Notation regarding integral curves and vector fields trying to solve some integral curve of vector field I am really confused about the notation used in many textbook.
I can see problem of this sort :
Find the integral curves of the following vector field :
\begin{align}
X(x,y) = x^2\frac{\partial}{\partial x} + xy\frac{\partial}{\partial y}
\end{align}
So to set up the notation in this setting, a curve should be $\gamma : \mathbb{R} \rightarrow \mathbb{R}^2$ and any vector field of the curve can be written as :
\begin{align}
X_{{\gamma},p} = \frac{\partial }{\partial x^i}.(x^i\circ\gamma)'(0)
\end{align}
Where $\gamma(0) = p$. So if we want to find the integral curve of the vector field we have to make sure at any point of the curve (any $t$) the tangent vector of the curve is the same as the vector field evaluated at $\gamma(t)$.
The problem is when equating the two equation we end up with something like $x^2 = (x^i\circ\gamma)'(0)$ which do not make sense. 
Are we abusing notation in this sort of problems where $x^2$ is actually the composition of the coordinate function $x$ with the curve at the specific point of interest ?
If this is the case, how inaccurate is the drawing of the vector field in 2D ? ( I can see in my textbook the drawing of the vector field where at each $(x,y)$ an arrow is drawn with direction $(x^2, xy)$.
I can see some other textbook which define the curve to be $\gamma(t) = (x(t), y(t))$ which again do not make sense as it is conflicting with the coordinate function definition $x : \mathbb{R^2} \rightarrow \mathbb{R}$.
I am asking this question because the books go ahead solving an ODE where they have no problem solving stuff like : $x' = x^2$ and $y' = xy$.
So is this abusing the notation ?
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}$If $x$ and $y$ denote the Cartesian coordinate functions on $\Reals^{2}$ and if $\gamma$ is a curve in $\Reals^{2}$, then
$$
\gamma(t) = ((x \circ \gamma)(t), (y \circ \gamma)(t)),
$$
so that
$$
\gamma'(t) = ((x \circ \gamma)'(t), (y \circ \gamma)'(t)).
$$
This notation is cumbersome in practice, so one usually writes
$$
\gamma(t) = (x(t), y(t)),
\tag{1}
$$
allowing $x$ to denote both a Cartesian coordinate in the plane and the first component of $\gamma$.
You're right: Technically (1) is abuse of notation. It would be more honest to introduce new symbols, writing
$$
\gamma(t) = (u(t), v(t)),
\tag{2}
$$
so that $(x \circ \gamma)(t) = u(t)$, $(y \circ \gamma)(t) = v(t)$, and (for your specific vector field $X$)
\begin{align*}
\gamma'(t) = (u'(t), v'(t)) &= u'(t)\, \frac{\dd}{\dd x} + v'(t)\, \frac{\dd}{\dd y}, \\
(X \circ \gamma)(t) &= u(t)^{2}\, \frac{\dd}{\dd x} + u(t) v(t)\, \frac{\dd}{\dd y}.
\end{align*}
The flow equation then becomes the ODE system
$$
u'(t) = u(t)^{2},\qquad
v'(t) = u(t) v(t).
$$
The thing is, this same reasoning holds if we write (1) instead of (2), with the conclusion
$$
x'(t) = x(t)^{2},\qquad
y'(t) = x(t) y(t),
$$
i.e., $x' = x^{2}$, $y' = xy$.
A: Relevant to the above is the following reference at Mathematics Stack Exchange:

How to properly apply the Lie Series

The OP's specific problem is found there at : $\;b)\; X = x^2\,\partial_x+xy\,\partial_y\;$ .
