# Computing the push forward of vector field $X = y^2 \partial/\partial x$ using Jacobians

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!

## 2 Answers

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, $$F_*X = \frac{v(u-1)^3}{u} \frac{\partial}{\partial u} + \frac{v^2(u-1)^2}{u^2} \frac{\partial}{\partial v}.$$

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