What are the components of a vector field on a manifold?

[Silviu]

Hello! I am not sure I understand the idea of vector field on a manifold. The book I read is Geometry, Topology and Physics by Mikio Nakahara. The way this is defined there is: "If a vector is assigned smoothly to each point on M, it is called a vector field over M". Thinking about the 2D Euclidean space, it would look like at each point there is a vector (tangent to a curve inside that manifold) pointing in a certain direction, going smoothly from a point to another. However, he says then that "each component of a vector field is a smooth function from M to R". I am not sure I understand why is this. The components of a vector field are the vectors and they map from F(M) to R, where F(M) is the set of all smooth functions on M and F(M) is not the same thing as M. Also he says that "A vector field X at p $\in$ M is denoted by $X|_p$, which is an element of $T_pM$". I am confused here, too. By the first definition, a vector field associates to each point in M a vector, in a smooth way. How can you have a vector field at a point, when you have just one vector at that point (I understand you can define the tangent vector space for every point, but this is not what we describe here, right?). Can someone clarify these for me please? Thank you!

 

[andrewkirk]

The components of a vector field are the vectors

I think the confusion may arise from this statement. The components of a vector field, with respect to a coordinate system, are not vectors but scalar fields. For instance, in the two-dimensional number plane with the usual Cartesian ( aka 'rectangular') coordinate system, the components of the vector field are two scalar fields, one that maps each point to the x coordinate of the vector at that point, and one that maps to the y coordinate.

How can you have a vector field at a point

A reference to 'the vector field at a point' is just a loose shorthand for 'the vector that is the value of the vector field at that point'.

 

[Silviu]

I think the confusion may arise from this statement. The components of a vector field, with respect to a coordinate system, are not vectors but scalar fields. For instance, in the two-dimensional number plane with the usual Cartesian ( aka 'rectangular') coordinate system, the components of the vector field are two scalar fields, one that maps each point to the x coordinate of the vector at that point, and one that maps to the y coordinate.
A reference to 'the vector field at a point' is just a loose shorthand for 'the vector that is the value of the vector field at that point'.

Thank you for your reply. However I am still confused. In your first statement: based on the example you gave, if we take all the vectors in the xy plane to form a vector field, then the x and y coordinates are the component of a specific vector within that vector field, but the vector field, is still formed of vectors. I am not sure I understand what you mean.

 

[fresh_42]

Hello! I am not sure I understand the idea of vector field on a manifold. The book I read is Geometry, Topology and Physics by Mikio Nakahara. The way this is defined there is: "If a vector is assigned smoothly to each point on M, it is called a vector field over M".

A vector field is an area of points with an attached vector at each of them. That they behave smoothly at those points makes sense, if one - as usual - wants to consider smooth maps and smooth manifolds. I think it should be defined without that condition and it should be mentioned explicitly if given, but without it you can't do very much with it. So smoothness comes in quite convenient and most physical applications are smooth manifolds. Continuity is normally required to be the least condition.

Thinking about the 2D Euclidean space, it would look like at each point there is a vector (tangent to a curve inside that manifold) pointing in a certain direction, going smoothly from a point to another. However, he says then that "each component of a vector field is a smooth function from M to R".

This refers to the coordinate functions of the vector. If the entire map $p \mapsto X_p$ is smooth, so are the single coordinates $p \mapsto (X_p)_i = \left. \frac{\partial}{\partial x_i}\right|_p$

I am not sure I understand why is this. The components of a vector field are the vectors and they map from F(M) to R, where F(M) is the set of all smooth functions on M and F(M) is not the same thing as M. Also he says that "A vector field X at p $\in$ M is denoted by $X|_p$, which is an element of $T_pM$". I am confused here, too. By the first definition, a vector field associates to each point in M a vector, in a smooth way. How can you have a vector field at a point, when you have just one vector at that point (I understand you can define the tangent vector space for every point, but this is not what we describe here, right?). Can someone clarify these for me please? Thank you!

If we have smooth functions, then we have tangent spaces as they are differentiable (by definition). Whether the tangent space is one-dimensional or of higher dimension depends on the function. In general, there is simply a tangent space attached to each point of differentiability, e.g. a plane if the function has two variables. In these cases, there is an entire tangent bundle attached, and a single vector only one tangent in a certain direction. So $\{(p,T_pM)\,\vert \,p \in M\}$ is called tangent bundle or short vector bundle. $X_p \in T_pM$ then denotes the directional derivative in direction $X$ at a point $p$. So $\{(p,X_p)\,\vert \,p \in M\}$ is still a vector field, since it contains only tangents in direction of $X$, but there are other tangents at $p$ if $M$ is of higher dimension than one.

 

[andrewkirk]

An example may help. Consider the vector field $\vec V:\mathbb R^2\to \mathbb R^2$ such that $\vec V(x,y)=-y\mathbf{\hat i} + x\mathbf{\hat j}$ where $\mathbb{\hat i}, \mathbb{\hat j}$ are the unit vectors in the x and y directions. This is a vector field whose flow rotates around the origin at a rate that is proportional to the distance from the origin.

At a specific point $(x',y')$ the 'value' of the vector field is a single vector $\vec V(x',y')=-y'\mathbf{\hat x} + x'\mathbf{\hat y}$, which has components $-y'$ and $x'$ in the usual basis. The vector field has two component fields, which are scalar fields $V^1:\mathbb R^2\to \mathbb R$ and $V^2:\mathbb R^2\to \mathbb R$ such that $V^1(x,y)=-y$ and $V^2(x,y)=x$. The scalar field $V^1$ maps each point in the plane to the $x$ component of the value of the vector field at that point, and the scalar field $V^2$ maps each point in the plane to the $y$ component of the value of the vector field at that point.