[MathMonkeyMan]

The joke I learned in a Physics course is "a vector is something that transforms like a vector," and "a tensor is something that transforms like a tensor." It's true, though.

The physicist's tensor is a matrix of functions of coordinates that transform in a prescribed way when the coordinates are transformed. It's a particular application of the chain rule from calculus.

I don't know why the word "tensor" is used in other contexts. Google says that the etymology of the word is:

> early 18th century: modern Latin, from Latin tendere ‘to stretch’.

So maybe the different senses of the word share the analogy of scaling matrices.

[ogogmad]

The mathematical definition is 99% equivalent to the physical one. I find that the physical one helps to motivate the mathematical one by illustrating the numerical difference between the basis-change transformation for (1,0)- and (0,1)-tensors. The mathematical one is then simpler and more conceptual once you've understood that motivation. The concept of a tensor really belongs to linear algebra, but occurs mostly in differential geometry.

There is still a "1% difference" in meaning though. This difference allows a physicist to say "the Christoffel symbols are not a tensor", while a mathematician would say this is a conflation of terms.

TensorFlow's terminology is based on the rule of thumb that a "vector" is really a 1D array (think column vector), a "matrix" is really a 2D array, and a "tensor" is then an nD array. That's it. This is offensive to physicists especially, but ¯\_(ツ)_/¯

[bjourne]

The problem with the physicist's definition is that the larger the N the less the geometrical interpretation makes sense. For 1, 2, and even 3-dimensional tensors there is some connection to geometry, but eventually it loses all meaning. Physicist has to give up and "admit" that an N-dimensional tensor really just is a collection of N-1-dimensional tensors.

[Koshkin]

> something that transforms

Well, they don't, it is their components that do (under a change of the coordinate system).