I am trying to solve the following problem. Let $M$ and $N$ be submanfiolds of $\mathbb{R}^2$ given by $M = \{(x,y) \in \mathbb{R}^2 : x > 0, x+y>0\}$ and $N = \{(u,v) \in \mathbb{R}^2 : u > 0 , v>0\}$ and let $F: M \rightarrow N$ be the diffeomorphism given by $F(x,y) = (F_1,F_2) = (1 + y/x, x+y)$. If $X$ is the vector field on $M$ given by $X = y^2 \frac{\partial}{\partial x}$, compute the vector field $F_*X$ on $N$.
My attempt is the following: To compute the push forward of a vector field under a certain map, we need to compute the Jacobian of the map. In this case $$JF = \left( \begin{matrix}\frac{\partial F_1}{\partial x} & \frac{\partial F_1}{\partial y} \\ \frac{\partial F_2}{\partial x} & \frac{\partial F_2}{\partial y} \end{matrix} \right) = \left( \begin{matrix} -\frac{y}{x^2} & \frac{1}{x} \\ 1 & 1 \end{matrix} \right)$$
So, with this computation, we can represent the vector field $X$ by the vector $X = (y^2,0)$ and so the Pushforward $F_*X = -\frac{y^3}{x^2}\partial u + y^2 \partial v$.
I would just want to make sure that this computation is correct. Thanks so much for your help!
The computation is okay, but in my opinion it is better to express the pushforward $F_* X$ in terms of the local coordinates on $N$, which is $(u,v)$. The mixture of functions given in terms of $x$ and $y$ and partial derivatives in terms of $u$ and $v$ looks odd.
I also think you have a typo: you might have meant $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial v}$ rather than just $\partial u$ and $\partial v$ in the computed expression for $F_* X$.
So, since $F : M \to N$ is a diffeomorphism, we can invert $F$ to express $x$ and $y$ in terms of $u$ and $v$:
$$ \begin{align} 1+y/x &= u\\ x+y&=v \end{align} \bigg\} \implies 1+(v-x)/x = u \implies x = v/u. $$ Therefore, $$ \begin{align} x &= v/u\\ y &= v(u-1)/u. \end{align} $$ Hence, $$ F_*X = -\frac{v(u-1)^3}{u} \frac{\partial}{\partial u} + \frac{v^2(u-1)^2}{u^2} \frac{\partial}{\partial v}. $$