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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!

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The computation is alright, 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 $v$:

$$ \begin{align} 1+x/y &= u\\ x+y&=v \end{align} \bigg\} \implies 1+(v-x)/x = u \implies x = v/u. $$ Therefore, $$ \begin{align} x &= v/u\\ y &= v(u-1)/u. \end{align} $$ Hence, $$ F_*X = \frac{v(u-1)^3}{u} \frac{\partial}{\partial u} + \frac{v^2(u-1)^2}{u^2} \frac{\partial}{\partial v}. $$

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The computation is correct, and is well explained in this previous answer: Pushforward of a vector field

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