Is there an intuitive reason for why tensors are so ubiquitous in physics? As a beginner, I'm able to see where different individual tensors come from in physics, but I'm trying to generate some intuition for why this object - defined by a fairly specific transformation law - is so widely applicable.
Here's one argument that I have - but not sure if this even makes sense:

*

*We'd like to define groups of physical quantities where the whole is invariant in some way under change of coordinates.

*Suppose that $ [ u_0, u_1, ..., u_n] $ transforms into $ [v_0, v_1, ..., v_n] $ after some coordinate transformation. One somewhat generic way to write the transformation law is $ v_i = f_i(u_0, ..., u_n, t_0, ..., t_n) $ where $ t $ is some factor that depends on the specifics of the coordinate transformation.

*If we write out the taylor series expansion of the above, we'll get terms that have quadratic or higher powers in $u_n$. Key: these should all be discarded based on dimensional analysis.

*This just leaves only the linear terms: $ f_i(u_0, ..., u_n, t_0, ..., t_n) = \sum_{n} {t_n}{u_n} $, which is the same as the tensor transformation law.

Is it appropriate to use dimensional analysis here in the context of taylor expansions? And overall - is there a more intuitive reason for the ubiquity of tensors in physics?
 A: I believe tensors appear in physics because they are representations of the existing groups of symmetries of space(time), such as Lorentz group or rotation group.
A: The reason why tensors appear so frequently in physics is really to do with the underlying mathematical framework being used. Typically, physical theories will be built on special types of spaces called manifolds. There are many different mathematical objects that can be constructed on a manifold, such as: functions, vectors, dual vectors, forms and of course tensors. In general, tensors have a very precise mathematical definition, but in local coordinates you can show that tensors have a uniquely defining transformation law under a change of coordinates. Tensors encapsulate in their definition many different mathematical objects, denoted by their rank.
I'm not sure whether this gives too much intuition about tensors themselves, but I would say that it is useful, in such circumstances to turn to a mathematical explanation. Perhaps try studying some differential geometry with applications to physics. I think that would give you some of the answers you are looking for.
