section 6_7 installed
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Scratch copy of Section 6.7 from Phil's "Wedge World" tensor notes, marked as already installed in the main document. It shows that the product of a k-tensor T and a k'-tensor S evaluates as T(v1..vk)S(vk+1..vk+k'), giving a (k+k')-tensor. The derivation is repeated in multi-index notation and extended to three and then N tensors, with total rank K = k1+...+kN.
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Section 6.7 scratch 11.10.15
this is installed, no need to keep
6.7 The Tensor Product of two or more functions in T(V)
Recall the expansion of two dual tensors shown in (6.6.4),
TS = Σii...ijj...jTii....i Sjj....j
(λi λi ..... λi λj λj ..... λj) .
(6.6.4)
Evaluate this functional at (v1,v2....vk, vk+1....vk+k') and then use (6.1.6) to get
(TS)(v1,v2....vk, vk+1....vk+k') = Σii...ijj...jTii....i Sjj....j
(λi λi ..... λi λj λj ..... λj)(v1,v2....vk, vk+1....vk+k')
= Σii...ijj...jTii....i Sjj....j
(v1)i (v2)i .. (vk)i (vk+1)j (vk+2)j .. (vk+k')j // (6.1.6)
= [ Σii...iTii....i (v1)i (v2)i .. (vk)i ] *
[Σjj...j Sjj....j (vk+1)j (vk+2)j .. (vk+k')j]
= T(v1,v2....vk) S(vk+1,vk+2....vk+k') . // (6.1.6) twice
We have then obtained the rule for multiplying two dual tensor functions,
(TS)(v1,v2....vk, vk+1....vk+k') = T(v1,v2....vk) S(vk+1,vk+2....vk+k') (6.7.1)
Recall comments (6.2.6) above, so Tk(V) is isomorphic to V*k and the elements of Tk(V) are k-tensors. Equation (6.7.1) combines the k-tensor function T ϵ Tk(V) and the k'-tensor function S ϵ Tk'(V) and generates a (k+k')-tensor function called (TS) where (TS) ϵ Tk+k'(V).
It is useful to repeat the above development in multi-index notation:
TS = ΣI,J TISJ λIλJ
(TS)(vI,vI') = ΣI,J TISJ λIλJ(vI,vI') = ΣI,J TISJ λI(vI)λJ(vI')
= ΣI,J TISJ (vI)I(vI')J = [ΣITI(vI)I] [ ΣJSJ(vI')J ]
= T(vI) S(vI') I = {i1, i2, .. ik} , I' = {ik+1, ik+2, .. ik+k'} . (6.7.2)
Then we can extend the idea easily to
TSR = ΣI,J,K TISJRK λIλJλK
(TSR)(vI,vI',vI") = ΣI,J,K TISJRK λIλJλK(vI,vI',vI")
= ΣI,J,K TISJRK (vI)I(vI')J(vI")K = [ΣITITI(vI)I] [ ΣJSJ(vI')J ] [ΣKRK(vI")K ]
= T(vI) S(vI')R(vI") I" = {ik+k'+1, ik+k'+2, .. ik+k'+k"} (6.7.3)
which we can then write out longhand as
(TSR)(v1, v2, ... vk+k'+k")
T(v1,v2....vk) S(vk+1,vk+2....vk+k')R(vk+k'+1,vk+k'+2....vk+k'+k") . (6.7.4)
This shows a combination of a k-tensor and a k'-tensor and a k"-tensor to produce a resulting
(k+k'+k")-tensor in the space Tk+k'+k"(V).
Finally in systematic notation,
Ti = tensor of rank ki Ii = multiindex range of ir values for tensor Ti
I1 = {i1, i2.....ik}, I2 = {ik+1, ik+2.....ik+k}, etc. (6.7.5)
we have
T1T2...TN = ΣII...I (T1)I(T2)I ...(TN)I λIλI...λI (6.7.6)
(T1T2...TN)(vI, vI ... vI)
= ΣII...I (T1)I(T2)I ...(TN)I (λIλI...λI)(vI, vI ... vI)
= ΣII...I (T1)I(T2)I ...(TN)I (vI)I(vI)I ... (vI)I
= [ΣI(T1)I(vI)I][ΣI(T2)I(vI)I] ... [ΣI(T2)I(vI)I]
= T1(vI) T2(vI) .... TN(vI)
so get the general result
(T1T2...TN)(vI, vI ... vI) = T1(vI) T2(vI) .... TN(vI) . (6.7.7)
This then is the tensor product of N ki-tensors to produce an K-tensor (T1T2...TN) where
K = k1+k2 + ... +kN. (6.7.8)