# The Unapologetic Mathematician

## Vector Fields

At last, we get back to the differential geometry and topology. Let’s say that we have a manifold $M$ with tangent bundle $\mathcal{T}M$, which of course comes with a projection map $\pi:\mathcal{T}M\to M$. If $U\subseteq M$ is an open submanifold, we can restrict the bundle to the tangent bundle $\pi:\mathcal{T}U\to U$ with no real difficulty.

Now a “vector field” on $U$ is a “section” of this projection map. That is, it’s a function $v:U\to\mathcal{T}U$ so that the composition $\pi\circ v:U\to U$ is the identity map on $U$. In other words, to every point $p\in U$ we get a vector $v(p)\in\mathcal{T}_pU$ at that point.

I should step aside to dissuade people from a common mistake. Back in multivariable calculus, it’s common to say that a vector field in $\mathbb{R}^3$ is a function which assigns “a vector” to every point in some region $U\subseteq\mathbb{R}^3$; that is, a function $U\to\mathbb{R}^3$. The problem here is that it’s assuming that every point gets a vector in the same vector space, when actually each point gets assigned a vector in its own tangent space.

The confusion comes because we know that if $M$ has dimension $n$ then each tangent space $\mathcal{T}_pM$ has dimension $n$, and thus they’re all isomorphic. Worse, when working over Euclidean space there is a canonical identification between a tangent space $\mathcal{T}_pE$ and the space $E$ itself, and thus between any two tangent spaces. But when we’re dealing with an arbitrary manifold there is no such canonical way to compare vectors based at different points; we have to be careful to keep them separate.

For each $U\subseteq M$ we have a collection of vector fields, which we will write $\mathfrak{X}_MU$, or $\mathfrak{X}U$ for short. It should be apparent that if $V\subseteq U$ is an open subspace we can restrict a vector field on $U$ to one on $V$, which means we’re talking about a presheaf. In fact, it’s not hard to see that we can uniquely glue together vector fields which agree on shared domains, meaning we have a sheaf of vector fields.

For any $U$, we can define the sum and scalar multiple of vector fields on $U$ just by defining them pointwise. That is, if $v_1$ and $v_2$ are vector fields on $U$ and $a_1$ and $a_2$ are real scalars, then we define

$\displaystyle\left[a_1v_1+a_2v_2\right](p)=a_1v_1(p)+a_2v_2(p)$

using the addition and scalar multiplication in $\mathcal{T}_pM$. But that’s not all; we can also multiply a vector field $v\in\mathfrak{X}U$ by any function $f\in\mathcal{O}U$:

$\displaystyle\left[fv\right](p)=f(p)v(p)$

using the scalar multiplication in $\mathcal{T}_pM$. This makes $\mathfrak{X}_M$ into a sheaf of modules over the sheaf of rings $\mathcal{O}_M$.

### Like this:

Posted by John Armstrong | Differential Topology, Topology

## 10 Comments	»

1. [...] is a coordinate patch, it turns out that we can actually give an explicit basis of the module of vector fields over the ring [...]

Pingback by | May 24, 2011 | Reply

2. thank you for this post.

Comment by tiredguy | May 25, 2011 | Reply

3. [...] know what vector fields are on a region , but to identify them in the wild we need to verify that a given function sending [...]

Pingback by | May 25, 2011 | Reply

4. [...] is a vector field on the manifold and let be any point in . Then I say there exists a neighborhood of , an [...]

Pingback by | May 28, 2011 | Reply

5. [...] a smooth vector field we know what it means for a curve to be an integral curve of . We even know how to find them by [...]

Pingback by | May 30, 2011 | Reply

6. [...] Fields on Compact Manifolds are Complete It turns out that any vector field on a compact manifold is complete. That is, starting at any point we can follow the vector field on [...]

Pingback by | June 1, 2011 | Reply

7. [...] know that any vector field can act as an endomorphism on the space of smooth functions on . What happens if we act by one [...]

Pingback by | June 2, 2011 | Reply

8. [...] be a smooth map between manifolds, with derivative , and let and be smooth vector fields. We can compose them as and , and it makes sense to ask if these are the same [...]

Pingback by | June 3, 2011 | Reply

9. [...] go back to the way a vector field on a manifold gives us a “derivative” of smooth functions . If is a smooth vector [...]

Pingback by | June 15, 2011 | Reply

10. [...] vector field defines a one-dimensional subspace of at any point with : the subspace spanned by . If is [...]

Pingback by | June 28, 2011 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 41, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9287415742874146, "perplexity_flag": "head"}