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Generalizing this, in the theory of differentiable manifolds, a vector (well, a "tangent vector") can be thought of as being an arrow rooted at a specific point

by kmill 5y ago

Generalizing this, in the theory of differentiable manifolds, a vector (well, a "tangent vector") can be thought of as being an arrow rooted at a specific point. In Euclidean space (affine spaces), you can always translate all vectors to an origin, so it's safe to ignore a vector's root, but in general there's not a canonical way to translate vectors. The issue is essentially curvature -- for example, if you took a tangent vector at the north pole of a sphere, dragged it down to the equator, dragged it along the equator for a bit, then dragged it back up to the north pole, the vector would have rotated. What this shows is that vectors at different points are not mutually comparable without more structure. I swept a detail under the rug, which is that the vectors are infinitesimal arrows, so their tip is not actually another point of the manifold. In an affine space, vectors can be regarded as non-infinitesimal arrows -- the arrows that you always find textbook illustrations on vector arithmetic. The arrows with a common root form a vector space.

gspr 5y ago
Right, but the individual vector spaces you speak of there are the tangent spaces of a manifold M. There's a vector space TpM attached to M at a point p. And the point you make is that there isn't necessarily a canonical way to relate vectors in TpM to vectors in TqM for p≠q. In this sense, we may well say that "a vector in TpM is anchored at p, and a vector in TqM is anchored at Q". I agree with that, obviously. However, within a fixed vector space, of which such a TpM is merely one example, there is no notion of vectors being "anchored".
- kmill 5y ago
  Given the definitions in a textbook on smooth manifolds, I agree with you -- the modern notion of a vector space has no notion of root or anchor since these words have no associated definitions. It's interesting to think about how we came to the modern viewpoint. Some of the underlying context behind my comment is that long ago manifolds were concretely subsets of R^n that could be locally parameterized. The tangent space at a point was the affine subspace of R^n that was tangent to the manifold there, and you could regard a tangent vector as an arrow rooted at that point inside that subspace. Even though we have our clean modern notion of an abstract vector space, somehow these old intuitions linger on in other disciplines In math we have a precise definition for a vector (an element of a vector space, full stop), but I don't think experts have the authority to be prescriptive about jargon outside their discipline. Of course, a student should be willing to drop preconceptions and to absorb correct usage.

by kmill 5y ago

gspr 5y ago

Right, but the individual vector spaces you speak of there are the tangent spaces of a manifold M. There's a vector space TpM attached to M at a point p. And the point you make is that there isn't necessarily a canonical way to relate vectors in TpM to vectors in TqM for p≠q. In this sense, we may well say that "a vector in TpM is anchored at p, and a vector in TqM is anchored at Q". I agree with that, obviously. However, within a fixed vector space, of which such a TpM is merely one example, there is no notion of vectors being "anchored".

kmill 5y ago
Given the definitions in a textbook on smooth manifolds, I agree with you -- the modern notion of a vector space has no notion of root or anchor since these words have no associated definitions. It's interesting to think about how we came to the modern viewpoint. Some of the underlying context behind my comment is that long ago manifolds were concretely subsets of R^n that could be locally parameterized. The tangent space at a point was the affine subspace of R^n that was tangent to the manifold there, and you could regard a tangent vector as an arrow rooted at that point inside that subspace. Even though we have our clean modern notion of an abstract vector space, somehow these old intuitions linger on in other disciplines In math we have a precise definition for a vector (an element of a vector space, full stop), but I don't think experts have the authority to be prescriptive about jargon outside their discipline. Of course, a student should be willing to drop preconceptions and to absorb correct usage.