Page:The Meaning of Relativity - Albert Einstein (1922).djvu/27

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PRE-RELATIVITY PHYSICS
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of rank we may obtain a tensor of rank by multiplying all the components of the first tensor by all the components of the second tensor:

(10)

Contraction. A tensor of rank may be obtained from one of rank by putting two definite indices equal to each other and then summing for this single index:

(11)

The proof is

In addition to these elementary rules of operation there is also the formation of tensors by differentiation ("erweiterung"):

(12)

New tensors, in respect to linear orthogonal transformations, may be formed from tensors according to these rules of operation.

Symmetrical Properties of Tensors. Tensors are called symmetrical or skew-symmetrical in respect to two of their indices, and , if both the components which result from interchanging the indices and are equal to each other or equal with opposite signs.

Condition for symmetry : .
Condition for skew-symmetry : .

Theorem. The character of symmetry or skew-symmetry exists independently of the choice of co-ordinates, and in