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On page 71 of The Absolute Differential Calculus by Levi-Civita, a very clear definition of a tensor is given in terms of how the coefficients of a multi-linear form transform, such that the product with its dual multi-linear form remains invariant:

An m-fold covariant is an m-fold system which is transformed in the same way as the coefficients of a multilinear form in point variables; an m-fold contravariant is one which is transformed in the same way as the coefficients of a multilinear form in dual variables; more generally, a mixed system or tensor is one which is transformed in the same way as the coefficients of a multilinear form in both point and dual variables (including also as particular cases both purely covariant and purely contravariant systems).

Were there other related definitions defined around this time?

How has today's definition of a tensor changed compared to its beginnings?

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    $\begingroup$ I think at some point physicists and mathematicians went different ways on this. Physicists still prefer to define things by their transformation properties. I think mathematicians normally prefer coordinate-independent descriptions of this kind of thing, so they wouldn't define a tensor in terms of how its coefficients transform. $\endgroup$
    – user466
    Jun 17, 2015 at 3:32
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    $\begingroup$ It isn't clear to me that "today's definition of a tensor" exists uniquely, even if we restrict to pure mathematics. For instance, while the definition one learns in elementary linear algebra is in terms of bilinear maps, I've known mathematicians who would rather define it based on, for example, tensor-hom adjunction. $\endgroup$
    – Logan M
    Jun 17, 2015 at 8:43
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    $\begingroup$ Not around that time, invariant approach began with Cartan in 1920s, but invariant definition in terms of multilinear forms on Cartesian products of copies of a space and its duals is equivalent to the transformational definition, which refers to expressions of such forms in coordinates, see hsm.stackexchange.com/questions/599/… $\endgroup$
    – Conifold
    Jun 17, 2015 at 22:55
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    $\begingroup$ @Conifold Are you sure it wasn't Levi-Civita who motivated the subject via invariance of the product of multi-linear forms with their duals first? Certainly in the chapter I've taken the quote from, it makes clear from the start the idea of the invariance of energy to coordinate transformations and extends this to the product of multi-linear forms and their duals in general. Secondly, Einstein gives a brief overview of tensors in his 1915 paper, and defines covariant tensors via the invariance of their product with the corresponding contravariant tensor. $\endgroup$
    – Larry Harson
    Jun 20, 2015 at 23:00
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    $\begingroup$ The beginnings of $m$-fold covariance and contravariance go back at least as far as Christoffel, for example "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", 1869. The Levi-Civita book which you mention was the 1926 translation of his 1925 book, based on ideas in the famous Ricci and Levi-Civita article in 1900 on absolute differential calculus. The tensorial ideas were already clear in Bianchi's works, which rested on Christoffel's papers. $\endgroup$
    – Alan U. Kennington
    Jun 28, 2015 at 7:46

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Were there other related definitions defined around this time?

In physics textbooks, vectors are introduced first, and then vector fields; given the complexity of the notion of tensors, one would suppose tensors would then be introduced and then tensor fields. Instead, a quite complex and obscure definition of a tensor field (and not a tensor) is presented without enough motivation to the curious - this is the definition you have more or less presented.

That the definition is obscure has been understood, with various physicists, such as Dirac, in his short book on GR, noticing the explanatory lacuna and who suggests an 'affine' description of a tensor would be worth looking at before moving to the notion of a tensor in general; but he doesn't attempt a description of an 'affine' tensor. Weinberg, in his book Gravitation and Cosmology, does make an attempt at it but the result is not much more clear than the original definition.

Moreover, one would suppose that Einstein himself, given the importance of tensors in his theory of gravity, would make an attempt to explain the notion in his book with Infeld, The Evolution of Physics; but whilst they pays careful attention to the notion of a vector - they, unfortunately, they do not discuss the notion of a tensor. This is simply, as I've already noted, that the actual definition used by physicists, isn't intuitive (in fact, it is, when thought through but nevertheless, it is long-winded and relies on an operational definition rather than an intrinsic definition).

The main alternative presentation of tensors is most often used by mathematicians rather than physicists, however, this tends to be increasingly not the case with more advanced work. It's presented, for example, in MacLanes book, Categories for the Working Mathematician (first published in 1971) by what is known as universal properties.

Given modules $A$ and $B$ over a commutative ring $K$, a tensor product is a universal element of the set $\text{Bilin}(A,B)$ of bilinear functions $\beta:A \times B \rightarrow C$ to some third $K$-module $C$. This set is (the object function of) a functor of $C$. To get a solution set for given $A$ and $B$, it suffices to consider only those bilinear $\beta$ which span $C$ (do not factor through a proper submodule of $C$). Then $C$ consists of all finite sums $\sum(\beta(a_i,b_i))$ ... [so] a tensor product $\otimes : A \times B \rightarrow A \otimes B$ exists. The usual (more explicit) construction is wholly needless, since all the properties of the tensor product follow directly from the universality.

He notes, that although Bourbaki, in principle, came close to this definition, but given that they didn't have the notion of a universal property to hand - they missed it.

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