A Modern Presentation of Grassmann's Tensor Analysis

A Modern Presentation of Grassmann's Tensor AnalysisAuthor(s): Helen BartonSource: American Journal of Mathematics, Vol. 49, No. 4 (Oct., 1927), pp. 598-614Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370842 .

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A Modern Presentation of Grassmann's Tensor Analysis.

BY HELEN BARTON.

1. Introduction.. Since Einstein advanced his Theory of Relativity in 1916, there has been a considerable increase in interest in what is known as Tensor Analysis. Mathematicians and physicists alike are realizing the power of this method of treatment; especially since it has led to. the association of physical quantities which hitherto had seemed unrelated. The idea of a tensor, however, far antedates Einstein, it being introduced by Grassmann in 1844.*

In this work of Grassmann's Die Ausdehnungslehre, the author gives a rather exhaustive algebraic treatment of these quantities, which he termed " extensive Gr6sse," but his ideas were often abstruse and vague, so that the work has never been used nor appreciated very much by mathematicians. The main part of this paper will be devoted to the development of explicit expressions for these "extensive Gr6sse," and for various combinations of them. It is hoped by this more definite treatment of the subject that Grassmann's work will be simplified and thereby become of greater value to the mathematical world.

One method of defining a tensor is by means of its components and the transformation of these components under a change of coordinates. If we form n functions of n ordered numbers in the x space

51 (1 * n) ~'02 (X$1

* * * $n * * * -

n (x xn)

arbitrary except that the ratios '1/1X, p2/x2, * *, *4/xn are of the same physical dimensions, and then form another set of n functions of the corresponding y's in the y space,

w re (nyle.. yn)

which are definecl'by the equation

* H. Grassmann, Die Ausdelmunngslehre, Werke, Bd. 1, Leipzig, 1896.

598



BARTON: A 'Modern Presentation of Grassmann's Tensor Analysis. 599

-r (a Oyr/Oxa *

then these n functions 4)1, p2, * pn are the n components of a contravariant tensor of rank one as presented in the x space, and 1, 2 * O,n are the n components of the same contravariant tensor as presented in the y space.

Similarly, if we start with n functions of the x coordinates, (p, *

On which are now subjected to the condition that each product crXr is of the same physical dimensions, and then form the corresponding functions of the y coordinates in, 'f2 , ** where 'lr --, aXal/yr then qp, , , * are the n components of a covariant tensor of rank one as presented in the x space, and O,, 2, * * n, An are the n components of the same covariant tensor as presented in the y space.,

Other tensors of each of these types, as well as combinations of the two, but of higher rank are defined in a similar way; but without going into further detail, it is evident that tensors, as viewed from this angle, are composite quantities, and their components presented in one space bear a certain definite relation to the components in another space.

Grassmann had a very different point of view. He was not interested in the presentation of these " extensive Gr6sse " with reference to any system of cordinates, nor in their transformations, but his work dealt with the various operations which could be applied to these quantities. To him, an " extensive Grosse" was a quantity built up from simple quantities which have both direction and magnitude. These "extensive Gr6sse," therefore, are defined by their parts or components in certain given directions. It follows from the definition that two " Grbsse " are equal if the corresponding components of the two are equal.

Grassmann made no reference as to whether these " extensive Gr6sse" were contravariantly or covariantly presented, and the reason for this is clear. In Euclidean space, where the coordinates are orthogonal, we have the funda- mental relation

(ds)2 =(dxl)2 + (dx2)2 +* + (dxn)2

= (dyl)2 + (dy2)2 + *+ (dyn)2.

If linear transformations only are considered, they are of the type xr =cry%, yr == Crax' and there is no distinction between the covariant and the contravariant presentation of a tensor.

* A Greek letter in an expression plays the r8le of an umbral symbol, i. e. it indicates that the expression is a summation of terms, each formed by assigning the -n numerical values in succession to the umbral symbol.



600 BARTON: A Modern Presentation of Grassmann's Tensor Analysis.

2. Outer Product of Tensors. We know from ordinary Vector Analysis that two vectors may be combined into a vector or outer product, to give a new vector of the same type but of rank two, and its components are given by a two row determinant. Thus the r1r2 component of the outer product of the vectors a and b is given by

ab]rlr2 arl ar2

[aljrl2 ] |bri br2 |

but this might also be written in the form

Arfr2 - [ab]rlr2 [ r1 2 ]aal b2.*

The symbol here used L 1 r2 ] is known as the generalized Kronecker symbol.

The symbol is, as its name indicates, a generalization of the ordinary

Kronecker symbol 'and we shall represent it in the form Fr1 r2 rt 1 SI S2 SM

where m is any integer 1, 2, , n and r1, r2, *, r. and sI, S2, * , SM

may take independently any of the n integral values. By definition, the symbol has but three distinct values:

[ r1 r2 rm] O S1 S2 Sm

if r,, r2, , rm and s1, S2, , sm are not arrangements of the same set of m distinct integers;

ri r2 **r. Sl S2 ***Sm

according as it takes an even or odd iumber of interchanges to bring r1, r2, * , rm into the same arrangement as s1, S2, * * * Sm.

It should be noted further that this symbol is itself a tensor, mixed in type-covariant of rank m and contravariant of rank m.t :Furthermore, this mixed tensor is non-metric, that is, it holds for all space, in which a point is merely a set of n ordered numbers.

If, however, we modify the symbol by fixing the order of one of the

sets of labels, for example, L1, 2. , n], as we do in defining the

* F. D. Murnaghan, American Mathematical Monthly, Vol. 32 (1925), pp. 233-241. t F. D. Murnaghan, Bulletin of the American Mathemaxtical Society, Vol. 31 (1925),

pp. 323-329.



BARTON: A Modern Presentation of Grassmann's Tensor Analysis. 601

complement of a tensor, it should be noted that it is no longer a mixed tensor of rank 2nm, but if it is multiplied by + gl/2 it becomes a covariant tensor of rank m, where g is a determinant whose elements are the components of a covariant tensor of rank 2,-obtained from the metrical form

(dS) 2 =g4X.

In normal space, where grs = 0 if r + s and grr = 1, the determinant g 1. Hence, since Grassmann was dealing, in effect, with points which were ex- pressed in normal coordinates, the factor gl/2 does not appear.

This symbol therefore lends itself to the formation of an outer product of any number of these vectors or tensors of rank one, thereby producing a tensor of higher rank,-equal to the number of these vectors contained in the product. But we need not restrict ourselves to the outer product of tensors of rank one. We may form the outer product of several tensors, each of different rank, and so obtain a tensor of still higher rank.

Thus if A is a tensor of rank p and B is a tensor of the same type as A and of rank q, and p + q 2 n (where n is the dimension of the space), then

[AB]rl1.*. rp 8s. ** Sq may be defined as

a rl. . . rp sl .. Sq [rjr2 ...

rP S1S2 ...

Sq ] Ac . , BP, .. q *~~~~~~~X (X2 * * * (XP #1#2 * * q

This is the explicit expression for the r1 * * * rp si *sq component of the outer product of A and B, and it is readily seen that in this case C is a tensor of rank p + q.

Furthermore, it should be noted that C, the tensor obtained from the outer product of two or more tensors is alternating. For

[AB]rlr2. . rp s. .. sq _[rl2i rPjs S A ]q A04 .a ap BP, . .3q.

[AB]r2rl ...rps... Sq _ r2r - rp sip sq Aa* . ap,B, . .B . q.

(Xl(2 ..aP Pi /3 -p

It is evident that all odd permutations in the first case will be even in the second, and vice versa; hence, from the definition of the generalized Kron- ecker symbol,

[AB]7rlr2* . sP 8... Sq - [AB]r2rl ... rp s8i. . .q

Furthermore, if A and B are each alternating, then many terms in the summation become identical and we may remove the numerical factor l/p! q!. Since, in this paper, we shall confine our attention to the case of alternating




tensors, we shall write as our explicit expression for the rjr2 * * rPSl * Sq

component of the outer product of the alternating tensors A and B

(1) [AB]rl- rp8 .. .8q = p! ! [ri- rpsi s; 3]Aai .. apBj O .

It is evident that this definition has no meaning if p + q > n, for in that case, the generalized Kronecker symbol vanishes identically.

With these definitions as a starting point, we shall proceed to prove a few theorems relating to the outer product of tensors.

We shall first prove that

[AB]-(( I)pr[BA]

where A is a tensor of rank p, B is a tensor of rank r, and p + r < n. By definition

[AB] * 1 [1112 ... IP p+1l*p+r 1Ac*t ap Bap+l ap+r p! r! L1a2 * *XP ap+1 * **p+r

[BA]I1 * * * Ip+r I r1112 Ir lr+i Ir+4 p ]Ba,.. aAarll .*a+ p[! r! 1a2 . . ar ar+ * ( Br+p

In order to compare similar terms, let us rewrite [BA] in a different manner.

[BA] 11* - - Iv+.r 1 [11 * * * * * Ip+r; Bap+l .. atp+rAaj ap. p! 1! Xp+l * **p+r (x * * ap

In order to determine the sign, we must rearrange the a's in the symbol so as to make them agree with those in the sign factor of [AB]. To do this, a, must be moved over fr positions to bring it to the first position, i. e. the sign will be (_ 1) r; similarly it will take r interchanges to bring a2 to the second position, etc. Therefore, the final sign, when all p of the a's have reached their proper positions will be (_ 1)rp. Hence

(2) [AB] = ( 1)pr[BA].

From this it follows that if C is a tensor of rank s, and D is a tensor of rank t, then

[ABCD] ( 1)st [ABDC] 1( st+rt [ADBC]

( )st+rt+pt [DABC] ( 1)8t+rt+Pt+r8+P8+Pr [DCBA]

We shall now prove that

[blb2' bin] determinant of a's * [aa2 ... am]



BARTON: A Modern Presentattion of Grassrnann's Tensor Analysis. 603

where bi, b2, * * *, bm is a series of simple " extensive Grosse " which are line- arly dependent upon m other simple " extensive Gr6sse" a1, a2, * am; i. e.

b1 = c1Pla., b2 a2P2ap2 * * b - xPmap

Any component a, of bi may be represented as bl6' a= 1P1ap,L1. By defi- nitioni

[b1b2 . . . bm]rLr2. .. r. Fr1 r? 1bl1-lb2oT2* bmorm L1 (T a

[r~i I r)] clkaplaol. a2 P2ap,02. . . a,.'f

[r i * *. rm] aClP1Ca2P2 **mP- * apfl? apf2 **. ap,0

(1/rn!) [> IZm] [t122I I ( ]ax PX2P2 **axm, P ap^l ap2e - ap .0'*

- (1 /rn!) [D n] D ( m ap,L ap2 * apm

Hence

(3 ) [b, b,]rl .. rm D D(a) - ***am- aiaia?2e amo'

= Determinant of a's [a1a2 am] t

So far, we have been concerned only with the outer product of tensors, the sum of whose ranks is less than or at most equal to n. But before proceeding to the case where this sum is greater than n, we shall define a new quantity known as the complement of a tensor.

If A is a tensor of rank p, then the complement of A is a tensor B of rank n - p (represented as I A = B). Any component of the complement of A is explicitly defined as

(4) (|A)8 ... 8n_p BS1. .. = (l/p!) [ AP PPA .

Before proceeding to the next proof, we shall introduce at this point several lemmas, concerning the generalized Kronecker symbol, as they will be needed in the following proofs.

* Formula 2.3 of Professor Murnaghan's paper, Americawn Mathemaatical Monthly. t Formula 3.4 of Professor Murnaghan's paper, America* Mathematical Monthly. 1 If A is a contravariant tensor of rank p, the complement of A is covariant of

rank n - p. The shift of labels from above to below is used to indicate the change of type of tensor.




LEMMA I.

( r, * * rm 1 Hli* -g (m-g) !Fr * rm s1 . . Sg tXl **m-g I L p . pm-g Si * * * Sg Pi pm-g

From the second factor on the left-hand side, it is evident that in order that the symbol shall not be zero, a* am-g must be some arrangement of the quantities pi, p2, * * pm-g. Any inversion in a particular set of values assigned to al, a2, * * -g in the one factor will cause a similar inversion in the other factor; hence, their product will have the same sign as if the arrangements of cl * * am-g and pi pm-g are identical. Since there are (m - g) ! such arrangements of (x, a25 , **mg we must introduce the factor (m - g) ! on the right.

LEMMA II.

(6) * S b S1 zm9 (m_g) ! [ S1***Sg tXl a Z- ri rin Si * ** S

It is evident that if the symbol is not to be zero, that s1) S2, 8 2 sg is contained in r1, . , rm; also b1, * * , bg is contained in r1, * , 'mr; therefore al, * , am-g may assume only such values of rl, * * *, r. as are not contained in sl, * * *, sg. Consequently, b1,.* , bbg must contain only thosei values of r, * *, rm not assumed by a,,* * *, am-g. i. e. those contained in S1, * * sg. Since there are (m - g) ! such arrangements of al, * ,ag the truth of (6) follows.

LEMMA III.

(7) [riAa ...aA...ant ______

.a . . amJ g!(rn g) !a,, * agJ ...rm

X g+1 Aal am (xg+l * * m

ri r. Aaz ..am

- m! A~'r. .rmt

[r S x X 7 l . . . arm

[ri rg F 1ru+. rrm Aaz.. am g !(m g)! Ar, * rm la gJ+L am+i

Both of these statements follow from the alternating character of these tensors. Therefore the truth of (7) follows.



BARTON: A Modern Presentation of Grassmann/s Tensor Analysis. 60O

LEMMA IV.

(8) [ri rXi .x][r * 3 1n Aa*. am+r B*h *8n- r - (rn ) [ am+r AJ *8*n-r

+ ( ![m... m][ ..n Aa, * am+-r B01 .. rn ! l . . . anz aXm+l *'*am i-r /31 **/n-r

From (7) we have

Fri - -rm r *A * Xr 1Aa-...am+r= (m+r) !rr rm1 Lx1 c. *'m (Xm+l * m+r m! r! LX*i* *.. n

(X [1 ]

i . . .

al"+r

Substituting this in the left-hand member of (8) we obtain

(m + r)! r, * w rm 1FAl X1r rn!r! (al . . .a(7m ILm+l am+r I

x [Xn* rpi . . QB ] Aai. am+r B1 ... 83n-r*

(m + r) !r . rm n [1 1 m!L~i~xmiLxm+w c'm+r i3~* /3n-r jAai . . amn+r BO,. rn (X .l . . *x anz m+l **amz+r /31 * nf-r

(by 5)

It is readily shown that the complement of the complement of a tensor is equal to ? the original tensor.

If A is a tensor of rank p, then B IA is a tensor of rank n -p, and C= B is a tensor of rank p. We shall prove therefore that

IA ?A. By definition,

. PP _ 1 r * * * Pp .Xl *Xn-p B cpl - - -Li nj [I BIaz. an-p (n-p) n

1 P pp al an-p

(n-p)!p! 1 . ni

X L* n AP,. Pp (by4) 1 * ** n-p Pi . . . PP

(-1)p(n-p) F" "PP AP1A.**Ppp (by 6) L!Pi * pp

()9p(n-p) AP) * Pp

(9) . 1A (_ J)p(n-p) A.




If A is of order p and B is of order r, and p + r > n, then as we noted above, our former definition of the outer product of two tensors (when p + r < n) would have no meaning. Hence, we define the outer product of AB, when p + r > n as a tensor of rank p + r - n and its 'explicit expression is found by taking the outer product of the complements of A and of B, and taking the complement of this outer product i. e. I [I A * B] [AB] by definition, when p + r > n. j A is a tensor of rank n - p; I B is a tensor of rank n-r; [IA B] is a tensor of rank 2n- p -r < n; [I A I B] is a tensor of rank p + r-nf < n.

To find the explicit expression for this outer product, we proceed directly.

[A. B] I]... 12-- 11 /n-pf- T1 T n-r] p! r!(n-p)! (n-r)! 21n*p*

x [n&nc 1;n]Aal.-ap B/31 .8r X [a1 * * n-p (;1 * * p ][1 - Tn-r /3; *PEr]

Taking the complement of this, we obtain

[ A *|B],1l.... 2n-p-r -- Cmn.1 * p+-r-n

p! r!(n- p) !(n -r) !(2n-p- r)!

rMl Mfp+r-n pi p2n-p-r 1-t ry L n-p T1 Tn-r

LI.*fnJ Lp p p2n-p-rl

* *y. : in-L ; * * nr ][1 * Aai.

*. av

B*. * .r

( ) ~ m~(n-p) (p+r_n) Ml' * p+r-n Ti * Tn-r 1 p! r! (n --r)I 1. * ap

x T *n nrp;..

Aaj * * * ap BO, .:.Or x [> 4n.r91. . .3A ..

BPr..3 (by 5 and 6).

10 ( 1) (n-p)(r-1) Ml P+r-n (10) r!(n- r)!(p+r- n)! Lcj .p+r-n J

x [P+r1+I * p nr] Aaz .. ap B61 z. . .,t By means of this expression, it is readily shown that the complement of

the outer product of two tensors is equal to + the outer product of their complements, i. e.,

I [AB] ?-+ [ A |B]




where A is a tensor of rank p, B is a tensor of rank r.

Casel. p+r < n.

I [AB] is of rank n -p- r. [j A B] is of rank 2n- p r n n -p r < n.

[AB] I=Ch __ l P+ip+r

Cl h * p+r 1 ll Ip+rl Aa:L apgl BL Or p !r! [(Xi **p ,81 /Pr B

I * D y. .n-P-r

p! r!(p + r) ! * Sn-p-rXl' * \p+r]

[a1 * zxp 1 * . ,B ]Aa:L * ap BAL . . . r,

rp! Es pr! Si . c B;. . Sn-p-ral a p B.l1 P r (by 5).

To find [j A I B] we must use the expression for the outer product of two tensors, the sum of whose ranks is greater than n.

A * ] B: (- 1) p(n-p-r) ral ... an-p-r (n-p-r)! (n-r) !r! LsI Sn-p-r J

X [an-p-r+1 ***n-p 1 */3n-r 1(I A)ai... anv(I B)13*...

-)p(n-p-r) (-1)... an-p-r 1 an-p-rl an-p ftl* n-r1

(n-p-r) !(n-r) !r!2p.! p L1 * I n]

Xl ... * * n-p '/i *P 9p] 1 fln-r 81 B r]

(-1) (p+r) (n- . n ] A 'v B6' .r

r! p! _sl***sn p ryl***yp ai***r

(by 5, 6, 7), .. [AB] _-- (- 1)(pw) (n-l) [lA* B].

Case II. p+r>n.

[I A I B] is a tensor of rank 2n - p- r < n.

[ABR] is a tensor of rank 2n - p -r. 10



608 BARTON: A Modern Presentation of Grassmann's Tensor Analysi,s.

I [AB] j [j A I B] by definition.

* [AB]== || [A |1B]. =U(n:L)(2-p-r)(n-1)[ A .B] (by9)

(l)~ ~~~~~~1 ( (p+r) (n-1)[ A* - B].

Hence the theorem holds when p + r < n and p + r > n. We shall next prove [A (BC)] [ABC] where A, B, C are tensors of

rank p, r, s respectively.

Case I. p + r + s < n.

(r + s) ! p! al,. aP: . A* .r+s Aa v.. ]Iaip[ ...BC 3r+

- rABC] * Zp+r+s

(r +s) p! pr! s! -al ap 91' * r+s_

X [Y r& 8 Aa:L.** ap T Pyr Ca1L. . . AJ

1 rl . . ~~~~lp+r+s AajL. . ap Boyi * ttyr Cal. 8 p! r! s! (Xi' *' xyi * Yr8i * s

=[ABC] 11 . p+r+jr Hence (12) [A (BC)] [ABC].

Case II. p+r+s > 2n.

Let A'= I A, B'=- B, Ca_= I C. Then A' is a tensor of rank n-p; B' is a tensor of rank n -r; C' is a tensor of rank n - s. [A'B'C'] is a tensor of rank 3n - p -r - s < n and therefore comes under Case I.

[A'(B'C')] i1 * * * gr = [A'B'C'] i1 * * * Z_-_r-a

[A'B'C'] (- 1)(3n-pr-8) (n-1) [I A B'' I C']. But A,'= A and

I A` - I I A (- I)P(n-')A.

i [A'BC,'] = ( ) (3n-p-r-8+p+r+8) (n-1) [ABC] =( 1)3n(n-1) [ABC] - [ABC].

Similarly I [A' (BIC)] (_ 1)(3n-pr8)(n-) [I A,. j(B'C')] 3n(n-1) [A (BC)] [A (BC)]

Since [A'(B'C')] [A'B'C'],

then [A (BC) ] = [ABC].



BARTON: A Modern Presentation of Grassmanr's Tensor Analysis. 609

The explicit expression for [ABC] when p + r + s > 2n may be derived as follows:

[A'B'C'] [ABC] 8L... 8p+r+a_-n

(-1) Qp+r+S-2n) (*-S Sp+r+8-2n ]l **n P fl

(3n (p+r-s) (n-r) ! (n-s) ! (n-p) !r !p!s ! .

r el en-P Si * an-r 1 *-l an-s n rn n r XAl * **sn-p-r-sJ Cle 'n-pa 'Xi LaP 9nr1 9r

x W n eAashaal Bl further prove v1 * * * n-s 71* 78

Cas. +r (p+r+s-2+s)(n-p)= _ Sp+r+s-2n+

(p+r+s 2n) !(nr) !(n-s)s ra.s+ o-r p+r+s-2n

(12A) X [~~~ap+r+.9 2nf+l ap+s-n fl .. Pr]

X [ n Aal * aP B891 . * * 0r Cyi y >s

ap+s-n+l ***ap yl * Ys

We shall further prove that

(13) [AB * AC] =+[ABC] A.

where A, B, C are tensors of rank p, r, s respectively.

Calsel1. p +r +s _n.

[l AB * |AC2]is atensor of rank s +ror n-p,

I [I AB I AC] is a tensor of rank p.

[ABAC]ril mp. ((p1)vs; M

[ml MP] P!(p+s)!r! a,cc-a

[Px **a+ 1***A+ [AB]a:l* ap+.r [AB]01* * p+,,

1-) Ps rMl.. MP nN p!3(p+s)!r!2s! [ml m ap [aP+1 (P+r 9 ]P+8

x[ *'' (X/p+r ][/ ' 'PP+8 ]A-:L* 'YPB..6tA . ..al. ..rAI P P1 P

''Y Si B p3 r 1 p pl. ..

*Ps

(_1)p(s+r) Fm MP n1 __ ___ ___ m s ! Lp ][1,l IP af Br pl

X A'Yi *YPA'*l* AP B63. *rCPl. . . P,




( l)P(s+r) [Ml *MP]A

p! Yl 'YP

X p ! r ! s ! [{1 *{p b; ..

8; pi *' **ps ]AAl ...

APB6: l **

rC ** Ps

(13A) (-1 ) p(s+r) [ABC] A1i **m.

Case II. p+r+s = 2n.

Let A' I A, B' I B, C' I C. Then [A'B'C'] is a tensor of rank 3n- p - r - s n. Hence we can apply the theorem for Case I.

[A'B' A'C1'] = (- (n-P) (n-) [A'B'C'] A'

| [A'B' A'C'] =( 1) (2n-p)(n-1) [I A' I B' A'. I C'] - (- 1) (2n+p+r+8) (n-1) [AB * AC]

= [AB AC].

Therefore to find the expression for [AB AC] when p + r + s 2n we shall find the complement of [A'B' A'C']

(n_p) [2 (nA) (n-s) !B11 '']np1]

r Pi** Pn- (Ti * '- * i .. f ***En-s

L1 * - - - - nZ

X Alai.. . Gn-v Al p.. pn-p BvO1 .n-r C/El en- (Case I)

(_ 1)p(n-1) (xi .. . an p p!2r!s!(n-p) !2(n-r) !(n-s)! LI1 iP

x Pi pn-p 1i Ofn-r En. ] [ * *.n n X L1 ~~~~~n j al an-p yl 'y.P

rl * * * n 1 l****n

X L pn-p1 *J *p- Lvi gn-r1 Si .. Sr

X~~~~~~ A[Y *'Y AA * * ff A t P gP B61 . .. ar Cvi *... Ps C1 * n-s Vif- VS


[A'B'* A'C']-([AB - AC] !(n- ) L nnP p p(n-1) Mnl rpki * np

p!2r!s! (n -p) !3 (n-r) ! (n-s) !1*** *nZ




xnX * P 1 * ] A * *n

rl nl rl ~ ~ n n l (Xi aXn-p yi 'ypJ L pn-p JJih * **#(p gi La Un-r 8i .. S8r

X n

AY:L -TP AA1 * -AP B61 . ar CV1 . Ps V

Cl- C n-s Vi' * ' *Vs

MI M ql qP1 AY1 * *P ap 1

p Y71 *- YP . r!s!(n-r) !(n-s)!

x [>1. * * Ifn-j 8 * * * r ] [gIn!r+i * Lp Vi* Vs]

X Alk AP B61 * * r CV1 Ps *V

(13B) Amr ni..mp [ABC].

In order to show that the above expression is equal to [ABC] when p + r + s = 2n, we shall first find [A'B'C'] whose rank n.

[A-B C] _ (XI ... an_p _91 *** n-r pi*** pn-s

(n-r) !(n-p) !(n-s) !i [ -* 3 n - n] X A l a, . . "-p Bt t. . . Cr p, . . . pn-s

But the complement of [A'B'C'] is [ABC]; i. e.

I [A'B'CG] =(-1)n(n-1)[ Al B* C'] = [ABC]

* [A* BC[ ] n!(n-r) !(n-p) !(n-s) !p!r!s! .1 * ' ] X [13 *I3an-p /31-/r pl ]As[. .1 Bn-r:Pi ... n *s n

Thenrprod1 of to "x ens- G rVss A, B as deie b ass-

(13C) (n-r)!(n-s)!r!s!tq[y -1Jn-r81-8r]

X [1 n ]AA1* AP gB61 .. *6,Cvl. P, r

Jn-r+l 'M *Jp Vi ... Vs

3. Inner Product of Tensors. In the study of simple vectors, we con- sider two varieties of products,--outer or vector products and inner or scalar products. So here, we form not only outer products of tensors, but also inne-r products.

The inner product of two " extensive Grosse " A, B, as defined by Grass-




mann, is the outer product of A into the complement of B; i. e. the inner product of A into B equals [A I B]. We note here that if A is of rank p and B is of rank r, and p > r, then the rank of [A I B] is equal to p- r; but if p < r, then the rank of [A I B] is n + p -r. In order to keep the two cases more nearly similar, we shall modify the above definition as follows.

If p 5 r, the inner product of A into B is defined as [A | B]. If p < r, the inner product of A into B is defined as I [A I B].

The rank in the second case is now r - p. We shall now proceed to derive the explicit expressions for the inner

product of two tensors A and B, in the following cases:

I. The ranks of the two tensors are equal. II. The rank of ,A is greater than rank of B.

III. The rank of A is less than rank of B.

Case 1. A is a tensor of rank p, B is a tensor of rank r and p r. Then the rank of [A B 1] is zero.

[A I B]81 . . . SP

(14) ( 1)~n1) ai. . apj .P Aa.. a , p

)(nf-p) [)1 P Aal ... ap B,01 . .. p

(14) = p! Aal. * ap Bal . .. ap

Similarly

A ~ ~ ~~~ pi ... * ** pjUi * * *n-p [B IA]81 ...8 p!p!(n-p)! s

- n Aal . B . ap

J)lp(n-1)Al) a a.

Hence [A I B]-[B I A] when p r. It is evident from (14) that Grassmann's definition of the inner product

of two tensors is in accord with the usual method of finding the inner product of two tensors, that of multiplying scalarly the contravariant presentation of one into the covariant presentation of the other.



BA1RTON: A Modern Presentation of Grassmrann's Tensor Analysis. 613

Case II. p > r.

By definition, the inner product of A into B is [A I B]. It is evident that the sum of the ranks of A and of I B is p + n - r which is greater than n, hence we must use formula 10 in writing the explicit expression for [A B]. Thus

[A I B]mi. (-nP-r (p-r)(n-P) Ml M r1

(p-r) !(n--r) ! ar apLr

[ap **r+i a]p * * *n-n

1l5) (-l) (p-r) !fr!2 [}l ap- I Lap;r+i .. pJ

X Aal ... ap Bz1...TYr Case III. p < r.

By definition, the inner product of A into B in this case is I [A * B]. The rank of this tensor is r -p.

[AlIB [i Sn. [A |B] 8 * ** 8nX_r P ! (n--r) ! r ! a, * ap i,; * * * &-,

X [a inr13 3 Aal * * ap , ...r


[A I B] C1. Zr-p

(n+pr) ! p! r ! (n-r) ! 11 ***Ir-p cri * an+,p-r_

[J1 *X ;h n p /nr

] -rb n * b 1 /r

(16) (_ I) r (-1 [I3i Pr-p ]Pr-p+l Pr ]Aal.. apB0 .

(16) ~~~(r-P) ! p!2 [11 Ir- p z B al * ap Btf

Let us now compare the inner product of A into B with that of B into A.

I. When p > r.

[A I B]1 ... m- - (- (p1)-r) (n-) (n-1) ml. mp-r1

[ABri(p-Xr) ! r !2 al ap-rJ

x fi Pr ]Aaczi.. ap BP, ... Pr.



614 BARTON: A Modern Presentation of Grcassmann's Tensor Analysis.

Now [B I A] will fall under Case III, therefore by 16, we may write at once

[B IA]mt ... m,- (p-r) r!2 Lm1 mpi

X qp-r+l' *x A al a,, Bgl r. ..B

The two expressions are of different types, but otherwise, they are identical except for sign. However, if we multiply [B I A] by (_ 1) r(p-1), the two are equal, since the exponents of (- 1) are congruent, modulus 2.

II. When p < r, the same relation holds, since in either case, it results in a comparison of formulas 15 and 16. Hence the inner product of A into B (- 1) r(p-1) inner product of B into A, provided we do not distinguish between contravariant and covariant.

4. Conclusion. In the preceding pages, we have endeavored to give a modern presentation of Grassmann's Die Ausdehnungslehre with special em- phasis on the development of the explicit expressions for various combinations of tensors.

Having defined, by means of the generalized Kronecker symbol, the outer product of tensors, the sum of whose rank < n, and also the complement of a tensor, we have derived the explicit expression for the outer product of tensors, the sum of whose ranks > n,

From these expressions as a basis, various relations between the outer products of tensors are readily derived.

We have also derived expressions for the inner product of two tensors, of ranks p and q, in the cases where p = r, p > r and p < r.*

* Since writing the above paper, we have read with interest a paper by C. L. E. Moore, " Grassmannian Geometry in Riemannian Space," Jou'rnal of Msathematics and Phys&is (Massachusetts Institute of Technology), Vol. V, No. 4 (June, 1926), which bears directly on this subject.



A Modern Presentation of Grassmann's Tensor Analysis

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