DifferentialForms - MIT Mathematics

1 Draft: March 28, 2018 Differential FormsVictor Guillemin & Peter J. HaineDraft: March 28, 2018 Draft: March 28, 2018 ContentsPrefacevIntroductionvOrganizatio nviNotational ConventionsxAcknowledgmentsxiChapter 1. Multilinear Quotient spaces & dual Alternating The space ( ) The wedge The interior The pullback operation on ( ) Orientations29 Chapter 2. Differential Vector fields and Integral Curves for Vector Differential Exterior The interior product The pullback operation on Divergence, curl, and Symplectic geometry & classical mechanics63 Chapter 3. Integration of The Poincar lemma for compactly supported forms on The Poincar lemma for compactly supported forms on open subsets of The degree of a differentiable The change of variables Techniques for computing the degree of a Appendix: Sard s theorem92 Chapter 4.

2 Manifolds & Forms on Tangent Vector fields & differential forms on manifolds109iiiDraft: March 28, Integration of forms on Stokes theorem & the divergence Degree theory on Applications of degree The index of a vector field143 Chapter 5. Cohomology via The de Rham cohomology groups of a The Mayer Vietoris Cohomology of Good Poincar Thom classes & intersection The Lefschetz The K nneth ech Cohomology194 Appendixa. Bump Functions & Partitions of Unity201 Appendixb. The Implicit Function Theorem205 Appendixc. Good Covers & Convexity Theorems211 Bibliography215 Index of Notation217 Glossary of Terminology219 Draft: March 28, 2018 PrefaceIntroductionFor most math undergraduates one s first encounter with differential forms is the changeof variables formula in multivariable calculus, the formula(1) |det | = In this formula, and are bounded open subsets of , is a bounded contin-uous function, is a bijective differentiable map, is the function , anddet ( )is the determinant of the Jacobian matrix.

3 ( ) [ ( )],As for the and , their presence in (1) can be accounted for by the fact that in single-variable calculus, with =( , ), =( , ), ( , ) ( , ), and = ( )a 1functionwith positive first derivative, the tautological equation = can be rewritten in the form ( )= and (1) can be written more suggestively as(2) ( )= One of the goals of this text ondifferential formsis to legitimize this interpretation ofequa-tion (1)in dimensions and in fact, more generally, show that an analogue of this formulais true when and are -dimensional related goal is to prove an important topological generalization of the changeof variables formula (1). This formula asserts that if we drop the assumption that be abijection and just require to be proper ( , that pre-images of compact subsets of to becompact subsets of ) then the formula (1) can be replaced by(3) ( )=deg( ) wheredeg( )is a topological invariant of that roughly speaking counts, with plus andminus signs, the number of pre-image points of a generically chosen point of.

4 1 This degree formula is just one of a host of results which connect the theory of differ-ential forms with topology, and one of the main goals of this book will explore some of theother examples. For instance, for an open subset of 2, we define 0( )to be the vector1It is our feeling that this formula should, like formula (1), be part of the standard calculus curriculum,particularly in view of the fact that there now exists a beautiful elementary proof of it by Peter Lax (see [6,8,9]).vDraft: March 28, 2018viPrefacespace of functions on . We define the vector space 1( )to be the space of formalsums(4) 1 1+ 2 2,where 1, 2 ( ). We define the vector space 2( )to be the space of expressions ofthe form(5) 1 2,where ( ), and for >2define ( )to the zero vector these vector spaces one can define operators(6) ( ) +1( )by the recipes(7) 1 1+ 2 2for =0,(8) ( 1 1+ 2 2)=( 2 1 1 2) 1 for =1, and =0for >1.

5 It is easy to see that the operator(9) 2 ( ) +2( )is zero. Hence,im( 1( ) ( )) ker( ( ) +1( )),and this enables one to define thede Rham cohomology groups of as the quotient vectorspace(10) ( ) ker( ( ) +1( ))im( 1( ) ( )).It turns out that these cohomology groups are topological invariants of and are, infact, isomorphic to the cohomology groups of defined by the algebraic topologists. More-over, by slightly generalizing the definitions inequations (4),(5)and(7)to(10)one candefine these groups for open subsets of and, with a bit more effort, for arbitrary manifolds (as we will do inChapter 5); and their existence will enable us to describe inter-esting connections between problems in multivariable calculus and differential geometryon the one hand and problems in topology on the make the context of this book easier for our readers to access we will devote therest of this introduction to the following annotated table of contents, chapter by chapterdescriptions of the topics that we will be 1: Multilinear algebraAs we mentioned above one of our objectives is to legitimatize the presence of the and in formula (1), and translate this formula into a theorem about differential a rigorous exposition of the theory of differential forms requires a lot of algebraicpreliminaries, and these will be the focus ofChapter 1.

6 We ll begin, inSections , byreviewing material that we hope most of our readers are already familiar with: the definitionof vector space, the notions ofbasis, ofdimension, oflinear mapping, ofbilinear form, andDraft: March 28, 2018 Organizationviiofdual spaceandquotient space. Then inSection will turn to the main topics ofthis chapter, the concept of -tensor and (the future key ingredient in our exposition ofthe theory of differential forms inChapter 2) the concept of alternating -tensor. Those tensors come up in fact in two contexts: asalternating -tensors, and asexterior forms, ,in the first context as a subspace of the space of -tensors and in the second as a quotientspace of the space of -tensors. Both descriptions of -tensors will be needed in our laterapplications.

7 For this reason the second half ofChapter 1is mostly concerned with exploringthe relationships between these two descriptions and making use of these relationships todefine a number of basic operations on exterior forms such as the wedge product operation(see ), the interior product operation (see ) and the pullback operation (see ).We will also make use of these results inSection define the notion of anorientationforan -dimensional vector space, a notion that will, among other things, enable us to simplifythe change of variables formula (1) by getting rid of the absolute value sign in the term|det |.Chapter 2: Differential FormsThe expressions inequations (4),(5),(7)and(8)are typical examples of DifferentialForms , and if this were intended to be a text for undergraduate physics majors we woulddefine differential forms by simply commenting that they re expressions of this type.

8 We llbegin this chapter, however, with the following more precise definition: Let be an opensubset of . Then a -form on is a function which to each assigns an elementof ( ), being the tangent space to at , its vector space dual, and ( )the thorder exterior power of . (It turns out, fortunately, not to be too hard to recon-cile this definition with the physics definition above.) Differential1-forms are perhaps bestunderstood as the dual objects to vector fields, and inSections elaborate onthis observation, and recall for future use some standard facts about vector fields and theirintegral curves. Then inSection will turn to the topic of -forms and in the exercisesat the end ofSection a lot of explicit examples (that we strongly urge readers ofthis text to stare at).

9 Then inSections will discuss in detail three fundamen-tal operations on differential forms of which we ve already gotten preliminary glimpses intheequation (3)andequations (5)to(9), namely the wedge product operation, the exteriordifferential operation, and the pullback operation. Also, to add to this list inSection discuss the interior product operation of vector fields on differential forms, a general-ization of the duality pairing of vector fields with one-forms that we alluded to earlier. (Inorder to get a better feeling for this material we strongly recommend that one take a lookatSection these operations are related to thediv, curl, andgradoperations infreshman calculus.) In addition this section contains some interesting applications of thematerial above to physics, inSection electrodynamics and Maxwell s equation, as wellas to classical mechanisms and the Hamilton Jacobi 3: Integration of FormsAs we mentioned above, the change of variables formula in integral calculus is a specialcase of a more general result: the degree formula; and we also cited a paper of Peter Laxwhich contains an elementary proof of this formula which will hopefully induce future au-thors of elementary text books in multivariate calculus to include it in their treatment of theRiemann integral.

10 In this chapter we will also give a proof of this result but not, regrettably,Lax s proof. The reason why not is that we want to relate this result to another result whichDraft: March 28, 2018viiiPrefacewill have some important de Rham theoretic applications when we get toChapter 5. To de-scribe this result let be a connected open set in and = 1 a compactlysupported -form on . We will prove:Theorem integral = 1 of over is zero if and only if = where is a compactly supported( 1)-form on .An easy corollary of this result is the following weak version of the degree theorem:Corollary and be connected open subsets of and a proper there exists a constant ( )with the property that for every compactly supported -form on = ( ).