Sorry if this is going to be really dumb, but I really don't understand how you operate with a vector field even though I carefully read my lecture notes. I will express what I don't understand by showing you an exercise that I don't really understand deeply.
Exercise. Find the flow of the vector field $X=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}\in \frak{X}(\mathbb{R}^2)$.
Okay, so I am even confused by this notation to begin with. I know that $\{\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\}$ is a basis in the tangent space. I also know that a vector field is a function from my manifold to its tangent bundle.So, I guess that what I am basically doing is writing $X$ using its basis components. I guess that I am also omitting arguments when writing like this and if I wanted to include arguments I would have to write $$X(f)=x\frac{\partial}{\partial x}(f)+y\frac{\partial}{\partial y}(f)$$ for a differentiable function $f$. Then I guess that I could go even further and evaluate all of this at some point $a$, obtaining something like $$X(f)(a)=x\frac{\partial}{\partial x}(f)(a)+y\frac{\partial}{\partial y}(f)(a).$$ What I don't get, though, is who are the $x$ and $y$ that I am multiplying with those "partial derivatives". This is something that I really don't understand.
Going back to the problem of finding the flow, here is the solution my instructor gave: we want to find $\gamma:(-\epsilon, \epsilon)\to \mathbb{R}^2$ such that $\gamma(0)=(x_0, y_0)$, where $(x_0, y_0)\in\mathbb{R}^2$ is fixed, and $\frac{d}{dt}\bigg|_{t=0}\gamma(t)=X_{\gamma(t)}$. This is basically just the definition of an integral curve, so I am okay with this. What I don't understand is how you write this equation in coordinates. I know that it should be (I think) $\frac{d \gamma^i}{dt}=X^{i}(\gamma(t)), i=\overline{1, 2}$, but I don't really understand why this gives me $$\begin{cases} \dot{\gamma^1}(t)=\gamma^1(t)\\ \dot{\gamma^2}(t)=\gamma^2(t) \end{cases}.$$ Intuitively I get it, we substituted that $x$ that I said that I don't understand where it's coming from with $\gamma^1(t)$ and we also substituted $y$ with $\gamma^2(t)$. But I don't undestand why we do this rigorously. What I am basically asking is how I compute $X^{i}(\gamma(t))$.
I would be extremely grateful if you could tell me if the things that I wrote are correct and if you could help me fill in the gaps where I said that I don't undestand what is going on. Thank you!