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International Journal of Scientific & Research, 7, Issue 4, April-2016 415 ISSN 2229-5518 Study of Grassmann 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 ,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 (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 .

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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 [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 , , 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 . k The of can be interpreted as the 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 can also푢 be ∧ 푣 computed using the 푘 … , 푉 , =∧ 1,2, … , k 푢 푣 If α P (V∧), 푉then α is said to be a -. 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 . 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 . 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 is defined as the quotient algebra of The exterior [8] is defined to be the unique R- the 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 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 , ( ) =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 . and are smooth 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 − , 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 푑휔linear 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 follow immediately. Suppose is a smooth < , > < , > function on and is a smooth field on 푋 ∧= 푌 푑푔 ∧ 푑푓 < , > < , > . Then 3 푓 푋 푑푔 푋 푑 3 ℝ 푋 ( ) = 0 � � = 푌 푑푔 = 푌. 푑 . ℝ ( ) = 0 푐푢푟푙 푔푟푎푑 푓 푋푔 푋푓 � R.H.S:� < � , >푋푔 푌푓 −<푋푓 , 푌푔> 4. Integration of Differential푑푖푣 푐푢푟푙 Forms푋 푌푔 푌푓 < [ , ] , > The calculus of differential forms [2], [6] provides a 푋 푌 휔 − 푌 푋 휔 = < , > < , > convenient setting for integration on , as we − 푋 <푌[ 휔 , ] , > explain in this 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 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 IJSER푌푋푓 proof 푋푔 of푌푓 − the푋푓 푌푔 theorem ( ) . 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 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 . 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 gives 푙푚 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 . … 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: 훽 퐻 ∘ 퐹 훾 �� 푠푔푛 휎 푏 푏 푏 = (�det푑푥 ⋀) ⋀푑푥… is smooth with open then is a 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) 퐺This formula∗ is especially significant in light of the 퐹 훼 퐷 퐹 푥 퐴�퐹 푥 �푑푥1⋀ ⋀푑푥푛 then푛 − we define b. 푛 훼 − change of variable formula 푛 − 퐺 ⊂ ℝ More generally, if is an dimensional manifold with ∫퐺 훼 ( ) = ( ) | det ( ) | (5) an say the image on an open set by : 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 sign around det ( ). We ∗ say ∗a map : G is orientation preserving when For an form on� 훼. If �it 휑takes훼 several coordinates Ω 푀 퐺 det∫ 퐹 훼 (1 ) > 0 for all . 퐷 퐹 푥 patches to cover define by writing as a sum of 퐶 퐹 Ω → forms, each푛 − supported훼 on푀 one patch. 푀 Proposition퐷 퐹 푥 1. If : 푥 ∈GΩ is a orientation preserving We need to show푀 that ∫this훼 definition of훼 is a diffeomorphism and an integrable1 form on then independent of the choice of on . 푀 = 퐹 Ω → 퐶 Thus suppose : and : ∫ are훼 both Proof. The wedge product훼∗ of ’s 푛extends − to a 퐺wedge coordinate patches so that = : is 푀an 퐺 Ω product on∫ form훼 as∫ follows.퐹 훼 If ( ) has the form (1) orientation preserving휑 퐺 → 푈diffeomorphism. ⊂ 푀 휓 −1Ω →We 푈 need ⊂ 푀 to check 푙 and if 푑푥 푘 that if is an form on supported퐹 휓 on∘ 휑 ,퐺 then → Ω 훽 ∈ ⋀ 퐺 = ( ) … ( ) = = ( ) 훼 푛 − 푀 푈 푙 ∗ ∗ ∗ ∗ 푖 푖1 푖푙 define 훼 � 훼 푥 푑푥 ⋀ ⋀푑푥 ∈ ⋀ 퐺 Thus the integral� 휑 훼 of �an휓 훼 form� 퐹 휓over훼 an oriented 푖 퐺 Ω 퐺 = ( ) ( ) … … dimensional manifold is well defined. , 푛 − 훼⋀훽 � 훼푖 푥 푏푗 푥 푑푥푖1⋀ ⋀푑푥푖푙 ⋀푑푥푖1⋀ ⋀푑푥푖푙 푛Proposition − 2. Given a compactly supported ( in ( 푖)푗, it follows that = ( 1) IJSER( ) 1) form of class on an oriented dimensional It is 푘+푙also readily verified that =푘푙 ( ) ( ). 푘 − Another⋀ 퐺important operator훼⋀훽 on∗ forms− is훽⋀훼 the∗ exterior∗ (of class C ) with1 boundary with its natural derivative : ( ) ( 퐹) defined훼⋀훽 as퐹 follows.훼 ⋀ 퐹 훽If orientation− 훽 2 퐶 푘 − 푀� 휕푀 ( ) is given푘 by (3)푘+1 then = (6)

푘 푑 ⋀ 퐺 → ⋀ 퐺 푀 휕푀 훽 ∈ ⋀ 퐺 = … Proof.∫ Using푑훽 ∫a partition훽 of unity and invariance of the , 푗 integral and the under coordinate 휕푏 푙 푗1 푗푘 If ( 푑훽) and �: 푙 푑푥G is⋀푑푥 a smooth⋀ ⋀푑푥map. Now transformations it suffices to prove this = { 푗 푙 휕푥 푘 : 0}. In that case we will be able to deduce (6) ��� 훽∗ ∈ ⋀ 퐺 퐹 Ω → from푘 the fundamental theorem of calculus.푀 If 푥= ∈ ( ) 1 =푑퐹 훽 … ℝ( 푥) ≤ … … with ( ) of bounded , 휕 ∗ ∗ 훽 푗 푙 푗1 푗푘 support, we have � 푙 �푏 ∘ 퐹 푥 � 푑푥 ⋀�퐹 푑푥 �⋀ ⋀�퐹 푑푥 � 푗 1 푗 푘 푗 + 푗 푙 (휕푥±) ( ( ))( ) … … 푏 푥 푑푥 ⋀ ⋀푑푥 ⋀ ⋀푑푥 푏 푥 = ( 1) … , ∗ ∗ ∗ 푗 푗1 푗푣 푗푘 푗−1 푗 Now� pull 푏back퐹 gives푥 퐹 directly푑푥 ⋀ that⋀ �퐹 푑푥 �⋀ ⋀�퐹 푑푥 � 휕퐵 1 푛 푗 푣 If > 1 we have푑훽 − 푑푥 ⋀ ⋀푑푥 휕푥푗 = = = ( 1) = 0 푖 푗 ∞ ∗ 푗 푖 휕퐹 푙 푖 푗−1 휕퐵 푙 and hence ( 퐹 푑푥) = � =푙 0푑푥, so only푑퐹 first sum in (A) 푗 푙 휕푥 And also� 푑훽 = 0−, where� ��: −∞ 푗 푑푥 is �the푑푥 inclusion. On contributes to ∗ . Meanwhile 푀 휕푥 푖 푖 ∗ = 1 푑 퐹 푑푥 푑푑퐹 the other hand for we have ∗ 푘 훽 푘 휕푀 → 푀� 푑퐹 훽 ∗ = … = ( ( ))( ) … 푗 ∞ 퐹 푑훽 1 푗 휕퐵 , 휕푏 ∗ ∗ ∗ 1 2 푘 푚 푗1 푗푘 � 푑훽 � ��−∞ 1 푑푥 � 푑푥 ⋀ ⋀푑푥 so �we have푚 퐹 푥 퐹 푑푥 �퐹 푑푥 �⋀ ⋀�퐹 푑푥 � 푀 = (휕푥0, ) = 푗 푚 휕푥 푙 푙 ( ) = ( ( )) 1 This proves Stokes’ formula.� 푏 푥 푑푥 � 훽 푗 ∗ 휕푀 휕 푗 푙 휕푏 푚 � 푙 �푏 ∘ 퐹 푥 � 푑푥 � 푚 퐹 푥 퐹 푑푥 푙 휕푥 푚 휕푥 IJSER © 2016 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 7, Issue 4, April-2016 418 ISSN 2229-5518 same components, only the basis vectors look different. Proposition 3. [4] There is no smooth retraction : Again the new basis vectors for are defined locally at of the close unit ball in onto its each single point on the manifold [5]. Then we may say boundary푛−1 . 푛 휑 퐵 → that geometrically it corresponds푑푓 to a local view of the 푆 푛−1 퐵 ℝ surfaces of constant푃 values for . Since they are locally Proposition푆 4. [4] If : B is a continuous map on the defined over an dimensional manifold, surfaces of closed unit ball in , then has a fixed point. constant values for some function,푓 say the temperature , Theorem 4. For푛 퐹 any퐵 → ( ) the formula are ( 1) ddimensional푚 − surfaces so called hyper ( ,…, ) ℝ 퐹 푘 surfaces since they are only one less than the푇 휔 ∈ Ω 푀 푚 − − = ( 1) ,…, ,…, entire space. The simplest 1 form is an algebraic 푑휔 푋1 푋퐾+1 representation of the set of hyper surfaces of the constant 푖−1 � − 푋푖 �휔�푋1 푋� 푋푘+1�� value for at the point. − 푑푇 + 푖 ( 1) ( , , ,…, ,…, ,…, 푖+푗 REFERENCES푇 � (− )휔 �푋푖 푋푗� �푋1 푋�횤 푋�횥 푋푘+1� defines푖<푗 a + 1 -form ( ). 푘+1 [1] H. Flanders’, Differential Forms with Applications Proof. To 푘show that 푑휔 is∈ aΩ ( +푀 1) form we need to to the Physical Sciences, Academic Press, 1962. show that [2] J.C. Islam, Modern for is anti-symmetric i.e.푑휔 for any푘 < − Physicist, World Scientific Publishing Co. Pt. Ltd., 1989. ( ,…, ,…, ,…, ) [3] M.A. Hossain and M.S. Ali, Study on Exterior 푑휔 = ( ,…, 푟 ,…,푠 ,…, ) 1 푟 푠 푘+1 Algebra Bundle and Differential Forms, Annals of Pure is multi-linear푑휔 푋 at each푋 point푋 ,i.e. 푋 is ( ) linear 1 푟 푠 푘+1 and , Vol. 5, No.2, 2014, 198-207. on . Note that − is푑휔 obviously푋 푋 linear.푋 ∞ 푋So for any [4] M. Nakahara, Geometry, and , 푑휔 ( ) 푑휔 퐶 푀 IOP Publishing Limited, 1990. 푀 ∞ ( , 푑휔,,…, ) = ℝ( − ,…, ) [5] M.S. Ali, K.M. Ahmed, M.R. Khan and M.M. 푓This ∈ 퐶can 푀be checked by a direct computation: Islam, Exterior algebra with differential forms on 1 2 푘+1 1 푘+1 푑휔 푓푋 푋 ( 푋 , ,,…,푓푑휔 푋) 푋 manifolds, Dhaka Univ. J.Sci., 60(2) (2012) 247-252. = ( ,…, ) [6] M. Shigeyuki, Geometry of Differential Forms, 1 2 푘+1 푑휔 푓푋 푋 푋 Japanese by Iwanami Shoten Publishers, Tokyo, 1998. + ( 1) 1 ,…,2 ,…, 푘+1 푓푋 �휔 푋 푋 � [7] P.P. Gupta and G.S. Malik, Tensors and 푖−1 푖 1 횤 푘+1 Differential Geometry, Pragati Prakashan, Meerut, U.P., � − 푋 �휔�푓푋 푋� 푋 �� + 푖>1( 1) ([ , ], ,…, ,…, India, 2000. 푖+푗 1 푖 2 횤 푘+1 [8] S.P. Novikov and A.T. Fomenko, Basic Elements �=− (휔 ,…,푓푋 푋 IJSER)�푋 푋� 푋 � of Differential Geometry and Topology, 1995. 푖>1 + ( 1) ( ) ,…, ,…, [9] S.S. Chern, W.H. Chen and K.S. Lam, Lectures 푓푑휔 푋1 푋푘+1 푖−1 on Differential Geometry, World Scientific Publishing Co. � − 푋푖푓 �휔�푋1 푋�횤 푋푘+1�� Pt. Ltd., 2000 푖>1( 1) ( ) ,…, ,…, 푖−1 − � =− ( 푋,…,푖푓 �휔�푓푋) 1 푋�횤 푋푘+1�� 푖>1

1 푘+1 5. Geometrical푓푑휔 description푋 푋 of Differential forms:

The geometrical notion of the gradient of the function, such as the temperature in a room, the ordinary 3- dimensional vector ( ) defines a vector field throughout the room, which is usually described by saying that it is the direction∇푇 of 푟⃗greatest change of , at the particular point with coordinates . However the surfaces of constant value of the temperature to which푇 the the direction is perpendicular. These�푟�⃗ are surfaces much like equipotentials for electrostatics,푇 where the temperature∇푇 does not change. It is altogether plausible that the surface on which the function does not change is the one that is perpendicular to the direction of its greatest change. We therefore take the analytic idea that is the generalization of the gradient of , i.e., . As in the case of tangent vectors and directional ,푑푓 the comparison is reasonable since the푓 two quantities∇푓 have the

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