Page:The Meaning of Relativity - Albert Einstein (1922).djvu/85

Rh Accordingly,

In particular, the derivatives $$\frac{\delta\phi}{\delta x_\alpha}$$ of a scalar $$\phi$$, are components of a co-variant vector, which, with the co-ordinate differentials, form the scalar $$\frac{\delta\phi}{\delta x_\alpha}dx_\alpha$$; we see from this example how natural is the definition of the co-variant vectors.

There are here, also, tensors of any rank, which may have co-variant or contra-variant character with respect to each index; as with vectors, the character is designated by the position of the index. For example, $$A_\mu^\nu$$ denotes a tensor of the second rank, which is co-variant with respect to the index $$\mu$$, and contra-variant with respect to the index $$\nu$$. The tensor character indicates that the equation of transformation is

Tensors may be formed by the addition and subtraction of tensors of equal rank and like character, as in the theory of invariants of orthogonal linear substitutions, for example,

The proof of the tensor character of $$C_\mu^\nu$$ depends upon (58).

Tensors may be formed by multiplication, keeping the character of the indices, just as in the theory of invariants of linear orthogonal transformations, for example,