Page:The Meaning of Relativity - Albert Einstein (1922).djvu/84

form of the equations which are co-variant with respect to arbitrary point transformations. The generalized calculus of tensors was developed by mathematicians long before the theory of relativity. Riemann first extended Gauss's train of thought to continua of any number of dimensions; with prophetic vision he saw the physical meaning of this generalization of Euclid's geometry. Then followed the development of the theory in the form of the calculus of tensors, particularly by Ricci and Levi-Civita. This is the place for a brief presentation of the most important mathematical concepts and operations of this calculus of tensors.

We designate four quantities, which are defined as functions of the $x_{\nu }$ with respect to every system of co-ordinates, as components, $A^{\nu }$, of a contra-variant vector, if they transform in a change of co-ordinates as the co-ordinate differentials $dx_{\nu }$. We therefore have

${A^{\mu }}'={\frac {\delta x_{\mu }'}{\delta x_{\nu }}}A^{\nu }$

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Besides these contra-variant vectors, there are also covariant vectors. If $B_{\nu }$ are the components of a co-variant vector, these vectors are transformed according to the rule

$B_{\mu }'={\frac {\delta x_{\nu }}{\delta x_{\mu }}}B_{\nu }.$

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The definition of a co-variant vector is chosen in such a way that a co-variant vector and a contra-variant vector together form a scalar according to the scheme,

$\phi =B_{\nu }A^{\nu }{\text{ (summed over the }}\nu {\text{)}}.$