Page:The Meaning of Relativity - Albert Einstein (1922).djvu/93

Rh vanishes for every value assigned to $$(A^\mu)$$. We therefore get

From this we arrive at the co-variant derivative of the co-variant vector by the same process as that which led to (71),

By interchanging the indices $$\mu$$ and $$\sigma$$, and subtracting, we get the skew-symmetrical tensor,

For the co-variant differentiation of tensors of the second and higher ranks we may use the process by which (75) was deduced. Let, for example, $$(A_{\sigma\tau}$$ be a co-variant tensor of the second rank. Then $$A_{\sigma\tau}E^\sigma F^\tau$$ is a scalar, if $$E$$ and $$F$$ are vectors. This expression must not be changed by the $$\delta$$-displacement; expressing this by a formula, we get, using (67), $$\delta A_{\sigma\tau}$$ whence we get the desired co-variant derivative,

In order that the general law of co-variant differentiation of tensors may be clearly seen, we shall write down two co-variant derivatives deduced in an analogous way:


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