What is the history of the term "tensor"?

Question

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")?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

A part of the tensor history must come from tenses (past present future) and how Aristotle defined time as the measure of change / motion / movement. So really descriptions of changes of the state (or unchanging stillness) whether static or dynamic.

EDIT:

Tensor:= Tense + or (REF-1, includes William Rowan Hamilton algebraic origin)

Tense:= Any of the forms of a verb which distinguish when an action or state of being occurs or exists.

Or:= Creates an agent noun

REF-1 https://en.m.wiktionary.org/wiki/tensor

“Borrowed from New Latin tensor (“that which stretches”), equivalent to tense +‎ -or. Anatomical sense from 1704. Introduced in the 1840s by William Rowan Hamilton as an algebraic quantity unrelated to the modern notion of tensor. The contemporary mathematical meaning was introduced (as German Tensor) by Woldemar Voigt (1898)[1] and adopted in English from 1915 (in the context of general relativity), obscuring the earlier Hamiltonian sense. The mathematical object is so named because an early application of tensors was the study of materials stretching under tension. (See, for example, Cauchy stress tensor on Wikipedia.)”