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 = gμν g dgμν = -gμν gdgμν.
The last form follows from the first when we remember that
gμν gμ′ν = δμ′μ , and therefore gμνgμν = 4,
consequently gμνdgμν + gμν dgμν = 0.
From (28), it follows that
"(29)"
Again, since gμν gνσ = δνμ , we have, by differentiation,