Author Department of Mechanical Engineering, Ben-Gurion University - PDF document

e Geometry of Continuum Mechanics Author

Department of Mechanical Engineering, Ben-Gurion University P .O.Box 653, Beer-Sheva 84105 Israel rsegev@bgumail.bgu.ac.il A. [ Arn74 ]e stress-energy tensor of field theory is defined and analyzed in a geometric setting where a metric is not available. e stress is a linear mapping that transforms the 3-form representing the flux of any given property, e.g., charge-current density, to the 3-form representing the flux of energy. e example of the electromagnetic stress-energy tensor is given with the additional structure of a volume element. Keywords. Differential forms, conservation laws, flux. 2001 PACS Class. 11.10.-z, 11.10.Cd

CHAPTER 1 Linear Forms and Generalized Forces In the special case where any one of the sequences contains the num- 0 1 j . the situation when in one (or both) of the sequences two or more indices situation when the two sequences do not contain the same elements, and We note two particular cases when the Levi-Civita symbol vanishes: the otherwise. 0 them by an odd number of transpositions, them by an even number of transpositions, 3 1.2. Path Integration and Work 1.3. Alternating Arrays ing symbol is defined by the switching of positions of two elements as a transposition . e alternat- provides a tool for working with alternating quantities such as the local 1.3.1. e Levi-Civita alternating symbol. e Levi-Civita symbol July 20, 2009 1.1. e Dual of a Vector Space e presentation below is similar to that in [ dR84 , pp. 17–18] representatives of forms. For a sequence of indices i 1 ,..., i r we will refer to    if the indices in the sequence ( i 1 ,..., i r )are distinct + 1     and the sequence ( j 1 ,..., j r )may be obtained from      ε i 1 ... i r j 1 ... j r = − 1 if the indices in the sequence ( i 1 ,..., i r )are distinct    and the sequence ( j 1 ,..., j r )may be obtained from         are equal (e.g., i p = i q ). We will sometimes use the notation i for the se- quence i 1 ,..., i r and we can write ε i e (somewhat degenerate) case where r = 1 is traditionally referred to as the Kronecker symbol (usually denoted by δ rather than ε ), { if i = j , ε i j = if i ̸ = j . bers { 1 ,..., r } and the other sequence is the ordered sequence (1 ,..., r ) the

1.3. ALTERNATING ARRAYS mation symbol or warn that the implicit summation is not performed. Clearly, if the two sequences contain the same elements, 4 otherwise. simple properties of the alternating symbol. In general, we will use the summation convention for repeated indices. However, in various instances, the summation convention cannot be used or it may cause confusion. In such cases we will explicitly write the sum- We first note that odd number of transpositions, (1.3.1) sides vanish. Assume that each of the two sequences contain distinct in- Hence, in the sum over the repeated i all terms vanish except for the term vanishing term and its sign depends only on the number of transpositions needed to arrive from the i -sequence to the j -sequence. Similarly, (1.3.2) that for nonvanishing terms the k -sequence contains the same elements (possibly in different order) as the j -sequence. us, each non-vanishing 0 0 even number of transpositions, (1 ,..., r )-sequence will be omitted in the notation. For example,   if ( i 1 ,..., i r ) may be obtained from (1 ,..., r ) by an + 1        − 1 if ( i 1 ,..., i r ) may be obtained from (1 ,..., r ) by an ε i 1 ... i r =       if ( i 1 ,..., i r ) cannot be obtained as a permutation of   { 1 ,..., r } , in particular, if two indices are equal. { ε i 1 ... i r ε j 1 ... j r ε i 1 ... i r j 1 ... j r = Assume that indices range in { 1 ,..., m } . We list below a number of ε ii 1 ... i m − 1 ij 1 ... j m − 1 = ε i 1 ... i m − 1 j 1 ... j m − 1 . If either i p = i q or j p = j q for some p ̸ = q , p , q = 1 ,..., m − 1, then both dices. en, since the m − 1 elements in the sequence ( i 1 ,..., i m − 1 ) belong to (1 ,... m ), they contain all the numbers 1 ,..., m except for one, say k . for which i = k , the missing element. It follows that in a non-vanishing term the elements of the sequence ( j 1 ,... j m − 1 ) also contain the elements of the 1 ,..., m except for k . is means that actually there is only one non- ( m − r ) ! ε i 1 ... i p j 1 ... j r ( m − r − p ) ! ε j 1 ... j r i 1 ... i p k 1 ... k r = k 1 ... k r . e elements of the given sequence ( j 1 ,..., j r ) determine the values that the repeated indices may assume—the m − r values required to complete them to { 1 ,..., m } —such that no two superscripts will be equal. is implies term in the sum on the repeated indices is ε j 1 ... j r k 1 ... k r , independently of the

1.3. ALTERNATING ARRAYS contain elements from the set (1.3.5) R 1.3.1. It is noted that if one requires that the sequences of of the expressions above assume simpler forms as the sequences cannot be permuted any more. Using parenthesis to indicate sequences that are ordered, e.g. , ( i ), we can write for example (1.3.6) 5 (1.3.4) any bijection on a finite set (1.3.7) However, once the elements of the sets are enumerated, the permutation (1.3.8) may be regarded as a permutation on the set e sign of the permutation p is defined by add up. Similar arguements lead to the slightly generalized rule, (1.3.9) (1.3.3) values of the i -indices. e number of such non-vanishing terms is the From the definition of the alternating symbol we also have numberofwaysyoucanassignthe m − r remainingvaluesforthe p repeated indices (choose p symbols out of m − r symbols), i.e., ( m − r ) ! /( m − r − p ) ! . In particular, for for r = 0, m ! ε i 1 ... i p ( m − p ) ! . i 1 ... i p = j 1 ... j r ε j 1 ... j r ε i 1 ... i r k 1 ... k r = r ! ε i 1 ... i r ε i j ε j k = r ! ε i k 1 ... k r , k . Once the indices i 1 ,..., i r and k 1 ,..., k r are given, for nonvanishing values of the alternating symbols, the j 1 ,..., j r indices should be obtained as per- mutations of these indices and there r ! such permutations that we have to j 1 ... j r ε j 1 ... j r n 1 ... n p k 1 ... ... k r + p = r ! ε i 1 ... i r n 1 ... n p ε i 1 ... i r k 1 ... ... k r + p . indices such as i 1 ,..., i r are of increasing order, i.e. , i p < i p + 1 , then some ε i ( j ) ε ( j ) k = ε i k . Permutation mappings. If the sequences i = ( i 1 ,..., i r ) and j = ( j 1 ,..., j r ) { 1 ,..., r } , then there is a bijection { 1 ,..., r } { 1 ,..., r } p : − → such that j = p ( i ), or j q = p ( i q ). It is noted that a permutation mapping is { a , b ,... } { a , b ,... } p : − → . { a 1 ,..., a r } { a 1 ,..., a r } p : − → , { 1 ,..., r } . Conversely, a per- { 1 ,..., r } { 1 ,..., r } induces a permutation of a sequence mutation p : → { j 1 ,..., j r } �− → { j p (1) ,..., j p ( r ) }. sign( p ) = ε p (1) ... p ( r ) = ε p (1) ... p ( r ) . 1 ...... r

1.3. ALTERNATING ARRAYS the components of an alternating array reverse their sign under any trans- k p ( r ) i r (1.3.11) Evidently, on the sum over all permutations of most one permutation for which the product does not vanish. Equation 6 1.3.2. Alternating Arrays and Anti-Symmetrization. An array of if (1.3.12) ternating symbol is an alternating array—the unit alternating array. us, position. Using an ordered sequence of indices, the definition may be k p (2) written as (1.3.13) If we want to use the summation convention for repeated indices, the equation above should be changed to or (1.3.14) as both the alternating symbol and the alternating array change sign under any permutation. by or i 2 (1.3.15) k p (1) i r i 1 i 1 In addition, the definition if the permutation symbol is equivalent to (1.3.10) i 2 k p ( r ) k p (1) Clearly, k p (2) j p (1) ... j p ( r ) ε = sign( p ) . j 1 ...... j r ∑ ∑ ε k 1 k 2 ... k r p sign( p ) ε ε ··· ε p sign( p ) δ δ ··· δ . i 1 i 2 ... i r = = { 1 ,..., r } above, there at (1.3.11) is usually referred to as the ε − δ -identity. If q : { 1 ,..., r } → { 1 ,..., r } is another permutation, then sign( q ◦ p ) = ε q ◦ p (1) ... q ◦ p ( r ) = ε q ( p (1)) ... q ( p ( r )) ε p (1) ... p ( r ) = sign( q )sign( p ) . 1 ... ... r 1 ...... r p (1) ...... p ( r ) Note that since the inverse permutation mapping p − 1 involves the same number of transpositions as p , sign( p − 1 ) = sign( p ). degree r , ω i 1 ... i r , i 1 ,..., i r ∈ { 1 ,..., m } is alternating or completely antisymmetric ω i 1 ... i r = ε j 1 ... j r i 1 ... i r ω j 1 ... j r , no sum on repeated indices , or alternatively, ω p ( i ) = sign( p ) ω i for any permutation p . Clearly, the al- ω j = ε ( i ) j ω ( i ) . r ! ε j 1 ... j r ω i 1 ... i r = 1 i 1 ... i r ω j 1 ... j r , ω i = 1 r ! ε j i ω j , Let A = ( A i 1 ... i r ) be any array, i.e. , not necessarily alternating. e array ( ) A induces an alternating array Alt A = Alt A j 1 ... j r r ! ε i 1 ... i r (Alt A ) j 1 ... j r = 1 (Alt A ) i = 1 r ! ε i j A i . j 1 ... j r A i 1 ... i r