Abstract
The properties of vectors and tensors derive essentially from two facts: (1) the linearity of the transformation relating representative matrices in two coordinate systems and (2) the transitivity property for successive coordinate transformations. Relative tensors are convenient to use in discussing surface and volume elements, the alternating tensor and vector products, and the invariant differential operators curl and div. The matrix notation is convenient for defining tensors of first and second rank; for tensors of third and higher rank, however, it is easier to revert to the more conventional component notation. A single contraction in a tensor product yields a tensor of a rank that is lower by two than the rank of the original product. There are various theorems, known as quotient theorems, which in a sense are converse theorems to the contraction theorems and are sometimes useful for the proof as per which a given array of components associated with an arbitrary but definite coordinate system represents a tensor.
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