Consider $M$ a smooth manifold of dimension $n$, then a vector at the point $a\in M$ can be defined without any reference to any coordinate system. Indeed, we define a vector $v\in T_aM $ usually as
- An equivalence class of curves
- A derivation on the ring of germs of smooth functions at $a$
Both definitions are independent of the coordinate system used. Because of that, any tensor of type $(k,l)$ is also independent of the coordinates used. They are intrinsic to the manifold. Following this idea, any pointwise association of those objects, i.e. vector fields and tensor fields, also do not depend on coordinates. Obviously real valued functions don't depend on coordinates also.
That said, I would realy like to know what is the true mathematically rigorous formulation of invariance. In Relativity physicists resort a lot to the notion of invariance, even in the Principle of Relativity we see it:
The laws of Physics are invariant on every inertial frame of reference
Now, $F = ma$ is an equality between vectors, but vectors are intrinsic objects independent of coordinates, so that this equality will be true no matter how do we choose to represent $F$ and $a$. Since real valued functions, vectors and tensors automatically do not depend on coordinates by definition, why anyone would even care with invariance? I mean, that would make that principle be empty. So invariance is probably defined in a somewhat different manner.
That said, what is the true mathematically rigorous definition of invariance that works for both classical mechanics and special/general relativity?
One way of saying that an operation on smooth manifolds, such as the operation of considering its tangent bundle or its space of differential forms, is really invariant under change of coordinates is to say that it's functorial with respect to smooth maps. This implies, but is strictly stronger than, saying that it behaves nicely under diffeomorphisms, which I think is something like what what invariance means in general relativity.
In some sense, in mathematics the trick is that if you never work in coordinates then you aren't even allowed to say things that aren't invariant, so you get invariance for free. Once you introduce coordinates, you've made an extra choice, and now you might have to worry about how what you're saying changes when you change that extra choice.