Page:The principle of relativity (1920).djvu/200

 { dg^{μν} = -g^{μα} g^{νβ} dg_{αβ}

(31) { [part]g^{μν}/[part]x_{σ} = -g^{μα} g^{νβ} dg_{αβ}

and

(32){ dg_{μν} = -g_{μα} g_{νβ} dg^{αβ}

{ [part]g_{μν}/[part]x_{σ} = -g_{μα} g_{νβ} [part]g^{αβ}/[part]x_{σ}.

The expression (31) allows a transformation which we shall often use; according to (21)

(33)  [part]g_{αβ}/[part]x_{σ} = [α σ β] + [β σ α]

If we substitute this in the second of the formula (31), we get, remembering (23),

(34) [part]g^{μν}/[part]x_{σ} = -(g^{μτ}{τ σ ν} + g^{ντ} {τ σ μ})

By substituting the right-hand side of (34) in (29), we get

(29a)   1/[sqrt](-g) [part][sqrt](-g)/[part]x_{σ} = {μ σ μ}.