Page:The Meaning of Relativity - Albert Einstein (1922).djvu/87

Rh multiply by $$g^{\mu\beta}$$ and sum over the $$\mu$$, we obtain, by the use of (62),

Since the ratios of the $$d\xi_\mu$$ are arbitrary, and the $$dx_\beta$$ as well as the $$dx_\mu$$ are components of vectors, it follows that the $$g^{\mu\nu}$$ are the components of a contra-variant tensor (contra-variant fundamental tensor). The tensor character of $$\delta_\alpha^\beta$$ (mixed fundamental tensor) accordingly follows, by (62). By means of the fundamental tensor, instead of tensors with co-variant index character, we can introduce tensors with contra-variant index character, and conversely. For example,

Volume Invariants. The volume element

is not an invariant. For by Jacobi's theorem,