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In this lecture series a definition of a tensor is given roughly as follows:

(Definition 1):

[we] define tensors as variants that transform from one coordinate system to another by a very special rule. A variant $T_i$ is called a covariant tensor if its values $T_i$ and $T_{i'}$ in the coordinate systems $Z^i$ and $Z^{i'}$ are related by $$T_{i'}= T_i J^i_{i'}$$ [then a similar definition for a contravariant tensor]

How is this reconciled with the definition that seems to me to be much more general:

(Definition 2):

Given vector space $V$, a tensor is a multilinear map $V^* \times ... \times V^*\times V \times ... \times V \to \mathbf{R}$

It seems to me that the first definition is extremely restrictive compared to the second one. Aren't there other types of linear transformations that tensors can do (acc. to the second definition) that are not specifically transformations between coordinate systems? (even if you ignore the fact that the first definition is restrictive in terms of the dimensions of the tensor).

What explains this difference?

1 Answers 1

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You're right, the first one is more restrictive, since it's only the definition of a $(0,1)$-tensor (corresponding to a linear map $T\colon V \to \mathbf{R}$ in terms of the second definition).

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    Thank you. I didnt even think of that, but I was thinking also of another point. Namely the fact that the first definition specifically demands that a tensor has to transform between coordinate systems. Aren't there other types of linear transformations that tensors can do that are not specifically transformations between coordinate systems? – 2017-02-04
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    By the way, don't you mean a (1,1) tensor? – 2017-02-04
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    No, $T_i$ has no (0) upper indices and one (1) lower index, hence type (0,1). – 2017-02-04
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    Regarding your first question, I'm not entirely sure what you mean. But see here for more info about different ways of thinking about tensors: http://math.stackexchange.com/questions/10282/an-introduction-to-tensors – 2017-02-04