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

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THE GENERAL THEORY
75

multiply by and sum over the , we obtain, by the use of (62),

(64)

Since the ratios of the are arbitrary, and the as well as the are components of vectors, it follows that the are the components of a contra-variant tensor [1] (contra-variant fundamental tensor). The tensor character of (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,

(65)
  1. If we multiply (64) by \frac{\delta x_\alpha'}{\delta x_\beta}</math>, sum over the , and replace the by a transformation to the accented system, we obtain

    The statement made above follows from this, since, by (64), we must also have , and both equations must hold for every choice of the .