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Since first stumbling across the word "tensor" over 50 years ago, in a book on analytical mechanics, I've been amazed at how slippery a concept it is. Nor have I ever seen a definition which gives me a clear intuition as to what kind of thing a tensor is (tho' I've read a lot of articles - especially on Wikipedia - involving the term). For comparison, a "vector" has the intuitive meaning of a carrier, and a "scalar" seems to mean a stretcher. Along those lines, I thought that perhaps "tensor" had to do with "tension"; but since tension is a term for a kind of force, which a vector already represents well, why would we need yet another term?

The same book on analytical mechanics also introduced me to the idea of the Hamiltonian, along with formulae defining it. On doing some research in the local university library (no Internet then!), I discovered Hamilton's original work on quaternions, a very exciting generalisation of complex numbers. Reading that work helped me understand the analytical mechanics book better, as it also used three mutually orthogonal unit vectors: i, j, k, just as with quaternions. Disappointingly however, Hamilton didn't use the word "tensor" there. [Edit: I'd forgotten he used "tensor" to name an analogue to the modulus of a complex number c = x+yi (a scalar measuring the length of a vector, or "size" of a hypercomplex number h = w+xi+yj+zk.)]

Altho' I've discovered that a tensor can take the form of a matrix, it seems that this isn't essential. But what is?

These are the things I still don't get:

  1. Just what is a tensor anyway?
  2. Who first used the term, and in what context?
  3. Has the meaning of the term changed and evolved over time to become more general?
  4. Has its definition also become more precise (like, for example, that of a 'continuous function")?
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    $\begingroup$ I am not posting an answer because I don't know the history, but "what is a tensor" has three or four different answers depending on the context, similar to "vector" meaning different things (list of numbers, element of vector space, vector but not pseudovector in a physics context) but perhaps even more thorny. $\endgroup$ – Mark S. Nov 25 '16 at 19:43
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    $\begingroup$ tensor - In new latin tensor means "that which stretches". The mathematical object is so named because an early application of tensors was the study of materials stretching under tension. If you separate a solid into two parts by a surface with normal vector $n$, the action that one piece applied to the other is described by a vector $T$ (the stress vector), $T$ is not always in the direction of $n$, to describe the relation, you need a $3 \times 3$ matrix and hence the born of stress tensor $\endgroup$ – achille hui Nov 25 '16 at 19:50
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    $\begingroup$ For the history, see Tensor : "was one of the family of terms introduced by William Rowan Hamilton (1805-1865) in his study of QUATERNIONS. VECTOR and SCALAR and VERSOR were among the others. The tensor is for quaternions what the MODULUS is for complex numbers. The term derives from the Latin tendĕre to stretch." $\endgroup$ – Mauro ALLEGRANZA Nov 25 '16 at 20:02
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    $\begingroup$ From Mauro's link, see the very next entry: "TENSOR, TENSOR ANALYSIS, TENSOR CALCULUS, etc. are 20th century terms associated with the ABSOLUTE DIFFERENTIAL CALCULUS developed by Ricci-Curbastroin the 1880s and -90s on the basis of earlier work by Riemann, Christoffel, Bianchi and others. ... [The word] tensor is due to the well-known Goettingen physicist Woldemar Voigt (1850-1919), who used it in his Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung of 1898 (OED and Julio González Cabillón)." $\endgroup$ – Rahul Dec 20 '16 at 15:50
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    $\begingroup$ Also, tensor does indeed have to do with tension: the stress tensor describes, in a sense, the distribution of tension in different directions in a solid body. $\endgroup$ – Rahul Dec 20 '16 at 16:08
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Look at the bottom of page 2 and top of page 3 of Conrad's "Tensor Products" paper for a discussion of the early usage of the term "tensor" in physics and mathematics.

Here is a brief historical review of tensor products. They first arose in the late 19th century, in both physics and mathematics. In 1884, Gibbs [4, Chap. 3] introduced the tensor product of vectors in $\mathbf R^3$ under the label "indeterminate product"$^*$ and applied it to the study of strain on a body. Gibbs extended the indeterminante product to $n$ dimensions two years later [5]. Voigt used tensors for a description of stress and strain on crystals in 1898 [14], and the term tensor first appeared with its modern meaning in his work.$^\dagger$ Tensor comes from the Latin tendere, which means "to stretch." In mathematics, Ricci applied tensors to differential geometry during the 1880s and 1890s. A paper from 1901 [12] that Ricci wrote with Levi-Civita (it is available in English translation as [8]) was crucial in Einstein's work on general relativity, and the widespread adoption of the term "tensor" in physics and mathematics comes from Einstein's usage; Ricci and Levi-Civita referred to tensors by the bland name "systems." In all of this work, tensor products were built out of vector spaces. The first step in extending tensor products to modules is due to Hassler Whitney [16], who defined $A \otimes_{\mathbf Z} B$ for any abelian groups $A$ and $B$ in 1938. A few years later Bourbaki's volume on algebra contained a definition of tensor products of modules in the form that it has essentially had ever since (within pure mathematics).

$^*$The label indeterminate was chosen because Gibbs considered this product to be, in his words, "the most general form of product of two vectors," as it was subject to no laws except bilinearity, which must be satisfied by any operation on vectors that deserves to be called a product.

$^\dagger$Writing $\mathbf i$, $\mathbf j$, and $\mathbf k$ for the standard basis of $\mathbf R^3$, Gibbs called any sum $a\mathbf i\otimes\mathbf i + b\mathbf j\otimes\mathbf j + c\mathbf k\otimes\mathbf k$ with positive $a$, $b$, and $c$ a right tensor [4, p. 57], but I don't know if this had any influence on Voigt's terminology.

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  • $\begingroup$ Conrad writes: "In 1884, Gibbs [4, Chap. 3] introduced the tensor product of vectors in R3 under the label indeterminate product and applied it to the study of strain on a body ... The label indeterminate was chosen because Gibbs considered this product to be, in his words, the most general form of product of two vectors, as it was subject to no laws except bilinearity, which must be satisfied by any operation on vectors that deserves to be called a product." So, "the most general form of a product of two vectors" would thus be the most basic (modern) meaning of the term tensor? $\endgroup$ – Yahya Abdal-Aziz Dec 20 '16 at 15:18
  • $\begingroup$ "Voigt used tensors for a description of stress and strain on crystals in 1898 ... the term tensor first appeared with its modern meaning in his work. Tensor comes from the Latin tendere, which means to stretch. In mathematics, Ricci applied tensors to differential geometry during the 1880s and 1890s. A paper from 1901 that Ricci wrote with Levi-Civita was crucial in Einstein’s work on general relativity, and the widespread adoption of the term tensor ... comes from Einstein’s usage; Ricci and Levi-Civita [called] tensors systems." So why then did Einstein call them tensors? $\endgroup$ – Yahya Abdal-Aziz Dec 20 '16 at 15:27
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    $\begingroup$ It is somewhat weird that Cauchy i snot mentioned in that text. He used the very first tensor when he was working in what is now called the Cauchy stress theorem, a result that describes the stress caused by a tension in materials. This was some 30 years before Gibbs. $\endgroup$ – Mariano Suárez-Álvarez Dec 20 '16 at 16:11
  • $\begingroup$ @MarianoSuárez-Álvarez, replying to you a year later, I have included references to Cauchy and Riemann in that paragraph now. $\endgroup$ – KCd Dec 20 '17 at 2:03
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The concrete coordinate notion is the tensor as a multidimensional array of numbers. The dimension of the array is known as the rank of the tensor. So a scalar is a zero rank tensor, a list of numbers aka vector is a rank 1 tensor, a 2 dimensional grid aka matrix is a rank 2 tensor, and higher rank things are just called tensors. Writing the tensor in terms of indexed components, the rank tells you the number of indices required.

A more powerful but more abstract definition of tensor is as an element of a tensor product. Since the dimension of a tensor product of vector spaces is the product of the dimensions of the spaces, these vectors can also be naturally organized as arrays once you choose bases, showing the equivalence of the two notions.

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  • $\begingroup$ Thanks (belatedly!) for this: it jibes well with my earlier understanding, as being the next thing after a matrix in the sequence: scalar, vector, matrix, tensor, …. Also, your more abstract definition: an element of a tensor product [of vector spaces] clearly makes it an element of a multidimensional space, without restricting the dimension to any particular value[s]. These two concepts - the element and the space - are very helpful in knowing what people might mean when using the word "tensor". So thanks again :-). $\endgroup$ – Yahya Abdal-Aziz Nov 13 '17 at 9:26
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My lecturer at university defined a tensor as anything which turns a vector into another vector, and quoted this as the Tensor Detection Theorem. You might like to try and google this phrase. I always understood that it was first used by Levi Civita, but I may be wrong.

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    $\begingroup$ That seems to be the least correct definition of a tensor I have ever seen. $\endgroup$ – Tobias Kildetoft Nov 25 '16 at 19:32
  • $\begingroup$ @TobiasKildetoft..wow! What's your definition, then? Perhaps I should go back to university and give the lecturer a good ticking off... $\endgroup$ – David Quinn Nov 25 '16 at 19:37
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    $\begingroup$ @tobias, actually it is a quite good one, and it was standard at some point. A vector is a rule that maps directions to scalars, tensors are rules that map directions to vectors. That is, in fact, how the very first tensor worth its name appeared: Cauchy stress tensor (from which the name tensor comes!) tells you what stress (a vector) is caused by a displacement in a material. The Cauchy stress theorem states that the rule describing this rule is tensorial, so that it is a tensor. $\endgroup$ – Mariano Suárez-Álvarez Nov 25 '16 at 19:50
  • $\begingroup$ Mariano Suárez-Álvarez, thanks! That's nice motivation for the term. Does this relate to Voigt's work on crystals (see above)? $\endgroup$ – Yahya Abdal-Aziz Dec 20 '16 at 15:48

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