Some special cases of Particular Importance.
A few auxiliary lemmas concerning the fundamental tensor. We shall first deduce some of the lemmas much used afterwards. According to the law of differentiation of determinants, we have
(28) dg = g^{μν} g dg_{μν} = -g_{μν} gdg^{μν}.
The last form follows from the first when we remember that
g_{μν} g^{μ´ν} = δ^{μ´}_{μ} , and therefore g_{μν}g^{μν} = 4,
consequently g_{μν}dg^{μν} + g^{μν} dg_{μν} = 0.
From (28), it follows that
(29) 1/[sqrt](-g) [part][sqrt](-g)/[part]x_{σ} = 1/2 log(-g)/[part]x_{σ} = 1/2 g^{μν} [part]g_{μν}/[part]x_{σ}
= -1/2 g_{μν} [part]g^{μν}/[part]x_{σ}.
Again, since g_{μν} g^{νσ} = δ^ν_μ , we have, by differentiation,
{g_{μσ} dg^{νσ} = -g^{νσ} dg_{μσ}
(30) { or
{g_{μσ} [part]g^{νσ}/[part]x_{λ} = -g^{νσ} [part]g_{μσ}/[part]x_{λ}
By mixed multiplication with g^{στ} and g_{νλ} respectively we obtain (changing the mode of writing the indices).
