So yesterday we noted that the big conceptual problem with partial derivatives is that they’re highly dependent on a choice of basis. Before we generalize away from this, let’s note a few choices that we are going to make.
First of all, we’re going to assume our space comes with a positive-definite inner product, but it doesn’t really matter which one. We’re choosing a positive-definite form with signature instead of a form with some negative-definite or even degenerate portion — where we’d get s or s along the diagonal in an orthonormal basis — because we want every direction to behave the same as every other direction. More general signatures will come up when we talk about more general spaces. But we do want to be able to talk in terms of lengths and angles.
Now this doesn’t mean we’ve chosen a basis. We can choose one, but there’s a whole family of other equally valid choices related by orthogonal transformations. Ideally, we should define things which don’t depend on this choice at all. If we must make a choice in our definitions, the results should be independent of the choice. Often, this will amount to the existence of some action of the orthogonal group on our structure, and the invariance of the results under this action.
Definitions which don’t depend on the choice are related to what physicists mean when they say something is “manifestly coordinate-free”, since we don’t even have to mention coordinates to make our definitions. Those which depend on a choice, but are later shown to be independent of that choice are a lesser, but acceptable, alternative. Notice also that this avoidance of choices echoes the exact same motives when we preferred the language of linear transformations on vector spaces to the language of matrices acting on ordered tuples of numbers.
But, again, we have made a choice of some inner product. But this doesn’t matter, because all positive-definite inner products “look the same”, in the sense that if we pick an orthonormal basis for each of two distinct inner products, there’s going to be a general linear transformation which takes the one basis to the other, and which thus takes the one form to the other. That is, the forms are congruent. So as long as we have some inner product, any inner product, to talk about lengths and angles, and to translate between vectors and covectors, we’re fine.