DOCSLIB.ORG

Study of Grassmann Algebra with Differential Forms Md

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Study of Grassmann Algebra with Differential Forms Md International Journal of Scientific & Engineering Research, Volume 7, Issue 4, April-2016 415 ISSN 2229-5518 Study of Grassmann Algebra with Differential Forms Md. Anowar Hossain1 and Md. Abdul Halim2 1Department of Natural Sciences, Stamford University Bangladesh Dhaka-1217, Bangladesh Email: [email protected] 2Department of Mathematics ,International University of Business Agriculture and Technology Dhaka 1230,Bangladesh Email: [email protected] Abstract. The aim of this paper is devoted to the study of an exterior algebra (Grassmann Algebra) and briefly discusses differential forms. Using this we have developed some important theorems and propositions. Finally we also represent the integration of differential forms with the help of Grassmann algebra. Keywords: Grassmann algebra, Exterior product. —————————— —————————— 1. Introduction 2. The Grassmann product is multilinear that is, … ( + … ) Exterior algebra [1] and differentials forms are two 1 = 2 ( … … ) 1 1 2 푟 important sections in differential geometry [7]. In 푣 ∧ ∧ 훼 푢 +훼 푢1(∧ ∧… 푣 … ) 1 1 mathematics, the exterior product or wedge product [3] of 3. The product is nilpotent that훼2 is,푣 any∧ ∧ 푢 ∧, ∧ = 0 2 2 vectors is an algebraic construction used in Euclidean 4. The set of all products훼 푣 ∧ …∧ 푢 ∧ is∧ linearly geometry to study areas, volumes, and their higher- independent. 푣 ∈ 푉 푣 ∧ 푣 푖1 푖푟 dimensional analogs. The exterior product of two 2. The exterior power 푒 ∧ ∧ 푒 vectors and , denoted by , is called a bivector. k The magnitude of can be interpreted as the area of The -th exterior power of , denoted P (V), is the vector the parallelogram푢 푣 with IJSERsides푢 ∧ and 푣 , which in three- subspace of ( ) spanned by elements of the form dimensions can also푢 be ∧ 푣 computed using the cross product 푘 … , 푉 , =∧ 1,2, … , k 푢 푣 If α P (V∧), 푉then α is said to be a -multivector. If, of the two vectors. Also like the cross product, the 1 2 푘 푖 exterior product is anticommutative, meaning that furthermore,푥 ∧ 푥α can∧ be∧ 푥expressed푥 ∈ as 푉 a푖 wedge product푘 of k = for all vectors and . Tensors elements∈ ∧ of V, then α is said to be decomposable푘 . products are not at all necessary for the understanding or For example, in , the following 2-multivector is not use푢 ∧ 푣of Grassmann−푣 ∧ 푢 algebra. As we shall푢 it is푣 possible to decomposable: =4 + build Grassmann algebra using tensors products as a tool. This is in fact a symplecticℝ form, since 0. 1 2 3 4 We develop the elementary theory of Grassmann algebra We can express훼 the 푒2-∧form 푒 푒 ∧ 푒 in polar coordinates on an axiomatic basis. Finally we discuss the integration by setting = , = we obtain훼 ∧ 훼 ≠ of differential forms by using exterior algebra. 푑푥 ∧ 푑푦= 3. Exterior푥 derivative푟푐표푠휃 푦of a 푟푠푖푛휃k-form Definition 1. The exterior algebra ( ) over a vector 푑푥 ∧ 푑푦 푟푑푟 ∧ 푑휃 space over a field is defined as the quotient algebra of The exterior derivative [8] is defined to be the unique R- the tensor algebra by the two-sided ∧ideal푉 generated by linear mapping from k-forms to (k+1)-forms satisfying the all elements푉 of the퐾 form such that . following properties: Symbolically, 퐼 1. is the differential of ƒ for smooth function ƒ. ( ) 푥 (⊗)/푥 푥 ∈ 푉 2. ( ) = 0 for any smooth function ƒ. the wedge product of two elements of ( ) is 3. 푑푓 ( ) = + ( 1) ( ) where defined by = ∧ 푉 ≔ ( 푇 푉 )퐼 α is a p-form.푑 푑푓 That is to say, d is a derivation푝 of degree 1 The exterior algebra′ ∧was ′ first introduced by Hermann∧ 푉 on the exterior푑 훼 ∧ algebra 훽 of푑훼 differential∧ 훽 − forms.훼 ∧ 푑훽 Grassmann in훼 ∧1844. 훽 훼 ⊗ 훽 푚표푑 퐼 The second defining property holds in more generality: in fact, ( ) = 0 for any form .The third defining Axioms of Grassmann Algebra: property implies as a special case that if ƒ is a function and 푑 a푑훼 -form, then (ƒ 푘) −= ƒ 훼 + ƒ because 1. The Grassmann product is associative that is, functions are forms of degree 0. ( ) = ( ) 훼 푘 푑 훼 푑 ∧ 훼 ∧ 푑훼 푓 ∧ 푔 ∧ ℎ 푓 ∧ 푔 ∧ ℎ IJSER © 2016 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 7, Issue 4, April-2016 416 ISSN 2229-5518 Theorem 2 (Poincare’s Lemma). [9] = 0, that is for = + + any exterior differential form , ( ) =2 0. 휕퐶 휕퐵 휕퐴 휕퐶 푑 � 휕푦 − 휕푧 � 푑푦 ∧ 푑푧 � 휕푧 − 휕푥� 푑푧 ∧ 푑푥 . Theorem 3. Suppose is a differential 1-form on a 휕퐵 휕퐴 휔 푑 푑휔 Let be the vector ( , , ), then the vector smooth manifold . and are smooth tangent vector �휕푥 − 휕푦� 푑푥 ∧ 푑푦 fields on . Then 휔 푋 퐴,퐵 퐶 , < , > 푀= 푋 < ,푌 > < , > < 휕퐶 휕퐵 휕퐴 휕퐶 휕퐵 휕퐴 [ , ] , 푀> formed by the� coefficients− of− is just− the� of the 푋 ∧ 푌 푑휔 푋 푌 휔 − 푌 푋 휔 − vector field , 휕푦denoted휕푧 by휕푧 휕푥 . 휕푥 휕푦 푋 푌 휔 < , > = < , > < 푑푎 푐푢푟푙 Proof. Given 3) Suppose = + + . [ ] 푋 푐푢푟푙 푋 , > < , , > (1) Then since both sides of (1푋) ∧are 푌 linear푑휔 with푋 respect푌 휔 to − ,푌 we 푎 퐴푑푦 ∧ 푑푧 퐵푑푧 ∧ 푑푥 퐶푑푥 ∧ 푑푦 may푋 휔 assume− that푋 푌 is휔 a monomial = , , = ; where and are smooth functions on휔 = 휕퐴 휕퐵 휕퐶 휔= 푑푎 � � 푑푥 ∧ 푑푦 ∧ 푑푧 where means휕푥 the휕푦 divergence휕푧 of the vector field 휔 푔 푑푓 < , 푓 > 푔 푀 L.H.S: 푑푖푣= (푋,푑푥, ∧)푑푦 ∧ 푑푧 ⇒ 푑휔 푑푔 ∧ 푑푓 . 푑푖푣 푋 푋 ∧ 푌 푑휔 From theorems, two fundamental formulas in a vector = < , > 푋 퐴 퐵 퐶 calculus follow immediately. Suppose is a smooth < , > < , > function on and is a smooth tangent vector field on 푋 ∧= 푌 푑푔 ∧ 푑푓 < , > < , > . Then 3 푓 푋 푑푔 푋 푑 3 ℝ 푋 ( ) = 0 � � ℝ = 푌 푑푔 = 푌. 푑 . ( ) = 0 푐푢푟푙 푔푟푎푑 푓 푋푔 푋푓 � 4. Integration of Differential푑푖푣 푐푢푟푙 Forms푋 R.H.S:� < � , >푋푔 푌푓 −<푋푓 , 푌푔> 푌푔 푌푓 < [ , ] , > The calculus of differential forms [2], [6] provides a 푋 푌 휔 − 푌 푋 휔 = < , > < , > convenient setting for integration on manifolds, as we − 푋 <푌[ 휔 , ] , > explain in this section due to the efficient way it keeps = 푋( 푌 푔) 푑푓 ( − 푌 ) 푋 푔[푑푓 , ] track of change of variables. − 푋 푌 푔 푑푓 A form on an open set has the for 푋 푔 푌푓 − 푌 푔 푋푓 − 푔 푋 푌 푓 = . + . 푛 + = . = ( ) … (1) 푘 − 훽 퐺 ⊂ ℝ Therefore푋푔 L.H.S푌푓 =푔 R.H.S푋푌푓 − 푌푔 푋푓 − 푔 푌푋푓 − 푔 푋푌푓 = ( ,…,푗푗 ) 푗1 푗푘 Here 훽 ∑ 푏 푥 푑푥is a ⋀ ⋀푑푥multi-index. We write This complete the푔 IJSER푌푋푓proof 푋푔of푌푓 −the푋푓 푌푔theorem ( ) . The wedge product used in (1) has the anti- 1 푘 □ commutative푘 푗 푗 property푗 푘 − 훽 ∈ Example 1. For a 1-form = + defined ⋀ 퐺 = 2 over . We have, by applying the above formula to each So that if is a permutation of {1, … , } we have 푙 푚 푚 푙 term (consider = and 휎 = 푢) 푑푥the following푣 푑푦 sum, 푑푥 ∧ 푑푥 −푑푥 ∧ 푑푥 ℝ 1 2 휎 푘 … = ( ) ( ) … ( ) (2) = 2푥 푥 푥 + 푦 2 In particular, an form on can be written 푖 푖 푗1 푗푘 푗휎 1 푗휎 푘 휕푢 휕푣 푑푥 ⋀푑푥 ( )푠푔푛 휎 푑푥 ⋀ ⋀푑푥푛 푑휎 �� 푖 푑푥 ∧ 푑푥� �� 푖 푑푥 ∧ 푑푦� = … = 푖=1 휕푥 + 푖=1 휕푥 If ( , )푛 then − we write훼 Ω ⊂ ℝ 휕푢 휕푢 1 푛 1훼 = 퐴 푥 (푑푥) ⋀ ⋀푑푥 (3) �휕푥+푑푥 ∧ 푑푥 휕푦 푑푦+ ∧ 푑푥� the퐴 right ∈ 퐿 side퐺 푑푥 being the usual Lebesgue integral. 휕푣 휕푣 ∫퐺 훼 ∫퐺 퐴 푥 푑푥 = 0 � 푑푥 ∧ 푑푦+ 푑푦 ∧ 푑푦+� 0 Suppose now is open and there is a 휕푢휕푥 휕푣휕푦 diffeomorphism : 푛 G. We define the pull back 1 = − 휕푦 푑푦 ∧ 푑푥 휕푥. 푑푥 ∧ 푑푦 of the form Ωin ⊂(1) ℝ as 퐶∗ 휕푣 휕푢 퐹 Ω → 퐹 훽 �휕푥 − 휕푦� 푑푥 ∧ 푑푦 Example 2. Suppose the Cartesian coordinates in are =푘 − ( ) … (4) given by ( , , ). 3 ∗ ∗ ∗ where 푗푗 푗1 푗푘 1) If is a smooth function on , ℝ 퐹 훽 ∑ 푏 �퐹 푥 ��퐹 푑푥 �⋀ ⋀�퐹 푑푥 � 푥 푦 푧 = = + + 3 then . ∗ 푗 푓 휕푓 휕푓 휕푓 ℝ 푗 휕퐹 푙 If = ( ) is an퐹 푑푥× �matrix푙 푑푥then by (2) and the The vector푑푓 formed휕푥 푑푥 by 휕푦its 푑푦coefficients휕푧 푑푧 ( , , ) is the 푙 휕푥 휕푓 휕푓 휕푓 formula for the determinant gives 푙푚 gradient of , denoted by . 휕푥 휕푦 휕푧 퐵 푏 푛 푛 2) Suppose 푓 = + 푔푟푎푑+ 푓 , where , , are smooth functions on . Then = + + 푎 퐴푑푥 퐵푑푦3 퐶푑푧 퐴 퐵 퐶 ℝ 푑푎 푑퐴 ∧ 푑푥 푑퐵 ∧ 푑푦 푑퐶 ∧ 푑푧 IJSER © 2016 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 7, Issue 4, April-2016 417 ISSN 2229-5518 which in turn follows from the chain rule. … If : is a smooth map and is a form on 푚 푚 1푚 푚 2푚 푚 푛푚 푛푚 then there is well defined form = on �� 푏 푑푥 �⋀ ��푏 푑푥 � ⋀ ⋀ ��푏 푑푥 � .represented퐻 푀 → ℝ in such coordinate charts훾 푘= − ( ∗ )ℝ . = 푚 ( ) 푚 …푚 ( ) ( ) ( ) Similarly if is a form on 푘 −as defined훾 above퐻 훾 ∗and 퐽 푗 1휎 1 2휎 2 푛휎 푛 1 푛 푀: is smooth with open훽 then퐻 ∘ 퐹 is훾 a �� 푠푔푛 휎 푏 푏 푏 = (�det푑푥 ⋀) ⋀푑푥… 휎 훽 − 푘 − 푚푀 ∗ Hence if : G is a map and is an form on well defined form on . 1 푛 We퐻 푈 define → 푀 the integral of 푈an ⊂ ℝ form over an퐻 oriented훽 as in (4) then 1 퐵 푑푥 ⋀ ⋀푑푥 푘 − 푈 퐹 Ω= →det ( 퐶) ( ) 훼 … 푛 − dimensional manifold as follows. First in an form supported on an open푛 set − given by (4) This퐺 formula∗ is especially significant in light of the 1 푛 then푛 − we define b. 푛 훼 − change of퐹 variable훼 formula퐷 퐹 푥 퐴�퐹 푥 �푑푥 ⋀ ⋀푑푥 More푛 − generally, if is an dimensional퐺 ⊂ ℝ manifold with 퐺 an orientation say∫ 훼 the image on an open set by ( ) = ( ) | det ( ) | (5) : carrying 푀the natural푛 − orientation of we can푛 set 퐺 ⊂ ℝ ∫퐺 퐴 푥 푑푥 ∫Ω 퐴�퐹 푥 � 퐷 퐹 푥 푑푥 (5) = The only difference between the right side of and 휑 퐺 → 푀 퐺 is the absolute value sign around det ( ).
Load more

Recommended publications
  • Topology and Physics 2019 - Lecture 2
    Topology and Physics 2019 - lecture 2 Marcel Vonk February 12, 2019 2.1 Maxwell theory in differential form notation Maxwell's theory of electrodynamics is a great example of the usefulness of differential forms. A nice reference on this topic, though somewhat outdated when it comes to notation, is [1]. For notational simplicity, we will work in units where the speed of light, the vacuum permittivity and the vacuum permeability are all equal to 1: c = 0 = µ0 = 1. 2.1.1 The dual field strength In three dimensional space, Maxwell's electrodynamics describes the physics of the electric and magnetic fields E~ and B~ . These are three-dimensional vector fields, but the beauty of the theory becomes much more obvious if we (a) use a four-dimensional relativistic formulation, and (b) write it in terms of differential forms. For example, let us look at Maxwells two source-free, homogeneous equations: r · B = 0;@tB + r × E = 0: (2.1) That these equations have a relativistic flavor becomes clear if we write them out in com- ponents and organize the terms somewhat suggestively: x y z 0 + @xB + @yB + @zB = 0 x z y −@tB + 0 − @yE + @zE = 0 (2.2) y z x −@tB + @xE + 0 − @zE = 0 z y x −@tB − @xE + @yE + 0 = 0 Note that we also multiplied the last three equations by −1 to clarify the structure. All in all, we see that we have four equations (one for each space-time coordinate) which each contain terms in which the four coordinate derivatives act. Therefore, we may be tempted to write our set of equations in more \relativistic" notation as ^µν @µF = 0 (2.3) 1 with F^µν the coordinates of an antisymmetric two-tensor (i.
    [Show full text]
  • LP THEORY of DIFFERENTIAL FORMS on MANIFOLDS This
    TRANSACTIONSOF THE AMERICAN MATHEMATICALSOCIETY Volume 347, Number 6, June 1995 LP THEORY OF DIFFERENTIAL FORMS ON MANIFOLDS CHAD SCOTT Abstract. In this paper, we establish a Hodge-type decomposition for the LP space of differential forms on closed (i.e., compact, oriented, smooth) Rieman- nian manifolds. Critical to the proof of this result is establishing an LP es- timate which contains, as a special case, the L2 result referred to by Morrey as Gaffney's inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the LP boundedness of Green's operator which we use in developing the LP theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the LP norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the /1-harmonic equation. 1. Introduction This paper contributes primarily to the development of the LP theory of dif- ferential forms on manifolds. The reader should be aware that for the duration of this paper, manifold will refer only to those which are Riemannian, compact, oriented, C°° smooth and without boundary. For p = 2, the LP theory is well understood and the L2-Hodge decomposition can be found in [M]. However, in the case p ^ 2, the LP theory has yet to be fully developed. Recent appli- cations of the LP theory of differential forms on W to both quasiconformal mappings and nonlinear elasticity continue to motivate interest in this subject.
    [Show full text]
  • NOTES on DIFFERENTIAL FORMS. PART 3: TENSORS 1. What Is A
    NOTES ON DIFFERENTIAL FORMS. PART 3: TENSORS 1. What is a tensor? 1 n Let V be a finite-dimensional vector space. It could be R , it could be the tangent space to a manifold at a point, or it could just be an abstract vector space. A k-tensor is a map T : V × · · · × V ! R 2 (where there are k factors of V ) that is linear in each factor. That is, for fixed ~v2; : : : ;~vk, T (~v1;~v2; : : : ;~vk−1;~vk) is a linear function of ~v1, and for fixed ~v1;~v3; : : : ;~vk, T (~v1; : : : ;~vk) is a k ∗ linear function of ~v2, and so on. The space of k-tensors on V is denoted T (V ). Examples: n • If V = R , then the inner product P (~v; ~w) = ~v · ~w is a 2-tensor. For fixed ~v it's linear in ~w, and for fixed ~w it's linear in ~v. n • If V = R , D(~v1; : : : ;~vn) = det ~v1 ··· ~vn is an n-tensor. n • If V = R , T hree(~v) = \the 3rd entry of ~v" is a 1-tensor. • A 0-tensor is just a number. It requires no inputs at all to generate an output. Note that the definition of tensor says nothing about how things behave when you rotate vectors or permute their order. The inner product P stays the same when you swap the two vectors, but the determinant D changes sign when you swap two vectors. Both are tensors. For a 1-tensor like T hree, permuting the order of entries doesn't even make sense! ~ ~ Let fb1;:::; bng be a basis for V .
    [Show full text]
  • Killing Spinor-Valued Forms and the Cone Construction
    ARCHIVUM MATHEMATICUM (BRNO) Tomus 52 (2016), 341–355 KILLING SPINOR-VALUED FORMS AND THE CONE CONSTRUCTION Petr Somberg and Petr Zima Abstract. On a pseudo-Riemannian manifold M we introduce a system of partial differential Killing type equations for spinor-valued differential forms, and study their basic properties. We discuss the relationship between solutions of Killing equations on M and parallel fields on the metric cone over M for spinor-valued forms. 1. Introduction The subject of the present article are the systems of over-determined partial differential equations for spinor-valued differential forms, classified as atypeof Killing equations. The solution spaces of these systems of PDE’s are termed Killing spinor-valued differential forms. A central question in geometry asks for pseudo-Riemannian manifolds admitting non-trivial solutions of Killing type equa- tions, namely how the properties of Killing spinor-valued forms relate to the underlying geometric structure for which they can occur. Killing spinor-valued forms are closely related to Killing spinors and Killing forms with Killing vectors as a special example. Killing spinors are both twistor spinors and eigenspinors for the Dirac operator, and real Killing spinors realize the limit case in the eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds of positive scalar curvature. There is a classification of complete simply connected Riemannian manifolds equipped with real Killing spinors, leading to the construction of manifolds with the exceptional holonomy groups G2 and Spin(7), see [8], [1]. Killing vector fields on a pseudo-Riemannian manifold are the infinitesimal generators of isometries, hence they influence its geometrical properties.
    [Show full text]
  • 28. Exterior Powers
    28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1.
    [Show full text]
  • Math 865, Topics in Riemannian Geometry
    Math 865, Topics in Riemannian Geometry Jeff A. Viaclovsky Fall 2007 Contents 1 Introduction 3 2 Lecture 1: September 4, 2007 4 2.1 Metrics, vectors, and one-forms . 4 2.2 The musical isomorphisms . 4 2.3 Inner product on tensor bundles . 5 2.4 Connections on vector bundles . 6 2.5 Covariant derivatives of tensor fields . 7 2.6 Gradient and Hessian . 9 3 Lecture 2: September 6, 2007 9 3.1 Curvature in vector bundles . 9 3.2 Curvature in the tangent bundle . 10 3.3 Sectional curvature, Ricci tensor, and scalar curvature . 13 4 Lecture 3: September 11, 2007 14 4.1 Differential Bianchi Identity . 14 4.2 Algebraic study of the curvature tensor . 15 5 Lecture 4: September 13, 2007 19 5.1 Orthogonal decomposition of the curvature tensor . 19 5.2 The curvature operator . 20 5.3 Curvature in dimension three . 21 6 Lecture 5: September 18, 2007 22 6.1 Covariant derivatives redux . 22 6.2 Commuting covariant derivatives . 24 6.3 Rough Laplacian and gradient . 25 7 Lecture 6: September 20, 2007 26 7.1 Commuting Laplacian and Hessian . 26 7.2 An application to PDE . 28 1 8 Lecture 7: Tuesday, September 25. 29 8.1 Integration and adjoints . 29 9 Lecture 8: September 23, 2007 34 9.1 Bochner and Weitzenb¨ock formulas . 34 10 Lecture 9: October 2, 2007 38 10.1 Manifolds with positive curvature operator . 38 11 Lecture 10: October 4, 2007 41 11.1 Killing vector fields . 41 11.2 Isometries . 44 12 Lecture 11: October 9, 2007 45 12.1 Linearization of Ricci tensor .
    [Show full text]
  • CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V
    CHAPTER 2 Clifford algebras 1. Exterior algebras 1.1. Definition. For any vector space V over a field K, let T (V ) = k k k Z T (V ) be the tensor algebra, with T (V ) = V V the k-fold tensor∈ product. The quotient of T (V ) by the two-sided⊗···⊗ ideal (V ) generated byL all v w + w v is the exterior algebra, denoted (V ).I The product in (V ) is usually⊗ denoted⊗ α α , although we will frequently∧ omit the wedge ∧ 1 ∧ 2 sign and just write α1α2. Since (V ) is a graded ideal, the exterior algebra inherits a grading I (V )= k(V ) ∧ ∧ k Z M∈ where k(V ) is the image of T k(V ) under the quotient map. Clearly, 0(V )∧ = K and 1(V ) = V so that we can think of V as a subspace of ∧(V ). We may thus∧ think of (V ) as the associative algebra linearly gener- ated∧ by V , subject to the relations∧ vw + wv = 0. We will write φ = k if φ k(V ). The exterior algebra is commutative | | ∈∧ (in the graded sense). That is, for φ k1 (V ) and φ k2 (V ), 1 ∈∧ 2 ∈∧ [φ , φ ] := φ φ + ( 1)k1k2 φ φ = 0. 1 2 1 2 − 2 1 k If V has finite dimension, with basis e1,...,en, the space (V ) has basis ∧ e = e e I i1 · · · ik for all ordered subsets I = i1,...,ik of 1,...,n . (If k = 0, we put { } k { n } e = 1.) In particular, we see that dim (V )= k , and ∅ ∧ n n dim (V )= = 2n.
    [Show full text]
  • The Language of Differential Forms
    Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects.
    [Show full text]
  • Mathematics 552 Algebra II Spring 2017 the Exterior Algebra of K
    Mathematics 552 Algebra II Spring 2017 The exterior algebra of K-modules and the Cauchy-Binet formula Let K be a commutative ring, and let M be a K-module. Recall that the tensor ⊗i algebra T (M) = ⊕i=0M is an associative K-algebra and that if M is free of finite rank ⊗k m with basis v1; : : : ; vm then T (M) is a graded free K module with basis for M given ⊗k by the collection of vi1 ⊗ vi2 ⊗ · · · vik for 1 ≤ it ≤ m. Hence M is a free K-module of rank mk. We checked in class that the tensor algebra was functorial, in that when f : M ! N is a K-module homomorphism, then the induced map f ⊗i : M ⊗i ! N ⊗i gives a homomorphism of K-algebras T (f): T (M) ! T (N) and that this gives a functor from K-modules to K-algebras. The ideal I ⊂ T (M) generated by elements of the form x ⊗ x for x 2 M enables us to construct an algebra Λ(M) = T (M)=I. Since (v+w)⊗(v+w) = v⊗v+v⊗w+w⊗v+w⊗w we have that (modulo I) v ⊗ w = −w ⊗ v. The image of M ⊗i in Λ(M) is denoted Λi(M), the i-th exterior power of M. The product in Λ(M) derived from the product on T (M) is usually called the wedge product : v ^ w is the image of v ⊗ w in T (M)=I. Since making tensor algebra of modules is functorial, so is making the exterior algebra.
    [Show full text]
  • 8 Brief Introduction to Differential Forms
    8 Brief introduction to differential forms “Hamiltonian mechanics cannot be understood without differential forms” V. I. Arnold In this chapter we give a very brief introduction to differential forms fol- lowing the chapter 7 of Ref.[26]. We refer the reader to [26, 9, 8] for more detailed (and more precise) introductions. 8.1 Exterior forms 8.1.1 Definitions Let Rn be an n-dimensional real vector space. Definition An exterior algebraic form of degree k (or k-form) is a function of k vectors which is k-linear and antisymmetric. Namely: ω(λ1ξ1! + λ2ξ1!!,ξ2,...,ξk)=λ1ω(ξ1! ,ξ2,...,ξk)+λ2ω(ξ1!!,ξ2,...,ξk), and ω(ξ ,...,ξ )=( 1)νω(ξ ,...,ξ ) i1 ik − 1 k with ( 1)ν = 1 ( 1) if permutation (i , . , i ) is even (odd). Here ξ Rn − − 1 k i ∈ and λ R. i ∈ The set of all k-forms in Rn form a real vector space with (ω + ω )(ξ)=ω (ξ)+ω (ξ),ξ = ξ ,...,ξ 1 2 1 2 { 1 k} and (λω)(ξ)=λω(ξ). Let us suppose that we have a system of linear coordinates x1, . , xn on n R . We can think of these coordinates as of 1-forms so that xi(ξ)=ξi - the i-th coordinate of vector ξ. These coordinates form a basis of 1-forms in the 65 66 Brief introduction to differential forms n 1-form vector space (which is also called the dual space (R )∗). Any 1-form can be written as a linear combination of basis 1-forms ω1 = a1x1 + ... + anxn.
    [Show full text]
  • Diff. Forms and Spinors
    “Äußerer Differentialkalkül für Spinorformen und Anwendung auf das allgemeine reine Gravitationasstrahlungsfeld,” Zeit. Phys. 178 (1964), 488-500. Exterior differential calculus for spinor forms and its application to the general pure gravitational radiation field By KLAUS BICHTELER (Received on 1 February 1964) Translated by D. H. Delphenich ____ Summary. – An exterior differential calculus for spinor forms that is quite analogous to the well- known tensorial Cartan calculus is developed that renders possible the direct spinorial calculation of the Weyl and Ricci tensors. As an application, the metric and curvature spinors of the most general space-time admitting a normal, sheer-free, null congruence are derived. The rather modest amount of calculation illustrates the advantages of the new method as compared to the techniques used thus far ( *). I. Spinor algebra Let T be the metric tensor algebra over the vector space of complex pairs that is endowed with the symplectic metric: A AC C AB ξ = ε ξC , ξA = ξ εCA , εAB = − εBA = ε , ε01 = 1. AB … Its elements are component matrices T CD … Let T′ be a second exemplar of that algebra whose elements will be written with primes in order to distinguish them. The spinor algebra S is the tensorial product of the two algebras. S is, in a natural way, also a A…… P ′ metric space whose elements are component matrices TB…… Q ′ . We would like to list some structural properties of S that we will need in what follows ( ** )( 1,2). (*) Remark by the editor: R. DEBEVER, M. CAHEN, and L. DEFRISE have, according to an oral commutation from one of them, developed a calculus for the calculation of curvature spinors that likewise works with complex forms.
    [Show full text]
  • Hodge Theory
    HODGE THEORY PETER S. PARK Abstract. This exposition of Hodge theory is a slightly retooled version of the author's Harvard minor thesis, advised by Professor Joe Harris. Contents 1. Introduction 1 2. Hodge Theory of Compact Oriented Riemannian Manifolds 2 2.1. Hodge star operator 2 2.2. The main theorem 3 2.3. Sobolev spaces 5 2.4. Elliptic theory 11 2.5. Proof of the main theorem 14 3. Hodge Theory of Compact K¨ahlerManifolds 17 3.1. Differential operators on complex manifolds 17 3.2. Differential operators on K¨ahlermanifolds 20 3.3. Bott{Chern cohomology and the @@-Lemma 25 3.4. Lefschetz decomposition and the Hodge index theorem 26 Acknowledgments 30 References 30 1. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory. In Section 2, we first describe a key motivation behind the Hodge theory for compact, closed, oriented Riemannian manifolds: the observation that the differential forms that satisfy certain par- tial differential equations depending on the choice of Riemannian metric (forms in the kernel of the associated Laplacian operator, or harmonic forms) turn out to be precisely the norm-minimizing representatives of the de Rham cohomology classes. This naturally leads to the statement our first main theorem, the Hodge decomposition|for a given compact, closed, oriented Riemannian manifold|of the space of smooth k-forms into the image of the Laplacian and its kernel, the sub- space of harmonic forms. We then develop the analytic machinery|specifically, Sobolev spaces and the theory of elliptic differential operators|that we use to prove the aforementioned decom- position, which immediately yields as a corollary the phenomenon of Poincar´eduality.
    [Show full text]