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

4.14. The Exterior Derivative The exterior derivative is a map by s.t. (i)( distributive ) (ii)( anti- Leibniz / derivation ) (iii)( Poincare lemma ) The Poincare lemma is a consequence of the irrelevance of the order of partial derivatives. Useful properties (see Ex.4.14) :

4.15. Notation for Derivatives Partial derivation is not a tensor operation: Let Then or Exception:are the components of the 1-form

4.16. Familiar Examples of Exterior Differentiation Details of vector analysis (in R 3 ) can be found in Frankel, §§ 2.5e & 2.9c. Let be a vector field in R 3 described by Cartesian coordinates. Its covector is the 1-form fieldwith → (valid only for Cartesian R 3 ) 

→ etc… Treating the forms as tensors, we have

→ Using the formula we have Alternatively: c.f. (Cartesian Coord)

Divergence: → Caution: For curvilinear coordinates, See Frankel §2.9

4.17.Integrability Conditions for Partial Differential Equations Consider the PDEs i.e.,with →(coordinate independent) → ( Integrability conditions ) General case: Frobenius’ theorem

4.18.Exact Forms A form α is exact if  β s.t. A form α is closed if  exact → closed But a closed form is guaranteed exact only locally. i.e., a closed form is exact in the neighborhood of a non-singular point that can be described in a single coordinate patch. Proof: See §4.19. Note that β is not unique since

Example: For (x, y)  (0,0) Let (0,0) be inside C 1, then α is closed in D  R 2. θ is multi-valued in D → α is not (globally) exact in D. α is (locally) exact in any region not enclosing C 1. α is globally exact in D. (x, y) = (0,0) is a singular point

Homology: study of topological invariants by means of chains. Chains are linear combinations of simplexes. –0-simplex = point. –1-simplex = closed interval. –2-simplex = triangle. –3-simplex = tetrahedron. –…–… Chains have simple boundaries. Chains form a vector space. Cohomology: homology of co-chains (duals of chains). Integration of forms makes use of cohomology theory. For an introduction, see Aldrovandi, Chap 2. For a more detailed treatment, see C.Nash & S.Sen, “Topology & Geometry for Physics”, Chaps 4 & 6.

4.19.Proof of the Local Exactness of Closed Forms Let U be a n-D region diffeomorphic to an open n-ball in R n. Then, with suitable scaling, the coordinates of a point in U can be written as a p-form closed in U. i.e. with Let  x U x U then

Let with → Sincewe have

→ where  k  means that the position of index k is fixed. Since  i.e.

4.20.Lie Derivatives of Forms Proof: Reminder: 1. L is a derivation on Ω p : 2.d is an anti-derivation: 3. Let Then see Choquet, pp. 206, 552

( all j  i k ) ( all i 1  i k ) ( i 1  i k in dx ) ( j  i k in ω ) →

→ 2nd term represents p terms resulted from interchanging j with i k.

For a 1-form ω : Alternative Derivation

4.21. Lie Derivatives and Exterior Derivatives Commute →  ω ω  In general, d commutes with all derivations.

4.22. Stokes' Theorem Called by Arnold the Newton-Leibniz-Gauss-Ostrogradski-Green-Stokes-Poincare theorem. Ampere & Kelvin also contributed. One of the most important theorems of all Mathematics: For proof, see Frankel, §3.3. For an introduction to homology, see Aldrovandi, Chap 3.

Consider an n-form ω over an open region U of an n-D manifold M. The boundary  U of U is an (n-1)-D submanifold that divides M into 2 disjoint sets U & CU. → Any smooth curve joining U & CU must contain a point in  U. Example: C 1 and C 2 are not boundaries, but C 1  C 2 and C 3 are. C 1  C 2 is disconnected. Let  U be smooth, orientable, and, for convenience, connected. An Indirect Proof

Let ξ be any vector field on M and Lie drag U along ξby ε. Let V be a patch of  U covered by coordinates { x 2, …, x n }. By Lie dragging V along ξ, one creates a region of coordinates { x 1 =ε, x 2, …, x n }. (ξ not tangent to  U )

Let The formula is also applicable a region where ξ is tangent to  U. In which case, δV = 0 and ω(ξ) = 0. Hence where

Let thenSee Frankel §3.1.f Lie dragging U along ξ by ε means where σ ε is the local transformation along the integral curves of ξ by ε.  If U is a p-D submanifold, then restricting the p+1 form dω to U gives 0. → E.g.then

 ξ ξ α = any (n  1)-form → Stokes theorem : Example: M = E 2 Let l = d/dλ be a vector tangent to  U. Restricting α to  U means confining the domain of α to l. →

4.23. Gauss' Theorem and the Definition of Divergence Let Then volume form

(divergence of vector fieldξ) Stokes theorem → Let be a 1-form normal to  U, i.e. Letthen → or ( Green’s theorem ) Ex

4.24. A Glance at Cohomology Theory Let Z p (M) be the set of all closed p-forms on M, i.e., Let B p (M) be the set of all exact p-forms on M, i.e., 2 closed forms are equivalent if they differ by an exact form, i.e., Let H p (M) be the set of all equivalent classes of closed p-forms on M, i.e., ( Quotient space ) = pth de Rham cohomology vector space of M dim H p (M) = b p = pth Betti number of M.

Local exactness: In any U diffeomorphic to an open n-ball in R n, → All closed forms are equivalent (& hence to the zero form) in U. i.e., If M is connected, then= space of constant functions = R Since there is no (  1)-form, where 0 is the zero function. → In general,for M with m connected components. ( Poincare lemma )

Some interesting results concerning S n : [ Proof: see M.Spivak, “A Comprehensive Introduction to Differential Geometry”,Vol 1, PoP (70) ] Comment: Homology & cohomology are theories on topological invariants. The study of cohomology in terms of differential forms was initiated by de Rham, hence the name de Rham cohomology. c.f. Ex 4.24 Ex

4.25.Differential Forms and Differential Equations → so that A solution y = g(x) defines a curve ( 1-D solution manifold S ) on state space M = R 2. i.e., aRaR In general, a particular solution to a set of DEs is a submanifold S whose tangent vectors at each point annul the set of differential forms representing the DEs. ( V annuls ω ) Locally, the space V p of annulling vectors at p is a vector subspace of T P (M). Globally, whether these V P s can be meshed to give a tangent bundle T(S) is determined by the Frobenius theorem (existence of solutions).

Example: Harmonic Oscillator Equivalent 1st order eqs: Equivalent set of forms: State space M is 3-D with (global) coordinates ( x, y, t ). Solution space S is 1-D, i.e., a spiral whose projection on the x-y plane is an ellipse. Ex. 4.32

4.26. Frobenius' Theorem (Differential Forms Version) Let B = {β i } be a set of forms of arbitrary degrees. The annihilator X P of B at p  M is the set of all vectors that annuls every β i in B. X P is a subspace of T P. The complete ideal I (B p ) of B at p is the set of all forms at p whose restriction to X P vanishes. Reminder: restriction of a form to V means confining its domain to V. See Choquet, §§ IV.3-4 A left ideal I of a ring X is a subring of X s.t. A right ideal I of a ring X is a subring of X s.t. An ideal is a left ideal that is also a right ideal. 

Let X P be spanned by basis {e 1, …, e m }, which is augmented by {e m+1, …, e n } to span T P. Let { ω m+1, …, ω n } be the dual basis to {e m+1, …, e n }, i.e., and i = m+1, …, n. Hence, the annihilator of { ω m+1, …, ω n } is also X p. If β  I (B p ), then Any q-form βon M can be written as unless  i j  { m+1, …, n } for some {γ i }. Hence, β  I (B p ) → Ex 4.30: Let { α j, j = 1, …, m } be a set of independent 1-forms. Then any form γ is in their complete ideal iff

Let B = {β i } be a set of fields of forms of arbitrary degrees. The complete ideal I (B) of B is the set of all fields of forms whose restriction to the annihilator X P vanishes  p  M. An ideal I is a differential ideal if γ  I → dγ  I. A differential system { α j = 0 } is closed if dα j  I ( { α j } ). i.e. dα j  α 1  …  α m = 0j = 1, …, m ( Ex ) Ex.4.31: a)The 1-forms of a closed system generate a differential ideal. b)On an n-D manifold, any lin. indep. set of n, or n  1, 1-form system is closed.

Frobenius theorem (on differential forms): Let {α i ; i = 1, …, m } be a set of lin. indep. 1-form fields in U  M (dimM = n). Then Proof: see §4.27 Interpretation: Consider the set of differential system Its solution is the submanifold S whose tangent space at each point is the annihilator of {α i }. Let I be the complete ideal generated by {α i }. If a solution exists, then dα i = 0, I is a differential ideal and {α i = 0 } is closed. Conversely, {α i } is not closed → no solution exists. Frobenius’ theorem: α i = 0 → dQ j = 0 ( S is then defined by Q j = const ). The {α i } are then said to be surface forming. Requirement of closedness is dual to that of the annihilator being a Lie algebra.

Frobenius theorem → f exists iff a is closed, i.e. da = 0. Example: PDE discussed in §4.17or Another example: Ex Example: Harmonic Oscillator By Ex 4.31, a set of two 1-forms in a 3-D manifold is closed. Frobenius theorem :  a,b,c,e,f,g s.t. Solution submanifold is 1-D on which f and g are constant. DEs described by higher forms may not be equivalent to those described by the 1-forms that generate the same ideal (see Choquet)

Alternative Formulation See Aldrovandi, § A set of independent 1-forms,is cotangent to a submanifold iff i.e., { θ i = 0 } is closed. i.e., E.g., S 2 can be specified by d r = 0.

4.27.Proof of the Equivalence of the Two Versions of Frobenius' Theorem Vector field version: A set of vector fields { V (i), i = 1, …, q }, which at every point form a p-D vector space, will mesh to form a p-D submanifold iff they are closed under the Lie bracket. 1-form field version: A set of 1-form fields { α (i), i = 1, …, p } can be written as iff they are closed. whereupon, { Q j = const } defines a (n  p)-D submanifold on which α (i) = 0. Reminder: An r-D subspace X P of the n-D T P (M) defines an (n  r)-D subspace of T * P (M) that is annulled by X P. An q-D subspace Ω P of the n-D T * P (M) defines an (n  q)-D subspace of T P (M) that annulls Ω P. → An r-D submanifold of M can be defined by either an r-D X P or an (n  r)-D Ω P annulled by X P.

Let S be an p-D submanifold of an n-D manifold M. → S is defined by n  p equations { Q k = const } so that n  p coordinates of a point on S can be set to 0. → { dQ k } are lin. indep. and T P (S) defines an (n  p)-D subspace Ω P of T * P (M) that is annulled by T P (S), i.e., Obviously, { dQ k } is a basis of Ω P. Any other basis { α (i) } of Ω P can be written as Let { V (i), i = 1, …, p } be a basis for T P (S). → 

If V (j) are closed under Lie bracket so that i.e., dα (i) are in the complete ideal → { α (i) = 0 } is closed. The converse is proved by retracing the argument. QED. →

4.28. Conservation Laws Let the solution of a set of equations be equivalent to finding surfaces that annul a set of forms { α i }. Lets.t. Then, for some suitable region U of a solution surface H,  σ s.t. and Stokes theorem →(conservation law)

Example: Harmonic Oscillator, M = { (x,y,t) } H is 1-D curve → γ = dσ are 1-forms on H→ σ is a 0-form (function) Eliminating dt, we have On H,  →  = energy since

4.29.Vector Spherical Harmonics Finite dimensional representations of SO(3) on L 2 (S 2 ) : basis = { Y lm ; m =  l, …, l } = Spherical harmonics Representations of SO(3) on L 2 1,0 (S 2 ) : basis = Vector spherical harmonics Let the metric tensor on S 2 be Then L 2 1,0 (S 2 ) is the set of all vector fields V on S 2 s.t. ω = volume 2-form The vector spherical harmonics are eigenvectors of l z and L 2.

Given Y lm, one gets the 1-form dY lm and its dual vector where A, B = {θ,φ } and Since g & d commutes with →is a basis vector of T (S 2 ) for given l, m. The other basis vector can be chosen as Other tensor spherical harmonics can be similarly constructed once covariant derivatives on S 2 are defined.

Let S 2 be a submanifold of M, say, E 3. Then with