Suppose $F : \mathbb{R^2} \rightarrow \mathbb{R^3} $ be a map given by $ F(x,y)= (u,v,w)$ s.t. $u=x, v=y, w=xy$. Now for an arbitrary point $p = (x_0, y_0) \in \mathbb{R^2}$, I need to compute $ (dF)_p\big( \frac{\partial}{\partial x} \big)\Big |_p$
Here $(dF)_p$ is the differential map at $p$, i.e. , $(dF)_p : T_pN \rightarrow T_{F_p}M$ for manifolds $N$ and $M$. I don't understand how to compute it. Can someone please help me in understanding such kinds of computations in general!
One way to define the differential map of $F\colon N\to M$ is as follows. Given $v\in T_pN$, let $\gamma\colon(-\epsilon,\epsilon)\to N$ be a curve in $N$ so that $\gamma(0)=p$ and $\dot{\gamma}(0)=v$. Then \begin{equation} dF_p(v) := \frac{d}{dt}(F\circ\gamma)|_{t=0}. \end{equation} So to compute $dF_p\left(\frac{\partial}{\partial x}|_p\right)$, you should first find a curve $\gamma\colon(-\epsilon,\epsilon)\to\mathbb{R}^2$ so that $\gamma(0)=p$ and $\dot{\gamma}(0)=\frac{\partial}{\partial x}|_p$, and then compute $\frac{d}{dt}(F\circ\gamma)|_{t=0}$.
Given $p=(p_1,p_2)\in\mathbb{R}^2$, you could let $\gamma(t)=(p_1+t,p_2)$, for example, and then compute $\frac{d}{dt}(F\circ\gamma)|_{t=0}$.