> On the other hand, the abstract formulation of the physical laws does not need such details, but only the high-level definitions using multi-linear functions,

> On the other hand, the abstract formulation of the physical laws does not need such details, but only the high-level definitions using multi-linear functions, and it is independent of any choice for the coordinate system. This is exactly the point. Abstract physical laws must be invariant to coordinate transformation. From a pedagogical point of view, perhaps this is less important when discussing anisotropic media, but critical when discussing general relativity. Hence, the first reason why many physicist book writer think it very important that covariance/contravariance of tensors be central to both their definition and their pedagogy as applied to physics. You have to convince the student that tensors are the right mathematical objects to describe reality because they preserve this invariance. The second reason is just as important. Physics is nothing without validating abstract physical laws by experiment. And that validation can not be done without computing predictions. Which in turn will require the right coordinate system, which will require covariance/contravariance of tensors. You can't just disregard these computations as unimportant or unnecessary from either a pedagogical point of view or a deeper philosophical one.