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 ( ).