Tagent vector and vector field difference

[weio]

Hi there

Can somebody please explain shortly the difference between a tangent vector and a vector field? I'm still new to differential geometry. I read couple of sources
that had mixed claims on which of them actually act on a given function f. so I'm kind of confused.

Much appreciated.

 

[Fredrik]

The tangent space T_pM of the manifold M at point p can be defined as the vector space spanned by the basis vectors

\frac{\partial}{\partial x^\mu}\bigg\lvert_p

where x is a coordinate system (a chart). (There is also a coordinate independent definition, but it will result in the same quantities being called tangent vectors, so it's equvalent to this one).

Let F denote the set of smooth (C^\infty) functions from M into R (the set of real numbers). The vectors in the tangent spaces are linear functions from F into R

In the physics literature, a vector field is sometimes defined as a function that takes each point p in a subset U of M to a vector in the tangent space at p. In the mathematics literature the definition is a little more complicated, but it's still pretty close to the sloppy physicist's definition.

I'm not 100% sure that I remember the mathematical definition 100% correctly, but I think this is at least very close to it:

The tangent bundle TM of the manifold M is defined by

TM=\big\{(p,v)|p\in M, v\in \bigcup_{q\in M}T_qM\big\}

The function \pi:TM\rightarrow M defined by

\pi(p,v)=p

is called the projection.

A vector field is a section of the tangent bundle. A section is a function X:M\rightarrow TM, such that

\pi(X(p))=p

 

[HallsofIvy]

Fredrik is completely correct.

In simpler, less precise, terms, a "vector field" is an assignment of a tangent vector at EVERY POINT.

In a certain sense, while a tangent vector IS a derivative (the gradient of a function), a vector field is a differential equation.

 

[weio]

Hi Fredrik, HallsofIvy, and all

First of all thanks for the explanation Fredrik and HallsofIvy, but just to make sure i understand this, i will try to give an example.

Let X_p be a tangent vector in an open neighborhood U of a point p \in R^n and let f be a C^\infty function in U. F^i are the smooth functions from M into R , and suppose that X_p = (X,p), where the components of the Euclidean vector X are a^1,...,a^n. Then for any function f, the tangent vector <br /> X_p operates on f according to


X_p(f) = \sum_{i=1}^n a^i ( \frac{ \partial} { \partial F^i } )(p).

which can be written as

X_p(f) = a^i ( \frac{\partial} { \partial F^i } )p.

And this equation is basically the vector field, which assigns the tangent vector to the point.

The quantities

( \frac{ \partial } { \partial F^1})p,..., ( \frac{\partial}{ \partial F^n})p

form the basis for a tangent space T_p(R^n) at the point p.

Please correct me if I'm wrong.

weio

 

[Fredrik]

"A tangent vector in an open neighborhood..." It sounds like you're talking about a vector field here. A tangent vector is always a member of the tangent space of the manifold at a particular point.

When I use a notation like X_p, this means a vector in the tangent space of M at p. It would be OK to call X a vector field, if X_p\in T_pM and X is the map p\mapsto X_p (physicists' version of a vector field) or the map p\mapsto (p,X_p) (mathematicians' version of a vector field).

It's OK to write

X_p=a^i\frac{\partial}{\partial F^i}\bigg\lvert_p

or

X_p(f)=a^i\frac{\partial}{\partial F^i}\bigg\lvert_p f

but if f appears on the left-hand side you have to keep it on the right-hand side too. Otherwise it looks like a number is equal to a map from the "set of functions from M into R" into R.

The derivative operators (that I prefer to write with the vertical bar and the p as a subscript) are a basis for the tangent space of M at p. T_p\mathbb R^n would be the tangent space of \mathbb R^n at p, but p is a point in M. It is however possible to use the function F to construct a vector space isomorphism from T_pM onto T_{F(p)}\mathbb R^n, but that's kind of off topic.

Note that the maps

p\mapsto\frac{\partial}{\partial F^i}\bigg\lvert_p

are vector fields (physicists' version), but the derivative operators

\frac{\partial}{\partial F^i}\bigg\lvert_p

are tangent vectors. A convinient notation is to denote those maps (the vector fields) by

\frac{\partial}{\partial F^i}

I just dropped the subscript that indicated a position on the manifold, just as I did for X_p. With this notation we have

X=a^i\frac{\partial}{\partial F^i}

This is an equation that describes a relationship between vector fields.

 

[weio]

Hi

I guess i could have written it as X_p(f) = a^i ( \frac {\partial f} { \partial F^i} )(p), but i missed the f, my bad. Anyways, I understand now, but I think I need more time to get used to the notion of vector fields and tangent vectors. Thanks for the clarification.

weio

 

[mathwonk]

a vector field is just a family of tangent vectors, one at each point of a given set (usually open) of the manifold. Thus a vector field defined on a one point set, is a just tangent vector!