Actually this is a quesion in Lee's book, Manifolds and differential geometry. I have problems working with projective spaces as manifolds.(e.g. what are curves in projective spaces ? I need to know the equivalence classes of curves to have an idea about tangent bundle... can I think of $\mathbb{R} P^2$ as the upper unit hemisphere, with half of the equator removed ? ... ) I know that the following standard charts on $\mathbb{R} P^2$ form an atlas : $$[x,y,z] \to (u_1,u_2)=(\frac{x}{z},\frac{y}{z}) \ \ \ on \ \ U_z=\{z \neq 0\}$$ $$[x,y,z] \to (v_1,v_2)=(\frac{x}{y},\frac{z}{y}) \ \ \ on \ \ U_y=\{y \neq 0\}$$ $$[x,y,z] \to (w_1,w_2)=(\frac{y}{x},\frac{z}{x}) \ \ \ on \ \ U_x=\{x \neq 0\}$$ I need to Show that there is a vector field $X$ on $\mathbb{R} P^2$ which in the last coordinate chart above has the following coordinate expression: $$w_1 \frac{\partial}{\partial w_1} - w_2 \frac{\partial}{\partial w_2} .$$ It is also asked that what are the coordinate expressions for the vector field $X$ in the other two charts ?
I already appreciate your help.
First let us change to an easier notation.
$$V_1 :[x,y,z] \to (a_1,b_1)=(\frac{x}{z},\frac{y}{z}) \text{ on } U_z=\{z \neq 0\}$$
$$V_2: [x,y,z] \to (a_2,b_2)=(\frac{x}{y},\frac{z}{y}) \text{ on } U_y=\{y \neq 0\}$$
$$V_3: [x,y,z] \to (a_3,b_3)=(\frac{y}{x},\frac{z}{x}) \text{ on } U_x=\{x \neq 0\}$$
You want to find a vector field $\Theta$ on $RP^2$ such that $\Theta|_{V_3} = a_3 \frac{\partial}{\partial a_3} - b_3 \frac{\partial}{\partial b_3}$.
All we have to do is find a definition on $\Theta$ on the charts $V_1$ and $V_2$ then check that they agree on intersections.
Let us start by seeing how to define $\Theta$ on $V_2$. We know that on $V_2 \cap V_3$, $a_3 = \frac{1}{a_2}$ and $b_3 = b_2a_3$. By the product rule $$0 = \Theta (a_3 a_2) = a_2\Theta(a_3) + a_3 \Theta(a_2)$$ so we know that $$\Theta(a_2) = -a_2$$ Likewise we can compute $$\Theta(b_2) = -2b_2$$ Therefore we can conjecture if such a $\Theta$ exists then $$\Theta|_{V_2} = -a_2\frac{\partial}{\partial a_2} - 2b_2 \frac{\partial}{\partial b_2}$$
Now do the same type of computation for $V_1$ and check that they agree on intersections.
If you've learned about homogenous coordinates these computations tend to be easier to do using that language.