1.0 IntroductionΒΆ

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: [4] , [6] , [1] , [2] , [7] , [3]

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 \(dx\wedge dy\).

[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.

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