this lies its importance. The proof follows from the equation defining tensors.
Special Tensors.
I. The quantities (4) are tensor components (fundamental tensor).
Proof. If in the right-hand side of the equation of transformation , we substitute for the quantities (which are equal to 1 or 0 according as or ), we get
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The justification for the last sign of equality becomes evident if one applies (4) to the inverse substitution (5).
II. There is a tensor skew-symmetrical with respect to all pairs of indices, whose rank is equal to the number of dimensions, , and whose components are equal to or according as is an even or odd permutation of 123 ...
The proof follows with the aid of the theorem proved above .
These few simple theorems form the apparatus from the theory of invariants for building the equations of pre-relativity physics and the theory of special relativity.
We have seen that in pre-relativity physics, in order to specify relations in space, a body of reference, or a space of reference, is required, and, in addition, a Cartesian system of co-ordinates. We can fuse both these concepts into a single one by thinking of a Cartesian system of co-ordinates as a cubical frame-work formed of rods each of unit length. The co-ordinates of the lattice points of