Tensor Products, Wedge Products and Differential Forms Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: June 4, 2016 Maple code is available upon request. Comments and errata are welcome. The material in this document is copyrighted by the author. The graphics look ratty in Windows Adobe PDF viewers when not scaled up, but look just fine in this excellent freeware viewer: http://www.tracker-software.com/product/pdf-xchange-viewer . The table of contents has live links. Most PDF viewers provide these links as bookmarks on the left. Overview and Summary.........................................................................................................................5 Notation....................................................................................................................................................7 1. The Tensor Product ..........................................................................................................................10 1.1 The Tensor Product as a Quotient Space ......................................................................................10 1.2 The Tensor Product in Category Theory.......................................................................................15 2. A Review of Tensors in Covariant Notation..................................................................................18 2.1 R, S and how tensors transform : Picture A..................................................................................18 2.2 The metric tensors g and g' and the dot product ...........................................................................23 n 2.3 The basis vectors en and e ...........................................................................................................25 n 2.4 The basis vectors un and u ...........................................................................................................26 2.5 The basis vectors e'n and u'n and a summary ................................................................................28 2.6 How to compute a viable x' = F(x) from a set of constant basis vectors en .................................31 2.7 Expansions of vectors onto basis vectors......................................................................................33 2.8 The Outer Product of Tensors and Use of ⊗.................................................................................36 2.9 The Inner Product (Contraction) of Tensors .................................................................................39 Dot products in spaces V⊗V, V⊗W, V⊗V⊗V and V⊗W⊗X.......................................................40 2.10 Tensor Expansions......................................................................................................................42 (a) Rank-2 Tensor Expansion and Projection .................................................................................42 (b) Rank-k Tensor Expansions and Projections..............................................................................43 2.11 Dual Spaces and Tensor Functions .............................................................................................45 (a) The Dual Space V* in Matrix and Dirac Notation ....................................................................46 (b) Functional notation....................................................................................................................47 (c) Basis vectors for the dual space V*...........................................................................................47 (d) Rank-2 functionals and tensor functions ...................................................................................51 (e) Rank-k functionals and tensor functions ...................................................................................54 (f) The Covariant Transpose ...........................................................................................................57 (g) Linear Dirac Space Operators ...................................................................................................57 (h) Completeness ............................................................................................................................64 3. Outer Products and Kronecker Products.......................................................................................66 3.1 Outer Products Reviewed: Compatibility of Chapter 1 and Chapter 2........................................66 3.2 Kronecker Products.......................................................................................................................68 1 4. The Wedge Product of 2 vectors built on the Tensor Product......................................................76 2 4.1 The tensor product of 2 vectors in V ...........................................................................................76 2 4.2 The tensor product of 2 dual vectors in V* .................................................................................80 2 4.3 The wedge product of 2 vectors in L ...........................................................................................83 2 4.4 The wedge product of 2 dual vectors in Λ ...................................................................................92 k 5. The Tensor Product of k vectors : the vector spaces V and T(V) ...............................................97 k 5.1 Pure elements, basis elements, and dimension of V ....................................................................97 k 5.2 Tensor Expansion for a tensor in V ; the ordinary multiindex ....................................................98 5.3 Rules for product of k vectors.......................................................................................................99 5.4 The Tensor Algebra T(V) ...........................................................................................................100 5.5 Comments about tensors .............................................................................................................102 5.6 The Tensor Product of two or more tensors in T(V)...................................................................102 k 6. The Tensor Product of k dual vectors : the vector spaces V* and T(V*)................................107 k 6.1 Pure elements, basis elements, and dimension of V* ................................................................107 k 6.2 Tensor Expansion for a tensor in V* ; the ordinary multiindex ................................................108 6.3 Rules for product of k vectors.....................................................................................................108 6.4 The Tensor Algebra T(V*) .........................................................................................................109 6.5 Comments about Tensor Functions.............................................................................................110 6.6 The Tensor Product of two or more tensors in T(V*).................................................................110 k 7. The Wedge Product of k vectors : the vector spaces L and L(V).............................................114 7.1 Definition of the wedge product of k vectors..............................................................................114 7.2 Properties of the wedge product of k vectors..............................................................................116 k 7.3 The vector space L and its basis................................................................................................119 k 7.4 Tensor Expansions for a tensor in L ..........................................................................................121 7.5 Various expansions for the wedge product of k vectors .............................................................124 k k 7.6 Number of elements in L compared with V .............................................................................126 7.7 Multiindex notation.....................................................................................................................127 7.8 The Exterior Algebra L(V) .........................................................................................................128 Associativity of the Wedge Product..............................................................................................129 7.9 The Wedge Product of two or more tensors in L(V) ..................................................................132 (a) Wedge Product of two tensors T^ and S^ ................................................................................132 (b) Special cases of the wedge product T^^ S^ .............................................................................134 (c) Commutivity Rule for the Wedge Product of two tensors T^ and S^ ......................................135 (d) Wedge Product of three or more tensors .................................................................................136 (e) Commutativity Rule for product of N tensors .........................................................................139 (f) Theorems from Appendix C : pre-antisymmetrization makes no difference...........................141 (g) Spivak Normalization..............................................................................................................143 8. The Wedge Product of k dual vectors : the vector spaces Λk and Λ(V)....................................147 8.1 Definition of the wedge product of k dual vectors......................................................................147