# The Unapologetic Mathematician

## Identifying Vector Fields

We know what vector fields are on a region $U\subseteq M$, but to identify them in the wild we need to verify that a given function sending each $p\in U$ to a vector in $\mathcal{T}_pM$ is smooth. This might not always be so easy to check directly, so we need some equivalent conditions. First we need to define how vector fields act on functions.

If $X\in\mathfrak{X}U$ is a vector field and $f\in\mathcal{O}U$ is a smooth function then we get another function $Xf$ by defining $Xf(p)=\left[X(p)\right](f)$. Indeed, $X(p)\in\mathcal{T}_pM$, so it can take (the germ of) a smooth function at $p$ and give us a number. Essentially, at each point the vector field defines a displacement, and we ask how the function $f$ changes along this displacement. This action is key to our conditions, and to how we will actually use vector fields.

Firstly, if $X$ is a vector field — a differentiable function — and if $(V,x)$ is a chart with $V\subseteq U$, then $Xx^i$ is always smooth. Indeed, remember that $(V,x)$ gives us a coordinate patch $(\pi^{-1}(V),\bar{x})$ on the tangent bundle. Since $\bar{x}$ is smooth and $X$ is smooth, the composition

$\displaystyle\bar{x}\circ X\vert_V=(x\circ I\vert_V;X\vert_V(x^1),\dots,x\vert_V(x^n))$

is also smooth. And thus each component $Xx^i$ is smooth on $V$.

Next, we do not assume that $X$ is a vector field — it is a function but not necessarily a differentiable one — but we assume that it satisfies the conclusion of the preceding paragraph. That is, for every chart $(V,x)$ with $V\subseteq U$ each $Xx^i$ is smooth. Now we will show that $Xf$ is smooth for every smooth $f\in\mathcal{O}V$, not just those that arise as coordinate functions. To see this, we use the decomposition of $X$ into coordinate vector fields:

$\displaystyle X=\sum\limits_{i=1}^nX^i\frac{\partial}{\partial x^i}$

which didn’t assume that $X$ was smooth, except to show that the coefficient functions were smooth. We can now calculate that $X^i=Xx^i$, since

$\displaystyle Xx^i=\sum\limits_{j=1}^nX^j\frac{\partial x^i}{\partial x^j}=\sum\limits_{j=1}^nX^j\delta^i_j=X^i$

But this means we can write

$\displaystyle Xf=\sum\limits_{i=1}^nXx^i\frac{\partial f}{\partial x^i}$

which makes $Xf$ a linear combination of the smooth (by assumption) functions $Xx^i$ with the coefficients $\frac{\partial f}{\partial x^i}$, proving that it is itself smooth.

Okay, now I say that if $Xf$ is smooth for every smooth function $f\in\mathcal{O}V$ on some region $V\subseteq U$, then $X$ is smooth as a function, and thus is a vector field. In this case around any $p\in U$ we can find some coordinate patch $(V,x)$. Now we go back up to the composition above:

$\displaystyle\bar{x}\circ X\vert_V=(x\circ I\vert_V;X\vert_V(x^1),\dots,x\vert_V(x^n))$

Everything in sight on the right is smooth, and so the left is also smooth. But this is exactly what we need to check when we’re using the local coordinates $(V,x)$ and $(\pi^{-1}(V),\bar{x})$ to verify the smoothness of $X$ at $p$.

The upshot is that when we want to verify that a function $X$ really is a smooth vector field, we take an arbitrary smooth “test function” and feed it into $X$. If the result is always smooth, then $X$ is smooth. In fact, some authors take this as the definition, regarding the action of $X$ on functions as fundamental, and only later talking in terms of its “value at a point”.

### Like this:

Posted by John Armstrong | Differential Topology, Topology

## 3 Comments	»

1. [...] the are the components of relative to the given local [...]

Pingback by | May 28, 2011 | Reply

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

Pingback by | June 2, 2011 | Reply

3. [...] as for vector fields, we need a good condition to identify tensor fields in the wild. And the condition we will use is [...]

Pingback by | July 7, 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": 51, "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": 0.9309941530227661, "perplexity_flag": "head"}