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:
- Just what is a tensor anyway?
- Who first used the term, and in what context?
- Has the meaning of the term changed and evolved over time to become more general?
- Has its definition also become more precise (like, for example, that of a 'continuous function")?