TITLE: Question about the physical intuition behind tensors
QUESTION [3 upvotes]: I would like to check something with you. I am a beginner in differential geometry and in general relativity so I may say wrong things.
This is what I understood from tensors.
Tensors are object for which coordinates follows a specific transformation when a change of basis is done.
If I define a tensor on a basis $A$, I automatically know its components on a basis $B$.
The philosophy behind the object is to be able to define quantities that are "absolute" and don't depend on coordinates.
For example $ds^2=dx^2+dy^2$ is also written in polar coordinates $ds^2=dr^2+r^2 d\theta^2$ but it represents physically the same distance : if I compute the scalar product between two vectors using either the first or the second basis, the result will be the same. 
So finally : can I say that tensors are mathematical object that are used to describe phenomenon that are absolute and don't depend on a given set of basis. For example if I take a n-contravariant and p-covariant tensor and I apply it on n vectors and p covectors the final result will not depend of any choice of basis.
Also : why do we only define tensors as object that transforms as $m'^i_j=m^k_l \frac{\partial x^l}{\partial y^j}\frac{\partial y^i}{\partial x^k}$ and not more general transformations ?
Is it because the tensors are constructed around the notion of vectors/covectors and we want that tensors applied on vector and covector give result independant of the coordinates. So all the tensors were constructed to ensure that applied on vector and covectors they give result indepent of coordinates. 
Indeed, we could imagine to create other objects that ensure basis-independant results with other object than vectors but it is just that we don't need them in practice (in general relativity we would'nt need them for example).
My last question is juste to be sure that I understand well... or not.

REPLY [2 votes]: You are essentially right. Your basic intuition is correct.
Think of tensors as objects constructed from vectors and covectors using the tensor product. It becomes clear that they have a meaning, in some sense, independent of the local coordinates that you choose. This is similar to having a vector in a vector space. If you choose a basis, you can then describe the vector using coordinates. If you choose another basis, then you can easily deduce the coordinates in that basis from the previous coordinates and the change of basis matrix. In some sense, the transformation laws are nothing but a curved version of that. And the vector has an existence, in some sense independent of the local coordinates we choose to describe it (getting a little philosophical).
Finally, you are right that there are other objects that can be described using other transformation laws (such as spinor fields, or connections/gauge fields for instance and so on). The tensor transformation laws are not the only transformation laws of interest.
Personally, I like to think of vector spaces abstractly, and then bases and coordinates later on. It makes the thoughts much clearer in some sense. Similarly, I like to think of tensors as smooth sections of the tensor product of a number of copies of the tangent bundle tensored with the tensor product of a number of copies of the cotangent bundle. This is how I think of them. But to prove things with them, you often need to compute something: here you either use covariant derivatives and Lie derivatives etc. (the more preferred method I guess nowadays, because it is a coordinate-free formalism), or you use local coordinates (tensors etc) or you use local orthonormal frames or coframes (let's call that the Cartan-Frenet method, if no one minds; this method makes the underlying Lie algebra more manifest), or if they apply, formulas for embeddings (Gauss-Codazzi etc.) and submersions (O'Neill etc.), or some more indirect method.
There is a lot to say about Differential Geometry. Enjoy learning it. It is a very nice subject.
Edit: to use a more physical language, tensors are fields on spacetime which vary in some sense in a way compatible to how the spacetime curves, much like, say, a tangent vector field varies on spacetime, or, say, a 1-form field varies on spacetime. This is the intuitive meaning of the transformation laws, and explains why tensors are so relevant to the geometry of spacetime. Examples of tensors include the Riemann curvature tensor, for instance, and the Ricci tensor, which is a contraction of the Riemann curvature tensor. The metric $g$ itself is a tensor. A connection is not a tensor, but the difference of two connections on a manifold is a tensor.
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