Question: Given, the Euclidean plane PP\mathcal{P}, with the point O∈PO \in \mathcal{P} (called “the origin”).

1. Is a Cartesian coordinate system for this plane centered at OO just an orthonormal basis for the cotangent space of the Euclidean point at OO, i.e. an orthonormal basis for (TOP)∗(T_O \mathcal{P})^*?

2. Are vectors with tail at OO in the Euclidean plane P\mathcal{P} just elements of the tangent space at OO, i.e. v∈TOPv \in T_O \mathcal{P}? And is a vector ww with tail at point A∈PA \in \mathcal{P} just an element w∈TAPw \in T_A \mathcal{P}?

3. That any point can be given a unique coordinate is just a reflection of the fact that P≃(TOP)∗\mathcal{P} \simeq (T_O\mathcal{P})^* as vector spaces, and the fact that every point can be associated with a position vector is just a way of expressing the fact that P≃TOP\mathcal{P} \simeq T_O\mathcal{P} as vector spaces? And the fact that position vectors can be associated with coordinates is just a reflection of the fact that, being finite-dimensional vector spaces, TOP≃(TOP)∗?T_O\mathcal{P} \simeq (T_O\mathcal{P})^*?

Incredibly long-winded context:

I was thinking recently that coordinate systems and “physical vectors” are naturally dual — the coordinates of a space don’t change under a linear transformation, whereas “physical vectors” do change under linear transformations. In contrast, coordinates change under (aptly named) changes of coordinates, whereas “physical vectors” do not change under changes of coordinates.

In other words, it seems like it makes the most sense to define coordinate systems to be covariant tensor fields of order 1, i.e. one-forms. Then the purpose of always having and considering cotangent and tangent spaces makes sense, because the former is the coordinate system with origin at a given point, and the latter is the space of “physical vectors” anchored at the point.

Moreover, the notions of coordinate system for a vector space only starts to get messy exactly when the notion of dual space gets messy — in infinite dimensions. The inner product at a point on a Riemannian manifold, which generates associated notions of length and angle, is defined on the tangent space, whose objects probably not coincidentally are “physical vectors”, the objects for which notions of length and angle make the most sense (compared to points in a coordinate space). Likewise, notions of integration and differentiation seem to correspond most naturally to cotangent spaces, which seems to correspond to how calculus is defined in the simplest possible setting — from a space of points to another space of points. And vector fields live in tangent spaces, and vector fields are most easily thought of in terms of “physical vectors”, again suggesting that tangent spaces are “physical vectors” and that cotangent systems are just coordinate systems.

At least one other person seems to have come to this conclusion independently (or maybe I read that answer before and my subconscious just understood it now). This seems to be how physicists think of everything, and implicitly the viewpoint behind much of differential geometry, or at least this mental model seems to make what these experts say seem much clearer now. For instance, these youtube videos about tensor calculus suddenly make much more sense when one thinks of “the covariant basis” and the “coordinate system” as literally the exact same thing. Watching this video again, it also makes a lot more sense if one thinks “coordinates = covectors”.

The only reason why one might not want to identify coordinate systems with cotangent spaces, is that one is often taught, when first introduced to “physical vectors”, to identify them with the coordinates of the heads of their arrows. But that shouldn’t discourage us, because obviously any finite-dimensional vector space is isomorphic to its dual vector space (as vector spaces). The isomorphism is so simple that we can completely ignore that it exists in elementary situations, but then suddenly one moves to differential geometry and gets confused because the distinction was pushed under the rug previously.

This answer also seems to suggest the same thing — a vector space has no “intrinsic coordinate system” — because one needs to specify a basis for the dual space in order to do so, i.e. a vector space has no “intrinsic coordinate system” for the exact same reason that it doesn’t have an “intrinsic basis” — one needs to specify both a basis for the vector space, and one for the dual vector space, the latter which we call a “coordinate system”. And the manner in which coordinates for points are found — via orthogonal projections, i.e. inner products — is just application of covectors to vectors, or at least so it would seem to me.

Related questions:

Computationnal geometry: vector, basis, point and coordinate system?

How can vector functions define coordinate systems?

What is (fundamentally) a coordinate system ?

Linear algebra revisited: What do we do when we set a coordinate system?

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1 Answer

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Not entirely, but you can find a direct connection. A covector field inhabits the cotangent space, a vector field inhabits the tangent space and a coordinate system is a set of nn scalar fields, where nn is the dimension of the manifold. This statement is valid for any differentiable manifold, without the need to identify an origin.

If one has specifically a flat manifold and designates an origin, one can canonically identify (points of) the manifold with (vectors of) the tangent space at the origin. A covector assigns to each vector a scalar value. Through the mentioned identification, nn linearly independent covectors become equivalent to assigning to each point a set of nn coordinates, and transformations necessarily behave equivalently. This picture only holds for “straight” coordinate systems though: it cannot produce curvilinear coordinate systems, but it applies regardless of the angles involved. To keep one’s implicit presumptions to a minimum, it is best to avoid assuming metric concepts such as angle and distance (“Euclidean”).

It may be noted that all this depends on a list of restrictions: the manifold must be flat, an origin must be selected, the (co)tangent space at only the origin is considered, not a general (co)vector field; the coordinates must be linear, parallel and uniformly spaced. But if one thinks in terms of differential behaviour, all of these are valid “locally”, and the insight is a useful one. So I’d say ‘no’ to all your questions, but in the limit of the tangent space at any given point, they become ‘yes’.

Thank you for double-checking my sanity — I appreciate it! This is sort of what I thought — given a point pp on a manifold MM, both TpMT_pM and (TpM)∗(T_pM)^* “sit” at pp, so a basis for (TpM)∗(T_pM)^* can be used to define a coordinate system for TpMT_pM via the “scalar product”. Of course in general the only point of (TpM)∗(T_pM)^* or TpMT_pM that one can identify with a point of MM is pp, but as you pointed out, the further identifications I mentioned depend crucially on the space being flat — obviously of course the isomorphisms with the tangent and cotangent spaces would not otherwise hold.

– William

2 days ago

Also the definition of a coordinate system as a set of nn scalar fields is a useful one and one I hadn’t thought of/known before. And with that definition, if one defines the coordinate system via the action of a basis of covectors on the tangent space, it is still obvious why the covectors are covariant and the vectors are contravariant. I still have to think about this some more, but this definitely helps a lot to clarify my thoughts — again I appreciate it.

– William

2 days ago

I guess what I found confusing is that no one ever bothers to say that the coordinate system is generated by the action of the dual frame on the original frame. For instance, while this excerpts on Wikipedia both mention the original frame and the coordinate system for them, they don’t mention that the coordinate system is generated by a dual frame, thus leaving no intuition as to why the coordinates should not change when a linear transformation is applied to the frame. en.wikipedia.org/wiki/Orthonormal_frame, en.wikipedia.org/wiki/Affine_space#Affine_coordinates

– William

2 days ago

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I’m forever confused by things because “no one ever bothers to say”. Even mathematicians tend to produce texts that induce one to assume things that are not universally true (which is essentially your point), and physicists are much worse. A basis need not even be a holonomic basis; it can be any set of nn linearly independent vector fields, just as an example that catches many (a slight diversion from what you’re saying). In general, of course, to think of coordinates as the action of a dual basis does not work.

– qman

2 days ago

When I say coordinate system I always have in mind “linear” coordinate systems (i.e. affine coordinate systems centered at the origin) i.e. something which is compatible with vector space structure. I know that the definition of coordinate system can be generalized to curvilinear coordinate systems and others, but I don’t know a word that applies only to the specific situation I have in mind. (Maybe “frame” comes close? I honestly don’t know.) Anyway the specific example I mentioned is the Cartesian coordinate system of the plane, which is certainly not curvilinear.

– William

yesterday