# The Unapologetic Mathematician

## Integral Curves and Local Flows

Let $\mathfrak{X}M$ is a vector field on the manifold $M$ and let $q$ be any point in $M$. Then I say there exists a neighborhood $V\subseteq M$ of $q$, an interval $I\subseteq\mathbb{R}$ around $0$, and a differentiable map $\Phi:I\times V\to M$ such that

\displaystyle\begin{aligned}\Phi(0,p)&=p\\\Phi_*\left(\frac{\partial}{\partial t}(t,p)\right)&=X\left(\Phi(t,p)\right)\end{aligned}

for all $t\in I$ and $p\in V$. These should look familiar, since they’re very similar to the conditions we wrote down for the flow of a differential equation.

It might help a bit to clarify that $\frac{\partial}{\partial t}(t,p)$ is the inclusion $\iota_{p*}\left(\frac{d}{dt}(t)\right)$ of the canonical vector $\frac{d}{dt}(t)\in\mathcal{T}_t\mathbb{R}$ which points in the direction of increasing $t$. That is, $\iota_p:I\to I\times V$ includes the interval $I$ into $I\times V$ “at the point $p\in V$“, and thus its derivative carries along its tangent bundle. At each point of an (oriented) interval $I$ there’s a canonical vector, and $\frac{\partial}{\partial t}(t,p)$ is the image of that vector.

Further, take note that we can write the left side of our second condition as

$\displaystyle\Phi_*\left(\iota_{p*}\left(\frac{d}{dt}(t)\right)\right)$

The chain rule lets us combine these two outer derivatives into one:

$\displaystyle\left[\Phi\circ\iota_p\right]_*\left(\frac{d}{dt}(t)\right)$

But this is exactly how we defined the derivative of a curve! That is, we can write down a function $c=\Phi\circ\iota_p:I\to M$ which satisfies $c'(t)=X(c(t))$ for every $t\in I$. We call such a curve an “integral curve” of the vector field $X$, and when they’re collected together as in $\Phi$ we call it a “local flow” of $X$.

So how do we prove this? We just take local coordinates and use our good old existence theorem! Indeed, if $(U,x)$ is a coordinate patch around $q$ then we can set $G=x(U)$, $a=x(q)$, and

$\displaystyle F=(X^1,\dots,X^n)\circ x^{-1}:G\to\mathbb{R}^n$

where the $X^i$ are the components $Xx^i$ of $X$ relative to the given local coordinates.

Now our existence theorem tells us there is a neighborhood $W\subseteq G$ of $a$, an interval $I$ around $0$, and a map $\psi:I\times W\to G$ satisfying the conditions for a flow. Setting $V=x^{-1}(W)$ and $\Phi(t,p)=x^{-1}\left(\psi(t,x(p))\right)$ we find our local flow.

We can also do the same thing with our uniqueness theorem: if $c$ and $\tilde{c}$ are two integral curves of $X$ defined on the same interval $I$, and if $c(t_0)=\tilde{c}(t_0)$ for some $t_0\in I$, then $c=\tilde{c}$.

Thus we find the geometric meaning of that messy foray into analysis: a smooth vector field has a smooth local flow around every point, and integral curves of vector fields are unique.

May 28, 2011