Differential Forms
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,
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Differential Forms
We first present the elementary calculus of differential forms, including the calculus of integration and a discussion of integral invariants. Then, in Sect. 4.3, we give an introduction to de Rham cohomology. Next, in Sects. 4.4 and 4.5, we present some elements of Riemannian geometry, discuss Hodge duality in detail and show how classical vector analysis can be understood in a coordinate-free way using the language of differential forms. In Sect. 4.6, we apply this framework to classical Maxwell electrodynamics. In Sect. 4.7, we give an introduction to the theory of Pfaffian systems and differential ideals. In particular, we derive an equivalent formulation of the classical Frobenius Theorem. Finally, we apply these notions to classical mechanics with constraints.
4.1 Basics Recall the following notions from Sect. 2.5. A differential k-form on a manifold M is a section in the vector bundle Λk T∗ M. We write Ω k (M) for the space of differential k-forms and ∗
Ω (M) = Γ
∗
T M ≡
∞
Ω k (M)
k=0
for the exterior algebra. Obviously, Ω 0 (M) = C ∞ (M) and Ω k (M) = 0 for k > dim M. The exterior product ∧ : Ω k (M) × Ω l (M) → Ω k+l (M) is given by (α ∧ β)(X1 , . . . , Xk+l ) 1 = sign(π)α(Xπ(1) , . . . , Xπ(k) )β(Xπ(k+1) , . . . , Xπ(k+l) ), k!l! π∈Sk+l
where α ∈ Ω k (M), β ∈ Ω l (M) and Xi ∈ X(M), cf. (2.4.17). The natural pairing of k-forms with k-vectors from multilinear algebra induces a C ∞ -valued pairing of G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_4, © Springer Science+Business Media Dordrecht 2013
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Differential Forms
k-differential forms with k-vector fields, given by α1 ∧ · · · ∧ αk , X1 ∧ · · · ∧ Xk = det αi (Xj )
(4.1.1)
for all αi ∈ Ω 1 (M) and Xi ∈ X(M). Then, by the definition of the exterior product, α(X1 , . . . , Xk ) = α, X1 ∧ · · · ∧ Xk for all α ∈ Ω k (M) and Xi ∈ X(M). For k ≥ r, we define the operation of inner multiplication of an r-vector field X with a k-differential form α by Xα, Y := α, X ∧ Y ,
Y ∈ Xk−r (M).
(4.1.2)
This operation is C ∞ (M)-linear in both arguments. Let us describe the above structures in a local chart (U, κ) on M. (U, κ) induces local frames {∂i } and {dκ i } of TM and T ∗ M, respectively, and by Example 2.4.5, the induced local frames in r TM and r T∗ M consist, respectively, of the local sections ∂i1 ∧ · · · ∧ ∂ir ,
dκ i1 ∧ · · · ∧ dκ ir ,
1 ≤ i1 < · · · < ir ≤ n.
It is common to use the following condensed notation. For a subset I ⊂ {1, . . . , n} of r elements define ∂I := ∂i1 ∧ · · · ∧ ∂ir ,
dκ I := dκ i1 ∧ · · · ∧ dκ ir ,
where i1 , . . . , ir denote the elements of I , ordered by magnitude, that is, I = {i1 , . . . , ir } and i1 < · · · < ir . Thus, the local frames under consideration consist, respectively, of the local sections ∂I and dκ I , where I runs through the subsets of {1, . . . , n} of cardinality r. In particular, by an extension of the summation convention, pairs of capital indices I are
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