Chapter 12. Tensors

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R (12.1) (12.2) (12.3) (12.4) ω P = 2 c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 types, covariant (0 • ( A vector. cotangent a with is tensor 1) (0, ( a that Then way as same tensor. written the 2) be in (1, can tensor which the this for of did components we the define can We ebi uei hr.Ltu rtcnie h opnnso a field of tensor components 2) the (1, consider a are For first chart us a chart. in Let a in chart. field a tensor in rule Leibniz map linear a is tensor • hsw n,i gemn ihteerirdefinition, earlier the with agreement in find, we Thus as field of tensor this functions are components The argument. every in linear is map the that • way a such in 44 hr the where etr n -omi rdc ftersetv components, respective the of product a is 1-form a and vectors q , ti osbet d esr ftesm ye u o fdifferent of not but type, same the of tensors add to possible is It ecnnwdfieteLedrvtv fatno edb using by field tensor a of derivative Lie the define now can We A oeseiltpso ( of types special Some Alternatively, esrhscmoet with components has tensor ) p A esrfield tensor pe nie.I scle a called is It indices. upper P × ∈ q − niae pout,i h es htisato ntwo on action its that sense the in ‘product’, a indicates | T tensor P  ⊗ · · · ⊗ M p dx A times b a i A 1 1 ⊗ ··· ··· P sarl iigatno tec point. each at tensor a giving rule a is A . b {z A a A dx q A p sa lmn fthe of element an is ij k = ( + P j ,v u, = A ⊗ : B T ,q p, ij k T A P b ; a ∂x 1 P dx 1 ω ∗ M ( ··· ∂ A ··· → M esr aeseilnms 1 0) (1, A names. special have tensors ) = ) ∂x b k } b a a i ∂ q 1 1 p  ⊗ ⊗ ··· ··· i ( = x , b ( A a dx q ,v u, ∂x | T p ij k nacat hsw a write can we Thus chart. a in ∂ q R P . ∗ A j contravariant u j ⊗ · · · ⊗ M , ⊗ oe nie.I scle a called is It indices. lower ; p, i ; ,q p, + v ω oi satnetvco.A vector. tangent a is it so q dx j ∂x = ) B ie (12.6) times ω )tno a components has tensor 0) ∂ esrhscomponents has tensor ) k esrpoutspace product tensor k ) ) k {z b a . . 1 u 1 , ··· ··· hpe 2 Tensors 12. Chapter i v b a , A q j p T ω . P k ∗ h components the M . p } − tensor (12.10) (12.11) (12.7) (12.8) (12.9) A . 2 2 c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 e n oe nie.Snetecmoet fatno edare field tensor a of up- components have remaining we the manifold, the Since the all on involving indices. functions terms lower the and for per stand dots the where Then ( h oue fvco ed n -om,adb suigLeibniz field assuming tensor by a and Consider 1-forms, products. and tensor fields for rule vector of modules the field vector space a some to save will readable. This more confusion. formulae the of make possibility and a is there unless write can we components, Equating Since to according change field tensor the of components i and A ecncluaeteLedrvtv fatno ed(ihrespect (with field tensor a of derivative Lie the calculate can We notation the use will we on, now From system coordinate i.e. charts, of change a Under A ij k £ u dx = i T 0 T A i r o qa ngnrl,w get we general), in equal not are ( = ⊗ ij k = + + dx dx T £ T T dx a j m u i a a ··· m m T ⊗ ⊗ ··· ··· ··· u 0 b i ··· ··· a m b b a)b sn h atthat fact the using by say) , 0 n ··· dx n n ∂x = ··· ∂ A b ∂ ( ∂ n m A j £ k i k m ∂x ) 0 ∂x 0 £ ⊗ ij k j u ⊗ · · · ⊗ = 0 ∂ ⊗ · · · ⊗ ∂ u 0 i = = i m ∂x m T 0 A ∂ dx a ) m ⊗ · · · ⊗ i k A A ··· k 0 ⊗ · · · ⊗ 0 ··· j i b ij k i k 0 = 0 , n 0 j ∂ ∂x 0 ∂x ∂x ∂ = n A ∂x ∂x n ∂x 0 ⊗ i k i i u 0 0 ⊗ 0 0 i i j ∂ 0 0 i dx 0 i i dx ∂ n ∂ ∂x 0 ∂x ( dx ∂x n i ∂x £ ⊗ ∂ i T ∂x a ⊗ 0 0 ⊗ u a 0 i j j m ⊗ · · · ⊗ i dx 0 0 ··· 0 0 dx j j dx ··· 0 ∂x ∂ = ⊗ ∂x b ∂x ∂x a ∂x n i a ∂x a dx ) ⊗ · · · ⊗ 0 for ∂x , 0 k k ∂x j j ⊗ · · · ⊗ 0 ⊗ · · · ⊗ 0 0 k k 0 dx £ 0 . j 0 i i dx 0 ∂x 0 u ∂ ⊗ j ∂x b ∂ sadrvtv on derivative a is i ⊗ . ∂x dx i and dx ∂x ∂ dx x ∂x 0 b k i b 0 b 0 k k → ∂ , 0 + i + f ∂x x · · · ∂ · · · for (12.19) (12.18) (12.17) (12.13) (12.12) (12.16) (12.15) (12.14) 0 k i 0 . , ∂x ∂f the 45 i c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 utn hs noteepeso o h i eiaiefor derivative Lie the for expression the into these Putting that know also we and 46 eaeigtedmyidcs efidtecmoet fteLie the of components the find we indices, derivative, dummy the relabeling £ ( £ u ∂ u m T ) = a m ··· ··· − b n ∂x ∂u = m i + − u ∂ i i T T ∂ , i a i m i ··· T ··· ··· ··· b a n b m ··· n ∂ ··· £ b ∂ i n u a u u m dx i − · · · − + a = · · · ∂u ∂x + a i T T hpe 2 Tensors 12. Chapter dx a m a m ··· ··· ··· ··· i b i i . n ∂ ∂ i u b u n i . (12.21) (12.20) T and