Vector fields in
A vector field on some open set is a function which associates a vector to each point of . That is, is a function from to . However, we give it a rather special interpretation. This distinction will become clear when we generalize. If is differentiable, then we say the vector field is differentiable.
Vector fields on manifolds
Suppose first that is a manifold of dimension embedded in . Then we would like to define a vector field on to be a function as before. But has a natural notion of tangent space at each point, so now we would like all the vectors to be tangent to . If we are to define a vector field as before, as simply a function from some open set of to , we must pick a basis for the tangent space at each point; the basis elements must, however, be differentiable. It is not obvious that we can always pick such a differentiable basis (think of the tangent spaces to the Möbius strip). The problem is that the tangent spaces form a fiber bundle, and this may not be trivial. We could get around this by shrinking until we could always construct such a basis, but then we would have to describe how to convert bases on the overlap. This can be done, and it is one way to approach the theory of differential manifolds; at this point one might as well do away with the ambient space .
Instead of doing this, we will take a coordinate-free approach. The first thing to notice is that given a tangent vector to a manifold, it makes sense to take a differentiable function on the manifold and ask what the directional derivative along the vector is. If we have two different tangent vectors, then we can find a function whose directional derivative along each vector is different. So we could identify tangent vectors with directional derivative operators. This is how we will define them in a general setting.
Let be a differential manifold of dimension , and let be a point on . Let be the space of differentiable functions defined in some neighborhood of . Then a tangent vector at is a linear operator on such that
if and agree on some neighborhood of , then .
We write the set of tangent vectors at as ; it is a vector space of dimension .
A vector field on an open set is a family of tangent vectors at each such that for every differentiable function on an open set , the function is differentiable.
What does this definition actually mean? Suppose we have a coordinate chart on some open set . Then we can define a differentiable function on by applying followed by extracting the th coordinate. So the function really extracts the coordinate in this coordinate system. Now, let be a vector field on . With some thought, we see that
or, in some sense
Each is by definition just a differentiable function on .
If we had chosen a different coordinate chart on an open set , we would obtain coordinate functions by analogy with the . Then in this coordinate system we would have
If and overlap, then on their overlap we can compare the components of in these two coordinate systems. A calculation will reveal
Observe that this transformation law means that we can’t compare vectors at different points in a coordinate-independent way, or at least, it will require some significant cleverness to transport a vector from one point to another. This is in fact possible with some extra information in the form of a connection on , allowing parallel transport of vectors along curves. The result will continue to depend on the curve, as you can imagine if you imagine trying to parallel-transport a vector from the North Pole to Baghdad to Mexico City and then back to the North Pole: it will have rotated. This is in fact a direct result of the curvature of the Earth.
Let be , and let
Then define a vector field by
We have a natural coordinate patch defined by the identity function; in this coordinate system, we can easily calculate that
this is precisely the vector field we had before, viewed in a new and more confusing light. Now, with a little imagination, we can see that even for fixed , the function is a different kind of object than ; while represents a rotation, a smooth map from to itself, while is a piece of information attached in an essential way to each point, perhaps representing a velocity vector field.
The Tangent Bundle
The tangent bundle is extremely useful in its own right, and other bundles of interest are also constructed from it. For example, the cotangent bundle is obtained by taking the dual vector space at each point. Sections of the cotangent bundle are one-forms, and since they are obtained by taking the dual, they transform according to the inverse matrix at each point. Higher wedge and tensor products of these bundles are used to construct tensors and differential forms.
Vector fields on a smooth manifold support an operation called the Lie bracket, making them into a Lie algebra; this construction produces an intimate link between Lie algebras and Lie groups, which are of great interest to physicists and mathematicians alike.
|Date of creation||2013-03-22 11:59:27|
|Last modified on||2013-03-22 11:59:27|
|Last modified by||mathcam (2727)|