Grad, Div, Curl, and all that… - WebRing

Gravitation and Cosmology
Proof of the theorem about determinants:
det(1 + G −1 ⎪1
+ (G
⎪
dG) =
⎪
⎪
−1 dG) 11
(G −1 dG) 21
= ∏
k
To this same order, then,
≈ 1 + ∑
k
(G −1 dG) 31
…
(G −1 dG) 12
1 + (G −1 dG) 22
(G −1 dG) 32
…
⎡ ⎣ 1 + (G −1 dG) kk ⎤ ⎦ + O⎛ ⎝ (G −1 dG) 2 ⎞ ⎠
(G −1 dG) 13
(G −1 dG) 23
1 + (G −1 dG) 33
…
⎛ ⎝ G −1 dG⎞ ⎠kk ≡ 1 + Tr⎛ ⎝ G −1 dG⎞ ⎠ .
d log [det(G(λ))] = log⎡ ⎣ 1 + Tr⎛ ⎝ G −1 dG⎞ ⎠ ⎤ ⎦ = Tr⎛ ⎝ G −1 dG⎞ ⎠
However, since G(λ) = e λA , clearly
dG(λ) = Ae λA dλ
and thus
d log [det(G(λ))] = Tr⎛ ⎝ G −1 dG⎞ ⎠ = Tr⎛ ⎝ e −λA A e λA ⎞ ⎠ dλ ≡ Tr(A) dλ ,
giving, by direct integration,
log [det(G(λ))] = λ Tr(A) + constant .
… ⎪
… ⎪
… ⎪
… ⎪
Since both sides must vanish when λ = 0, the constant is zero, giving at last (with λ = 1)
det(G) = exp [Tr(A)] = exp ⎡ ⎣ Tr(log(G))⎤ ⎦ .
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