> The main problem with most introductions to the topic are that they deal with coordinate systems where the basis vectors are orthogonal, so the covariant and contravariant components are the same. You need to deal with non-orthogonal basis vectors, because then you realize that naturally there are two ways of defining basis vectors at a given point in space.
Or, preferably, not with basis vectors at all, but that notion seems to do violence to a physicist's way of thinking about linear algebra—I've never quite understood why the physical approach to the subject is so bound up in co-ordinates, when it seems like physicists would be one of the groups most likely to benefit from fully grokking a co-ordinate-free approach.
Because most physicists are ultimately interested in predicting the results of experiments with actual numbers. And that demands the use of coordinates.
> Because most physicists are ultimately interested in predicting the results of experiments with actual numbers. And that demands the use of coordinates.
Of course, at some point, you need numbers! But there's no reason that those numbers need to infest the whole computation; you can de-coordinatise as soon as possible and re-coordinatise as late as possible, and in between you not only can but must think in a coordinate-free fashion.
(I am a mathematician, not a physicist, and am not presuming to tell physicists how to do their job—this isn't an argument about whether they should do things this way. I'm just pointing out that they could, and that it seems not only possible but advantageous. But of course long-established knowledge of what actually works in physics beats this tyro's guess at what could.)
Or, preferably, not with basis vectors at all, but that notion seems to do violence to a physicist's way of thinking about linear algebra—I've never quite understood why the physical approach to the subject is so bound up in co-ordinates, when it seems like physicists would be one of the groups most likely to benefit from fully grokking a co-ordinate-free approach.