418. Calculations on Back Pages of Travel Diary [ca. 9–22 January 1923][1] [p. 51] ∂xτ -------( gμνgμν) 0= gμα ∂gμν ∂xτ -----------gνσ ∂gνσ -----------gμνxτ∂ –= ∂gνσ ∂xσ ----------- = ∂gασ ∂xτ -----------δν - α gαμgσν-----------μν ∂g ∂xτ –= gαμgσν μτ ν ντ μ + –= ∂gασ ∂xτ ----------- - gαμΓμτ σ gσνΓντα = ∂gασ ∂xσ ----------- - gαμΓμσ σ gσνΓνσα = ∂Aν ∂xβ -------- - Γαβ ν + BβCν Inv Tens. = Γαβ ν AαBβCν Inv. ∂Aν ∂xβ --------BβCν - –= Γαβ'Aα'Bβ'Cν' ν Inv. ∂Aν' ∂xβ' --------- -Bβ'Cν' –= ∂Aν ∂xβ -------- - Γαβ ν + BβCν ∂An' ∂xb' --------- - Γab'Aa n '+ Bb'Cn' = ∂xβ ∂xb' -------- - ∂xβ ------- - ∂An ∂xν -------- - dxν ' ∂x ∂xν --------Aν-n ∂xβ ∂xb' ------------------------Aν - ∂2xn' ∂∂ ∂xb' --------------------------nAν∂'x∂-βx∂ ∂xν ∂xβ + Γαβ ν AαBβCν ∂xβ ∂xb' ------------------------AνBb'Cn' - ∂2xn' ∂∂ Γab'Aa'Bb'Cn'n + = ∂x ∂xa' --------Aa'-ν Γαβ ν ∂xα ∂xa'∂xb' ------------------------ - ∂xβ - ∂xn-' ∂xν ∂xβ ∂xb'∂xa' -------------------------------- - ∂xν - ∂2xn' ∂∂ Γab n '+ = '∂x ∂x ∂xτ -------------------------ρ-a-b ' ∂xσ ∂x ∂xn' Γστ ρ ∂2xn' xσ∂xn' ∂∂ ------------------------ ∂xρ - Γab'…An' n δνβδνσ ∂xν ∂xn'∂xβ --------------- - - ' ∂x ∂xν ---------n + = Γστ ρ ∂2xn' ∂∂ ----------------An' Γab'An'∂xn-'Aρ n ∂xρ -------- + = ∂x ∂xa' '=aA-ρ-------- dAν δAνVektor ∂Aν ∂xβ -------- - Γαβ ν + dxβ = δAν Γαβ ν Aαdxβ –= dAν ∂Aν ∂xβ --------dxβ - = [eq. 1] 0 0 [2]
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