There are many approaches to introduce differential forms: alternating linear
mappings, completely antisymmetric tensors, sections of cotangent bundles or
free vector spaces to name a few. The following references give witness of the
different methods and are ordered by nondecreasing
difficulty and rigor: , , , , ,
Here we have to consider differential forms in local coordinates in order to
perform some nonsymbolic computations:
\[\omega(x) = \sum_{1\leq j_1<\ldots<j_p\leq n}
\omega_{j_1,\ldots,j_p}(x_1,\ldots,x_n) \,
dx_{j_1}\wedge\ldots\wedge dx_{j_p}\]
The expression above is the local representation of the p-form
\(\omega\) in an
open set of \(\mathbb{R}^n\). There are many other ways to express this
fact: some authors take the sum over
\(\{j_k=1,\ldots,n ; k =1,\ldots,p\}\)
then one has to multiply by the factor \(\frac{1}{p!}\),
others write it as a p-covector field, but all these have the same
meaning. The wedge product \((\wedge)\) basically denotes the
antisymmetric tensor product:
\[\alpha\wedge\beta=\frac{1}{2}\, \left(a\otimes b - b\otimes a \right)\]
and is, although standard today, quite unhandy and there are different
notations in the literature.
In the book of H.Flanders [4] it is not used at all while H.Whitney used the
symbol \(\vee\) in his representation [7].
Notation: The domain DeRhamcomplex uses dx*dy for \(dx\wedge dy\).
FriCAS :: DeRhamComplex