Abstract
The chapter starts with a presentation of the elementary calculus of differential forms. This includes the exterior derivative, integral invariants, the theory of integration and the Stokes Theorem, as well as an introduction to de Rham cohomology. Next, we discuss elements of Riemannian geometry and Hodge duality. As an application, we show how classical Maxwell electrodynamics can be understood in a coordinate-free way using the language of differential forms. The final two sections are devoted to an introduction to the theory of Pfaffian systems and differential ideals and to the application of these notions to classical mechanical systems with constraints. In particular, we derive a formulation of the classical Frobenius Theorem in terms of differential ideals.
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Notes
- 1.
We caution the reader that for r≥2 this sum runs over a linearly dependent system of sections. In particular, the left hand side does not determine the coefficient functions on the right hand side uniquely. There is however a unique choice if we limit our attention to functions which are totally antisymmetric in the indices i 1,…,i r , and this choice is given by (4.1.6).
- 2.
One may also write, for example, \(\mathrm{d}\rho^{i_{1}} \wedge\cdots\wedge\mathrm{d}\rho^{i_{r}} = A^{i_{1}}_{j_{1}} \cdots A^{i_{r}}_{j_{r}} \mathrm{d}\kappa^{j_{1}} \wedge\cdots\wedge\mathrm{d}\kappa^{j_{r}}\), keeping in mind that on the right hand side the sum runs over a linearly dependent system rather than a basis.
- 3.
For our purposes, the Riemann integral would do as well.
- 4.
Beware that the interior and the boundary of a manifold with boundary are defined by the manifold alone, whereas the interior and the boundary of a subset of a topological space are defined with respect to the ambient space. By chance, in this example, these two notions coincide.
- 5.
In the sense of Definition 4.2.8.
- 6.
Nevertheless, in [181], F α is referred to as the characteristic distribution of α, which is consistent with the notion of distribution used there.
- 7.
For i=1, the group multiplication is induced from an appropriate composition of closed paths. For i>1 there is an analogous construction, see [76] for a detailed presentation. In contrast, π 0(M,x 0), which is in bijective correspondence to the set of connected components of M, in general does not carry a group structure.
- 8.
If α vanishes on N, then φ ∗ α vanishes, too. Note that the converse is, of course, not true in general.
- 9.
For a proof see [302], Sects. 4.17, 5.36 and 5.45.
- 10.
Denoted by the same letter.
- 11.
Some authors prefer to call t=r−s the signature.
- 12.
- 13.
The proper sign would be obtained for the signature (− + + +).
- 14.
Named after the Dutch physicist Hendrik Lorentz (1853–1928).
- 15.
As usual in the physics literature we use Greek indices here.
- 16.
Named after the Danish mathematician and physicist Ludvig Valentin Lorenz (1829–1891).
- 17.
More generally, the Coulomb gauge condition can also be imposed if charges and currents are present, but A 0 cannot be set equal to zero then. Instead, it is determined dynamically.
- 18.
The equation df=0 is of geometric nature, it does not follow from a variational principle.
- 19.
The mathematical tools needed for understanding the following interpretation will be presented in detail in part II of this book.
- 20.
It is common to refer to (4.7.2), rather than to Δ, as a Pfaffian system and to the 1-forms ϑ j as Pfaffian forms.
- 21.
In mechanics, this principle is usually spelled out by saying that the constraining forces do no virtual work.
- 22.
We identify vectors and covectors on ℝ3N via the Euclidean metric.
- 23.
E.g. if the original external forces possess a potential.
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Rudolph, G., Schmidt, M. (2013). Differential Forms. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_4
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