I'm working on a differential equations project where I need to use the "Existence and Uniqueness Theorem for a Vector Field's Integral Curves" to prove the Existence and Uniqueness Theorem for 1st order linear differential equations in $\mathbb{R}^{2}$. The instructions for this project are very vague and poorly written, and I'm having trouble making sense of the part about integral curves of vector fields. I'll state the vector integral curve theorem here. If anyone can dumb down this theorem for me or give any pointers on where to start, that would be greatly appreciated.
For a vector field $\textbf{v}(x)$ on a subset $U \in \mathbb{R}^{n}$:
Given any point $p \in U$, and any initial "time" $t_0 \in \mathbb{R}$, there exists some $\epsilon > 0$ such that there is a unique, continuously differentiable function $\gamma: (t_{0}-\epsilon, t_{0}+\epsilon)\to U$ defined on the interval $(t_{0}-\epsilon, t_{0}+\epsilon)$ around $t_0$ such that:
- $\gamma'(t)=\textbf{v}(\gamma(t))$
- $\gamma(t_0)=p$
The way I'm trying to visualize this right now is that I can choose some point in this vector field's domain, then pick some curve passing arbitrarily close to that point. The rate of change of this curve will be equal to the vector field composed with the equation for the curve. Additionally, evaluating the curve at point $t_0$ will be equal to the point $p$ because the curve passes arbitrarily close to $p$ (I think).
If I'm thinking about this right, how do I start connecting it to the existence and uniqueness theorem for 1st order diff eqs?
I'm not sure if this is not a forest-tree problem. You are just using a different language for the same underlying objects and facts, a coordinate centered view vs. the dynamic of geometric objects (here only points and tangents). It might be that the language of (tangential) vector fields and their integral curves has a more natural connection to manifolds.
$$ x'=v(x),~~ v(t_0)=p $$ with $x$ and $v(x)$ in $\Bbb R^n$ is also a first-order system of differential equations with initial value, short an IVP.
Depending on the properties of $v$, the existence theorem is the theorem of Peano or the Picard-Lindelöf theorem ($v$ being Lipschitz) or its specialized version as Cauchy-Lipschitz theorem ($v$ being continuously differentiable, thus locally Lipschitz).
One can recast every non-autonomous system as autonomous system with the time as additional component of the state. This gives a slight restriction of the general claim of Picard-Lindelöf, as there the ODE function only needs to be continuous in time, while being Lipschitz in the state vector.