The Principle of Relativity

by Albert Einstein; H. Minkowski

= ∂/∂x_ν (F_σμ F^μν) - F^μν ∂F_σμ/∂x_ν.

On account of (60) the second member on the right-hand side admits of the transformation—

F^μν ∂F_σμ/∂x_ν = -½ F^μν ∂F_μν/∂x_σ

= -½ g^μα g^νβ F_αβ ∂F_μν/∂x_σ.

Owing to symmetry, this expression can also be written in the form

= -1/4 [g^μα g^νβ F_αβ ∂F_μν/∂x_σ

+ g^μα g^νβ ∂F_αβ/∂x_σ F_μν],

which can also be put in the form

- 1/4 ∂/∂x_σ (g^μα g^νβ F_αβ F_μν)

+ 1/4 F_αβ F_μν ∂/∂x_σ (g^μα g^νβ).