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.
The tangent space [tex]T_pM[/tex] of the manifold M at point p can be defined as the vector space spanned by the basis vectors [tex]\frac{\partial}{\partial x^\mu}\bigg\lvert_p[/tex] 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 ([tex]C^\infty[/tex]) 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 [tex]TM=\big\{(p,v)|p\in M, v\in \bigcup_{q\in M}T_qM\big\}[/tex] The function [tex]\pi:TM\rightarrow M[/tex] defined by [tex]\pi(p,v)=p[/tex] is called the projection. A vector field is a section of the tangent bundle. A section is a function [tex]X:M\rightarrow TM[/tex], such that [tex]\pi(X(p))=p[/tex]
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.
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 [tex] X_p [/tex] be a tangent vector in an open neighborhood [tex] U [/tex] of a point [tex] p \in R^n[/tex] and let [tex] f [/tex] be a [tex] C^\infty [/tex] function in [tex] U [/tex]. [tex] F^i [/tex] are the smooth functions from [tex] M [/tex] into [tex] R [/tex] , and suppose that [tex] X_p = (X,p) [/tex], where the components of the Euclidean vector [tex] X [/tex] are [tex]a^1,...,a^n [/tex]. Then for any function [tex] f [/tex], the tangent vector [tex] X_p [/tex] operates on [tex] f [/tex] according to [tex] X_p(f) = \sum_{i=1}^n a^i ( \frac{ \partial} { \partial F^i } )(p). [/tex] which can be written as [tex] X_p(f) = a^i ( \frac{\partial} { \partial F^i } )p. [/tex] And this equation is basically the vector field, which assigns the tangent vector to the point. The quantities [tex] ( \frac{ \partial } { \partial F^1})p,..., ( \frac{\partial}{ \partial F^n})p [/tex] form the basis for a tangent space [tex] T_p(R^n) [/tex] at the point p. Please correct me if i'm wrong. weio
"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 [tex]X_p[/tex], this means a vector in the tangent space of M at p. It would be OK to call [tex]X[/tex] a vector field, if [tex]X_p\in T_pM[/tex] and [tex]X[/tex] is the map [tex]p\mapsto X_p[/tex] (physicists' version of a vector field) or the map [tex]p\mapsto (p,X_p)[/tex] (mathematicians' version of a vector field). It's OK to write [tex]X_p=a^i\frac{\partial}{\partial F^i}\bigg\lvert_p[/tex] or [tex]X_p(f)=a^i\frac{\partial}{\partial F^i}\bigg\lvert_p f[/tex] 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. [tex]T_p\mathbb R^n[/tex] would be the tangent space of [tex]\mathbb R^n[/tex] 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 [tex]T_pM[/tex] onto [tex]T_{F(p)}\mathbb R^n[/tex], but that's kind of off topic. Note that the maps [tex]p\mapsto\frac{\partial}{\partial F^i}\bigg\lvert_p[/tex] are vector fields (physicists' version), but the derivative operators [tex]\frac{\partial}{\partial F^i}\bigg\lvert_p[/tex] are tangent vectors. A convinient notation is to denote those maps (the vector fields) by [tex]\frac{\partial}{\partial F^i}[/tex] I just dropped the subscript that indicated a position on the manifold, just as I did for [tex]X_p[/tex]. With this notation we have [tex]X=a^i\frac{\partial}{\partial F^i}[/tex] This is an equation that describes a relationship between vector fields.
Hi I guess i could have written it as [tex] X_p(f) = a^i ( \frac {\partial f} { \partial F^i} )(p) [/tex], but i missed the [tex] f [/tex], 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
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!