1. 4. Differential forms A. The Algebra and Integral Calculus of Forms 4.1 Definition of Volume – The Geometrical Role of Differential Forms 4.2 Notation and Definitions for Antisymmetric Tensors 4.3 Differential Forms 4.4 Manipulating Differential Forms 4.5Restriction of Forms 4.6 Fields of Forms 4.7 Handedness and Orientability 4.8 Volumes and Integration on Oriented Manifolds 4.9 N-vectors, Duals, and the Symbol  ij…k 4.10Tensor Densities 4.11 Generalized Kronecker Deltas 4.12 Determinants and  ij…k 4.13 Metric Volume Elements.

2. B. The Differential Calculus of Forms and Its Applications 4.14 The Exterior Derivative 4.15 Notation for Derivatives 4-16 Familiar Examples of Exterior Differentiation 4.17Integrability Conditions for Partial Differential Equations 4.18Exact Forms 4.19Proof of the Local Exactness of Closed Forms 4.20Lie Derivatives of Forms 4.21 Lie Derivatives and Exterior Derivatives Commute 4.22 Stokes' Theorem 4.23 Gauss' Theorem and the Definition of Divergence 4.24 A Glance at Cohomology Theory 4.25Differential Forms and Differential Equations 4.26 Frobenius' Theorem (Differential Forms Version) 4.27Proof of the Equivalence of the Two Versions of Frobenius' Theorem 4.28 Conservation Laws 4.29Vector Spherical Harmonics

3. Concepts that are unified and simplified by forms Integration on manifolds Cross-product, divergence & curl of 3-D euclidean geometry Determinants of matrices Orientability of manifolds Integrability conditions for systems of pdes Stokes' theorem Gauss' theorem … E. Cartan

4. 4.1. Definition of Volume – The Geometrical Role of Differential Forms 2 vectors define an area (no metric required). Different pairs of vectors can have same area. area(, ) is a ( 0 2 ) skew-tensor such that → Ex. 4.1 For vectors in the x-y plane: area( a, b + c ) area( a, b ) + area( a, c )

5. ←←← Proof is to show that regions of the same colors have equal areas.

6. 4.2. Notation and Definitions for Antisymmetric Tensors is completely antisymmetric if Totally antisymmetric part Index-notation: A ( 0 p ) tensor satisfies

7. A skew ( 0 p ) tensor on an n-D space has at most independent components where In general,

8. 4.3Differential Forms p-form = completely antisymmetric ( 0 p ) tensor ( p = degree of form). 0-form = scalar function.1-form = covariant vector. Wedge (exterior or Grassmann) product  Letbe p- & q- forms, resp. Then ( see Choquet, § IV.A.1 ) = (vector) space of all p-forms at x  M by is defined by where π is a permutation of (1,2,…, p+q). = (module) space of all p-form fields on M

9. (Antisymmetric) Properties of  ( proofs are straightforward ) : if p = q = 1 (Associative) (Bilinear) where f is a constant for forms, or a function for fields.

10. be the vector & its dual 1-form bases, resp.Let & Then is a basis for 2-forms. A more convenient notation for a basis for p-forms is For a 2-formwe have (& analogously for fields),

11. Grassmann algebra = { all p-forms, +,  ;  or L(M) } Ex. 4.8: Show that if E.g., It is a graded algebra with Dim = where L(M) = space of all C  functions on M.

12. 4.4 Manipulating Differential Forms Attention: signs Letbe p- & q-forms, resp. Then Proof:Letbe 1-forms such that Then

13. Proof using basis:

14. Contraction : Letbe a vector &a p-form. i.e.,Define Example:whereare 1-forms [ p! terms] Usually, this is called the interior product : See Frankel, § 2.9.

15. → In general = (p–1)-form with components Alternatively,

16. [ p! terms] → = (p–2)-form with components

17. 4.5. Restriction of Forms A p-form is a ( 0 p ) tensor → its domain is The restriction (section)to a subspace (hyperplane) W of V is → →is 1-D (annulled by W) with domain W.

18. 4.6. Fields of Forms A field Ω p (M) of p-forms on a manifold M = a rule that gives a p-form at each point of M. Ditto p-vector field. A submanifold S of M picks a subspace V P of T P  P  S. → Restriction of p-form field to S = restriction of p-form at P to V P  P  S.

19. 4.7. Handedness and Orientability See. Frankel, §2.8e Letbe a basis for T P (M) chosen as right-handed. Any other basis related to it by a transformation with J > 0 is also right-handed. → Hence: Alternative definition: also represents the same handedness as 

20. Relative handedness is independent of choice of M is (internally) orientable if it is possible to define handedness continuously over it, i.e.,  a continuous basis with the same handedness everywhere on M. i.e., E.g.  n is orientable. The Mobius band is not. Absolute handedness is fixed by the choice of the coordinate charts.

21. 4.8. Volumes and Integration on Oriented Manifolds Integration of a function on an n-D manifold M with coordinates { x i }. A parallelepiped / cell can be denoted by n vectors: “Volume” of cell is: Given f : M → , we construct a 1-D n-form field Integration of f over cell : Integration of f over U  M : Caution: volume in the usual geometric sense requires a metric.

22. Change of Variables is independent of coordinates up to an overall sign. E.g., M is 2-D : Changing coordinates → 

23. Thus, the assignment where J is the Jacobian that link (  λ,  μ ) to the basis defined to be right-handed. However, further development in the integration of a pseudo-form is slightly different from that of a true form (see Frankel). works only if U is orientable (with ω right-handed). For U non-orientable, one replaces ω with a pseudo (odd) form To include the case where ω is left-handed, we set where o(λ,μ) =  1 is the orientation (handedness) of (  λ,  μ ). (de Rham)

24. Integration on Submanifold is defined only for n-forms on an n-D manifold M, or p-forms over a p-D submanifold S. Relation between the orientabilities of M and S ? ( Domain must be internally orientable ) Let M be orientable and a right-handed n-form at P  S. the p-formis a right- handed restriction ofto S not tangent to S at P,Given n–p independent normal vectors determines an external orientation for S at P. S is externally orientable if it is possible to define an external orientation continuously over it. If U  M is orientable, then S  U is either both internally and externally orientable, or it is neither. Otherwise, S may be one but not both.

25. Mobius strip embedded in  3. M is not externally orientable in R 3. A curve is always internally orientable → it can't be externally orientable inside a nonorientable submanifold  C 1 is not orientable in M But C 2 is both internally & externally orientable in M

26. 4.9. N-vectors, Duals, and the Symbol ε ijk… Caution: The following differs substantially from Schutz’s version. See Frankel, §§ 2.8e & 14.1a, Choquet, §V.A.4, and Schutz, §4.13. Let M be an n-D Riemannian manifold with a metric tensor g. The volume pseudo n-form is therefore defined as Under a coordinate transformation Λ,( see §2.29 ) →where i.e.,  g transforms like the component of an n-form. In an orthonormal basis→ so that Euclidean metric in spherical coordinates (r,θ,φ) :

27. The inner product  |  on T x (M) induces an inner product on Λ p x (M) & Ω p x (M): s.t. Choquet, §V.A.4. →

28. Proof:

29. Possible maps given g: where the Hodge (dual) star operator * is the map: bys.t. Let → dim = C n p = C n n  p

30. → is the contraction of the ( 0 n ) tensorτwith the ( p 0 ) tensor β. Hence, Caution: 1.Schutz uses a “contraction”   that absorbs the p! factor (see eq.4.20). 2.Schutz is also the only author who defined the Hodge star  as transforming a p-vector to a (n  p)-form & vice versa, i.e., where 3. In the presence of a metric tensor, τshould always be the volume n-form.

31. Example: Cross Products in  3 Then be vectors &Letthe associated 1-forms. Let→ → The cross product exists only in  3, where

32. Vector Analysis in  3 Ex. Calculate the quantities above for a monolinic coordinate system. the corresponding vectors.Letbe 1-forms & Then where

33. Let T be a q-vector, then 

34. T = q-vector

35. → s = signature of g. Let then (c.f. eq.4.24) i.e.,

36. Let T be an q-vector, then is an (n  q)-form.  Similarly, ~

37. Let ω be an n-form on an n-D manifold. → whereη= canonical form of g. (wrt orthonormal dual bases) → s is invariant under coord.transf.

38. 4.10.Tensor Densities Consider the n-form on an n-D manifold M: Under a coordinate transformation we have → is called a scalar density of weight +1, e.g.,  |g|.

39. In general, a quantity that transforms like a tensor multiplied by J W is called a tensor density of weight W. i.e., → T is a ( p q ) tensor density of weight W. Note that J is the jacobian of the transform of forms (covectors). W = 0 ~ ordinary tensors Example: Let T be a ( 2 0 ) tensor with components T i j. Its density over a 3-D volume is the components of a 3-form with so that ω is a tensor-valued form.

40. 4.11. Generalized Kronecker Deltas It is straightforward to verify that the following defines a ( p p ) antisymmetry tensor in an n-D space that takes the same form wrt all orthonormal bases. The Levi-Civita symbol represents the special valueIt’s not a tensor. An equivalent definition is is called a Kronecker tensor, or a p-delta symbol. where i k, j k = 1, …, n. i k, j k = 1, …, n. Choquet denotes it as

41. Contraction of a p-delta in an n-D space: Contracting q indices of ε(~ n-delta in n-space) :

42. Example: (valid only for Cartesian coord in  3 )

43. 4.12. Determinants and ε ij…k Already discussed in previous sections.

44. 4.13. Metric Volume Elements Already discussed in previous sections.