I'm looking at the real projective plane $\mathbb RP^2$, with homogeneous coordinates $(x:y:z)$. I want to find the transition matrix between bases associated with different coordinate charts.
I've arrived at an answer, which I think is correct, but I'm not sure, and the answer doesn't seem to be in the form the questions suggests it should be.
Here's what I've done:
First, I split $\mathbb RP^2$ up into three charts: $x \neq 0$, $y \neq 0$ and $z \neq 0$.
In the chart $x \neq 0$, define $\varphi_1 : \mathbb R^2 \to \mathbb RP^2$ where $\varphi_1(u_1,v_1) = (1:u_1:v_1)$ and $$\varphi_1^{-1}(x:y:z) = \left(\frac{y}{x},\frac{z}{x}\right) $$
In the chart $y \neq 0$, define $\varphi_2 : \mathbb R^2 \to \mathbb RP^2$ where $\varphi_2(u_2,v_2) = (u_2:1:v_2)$ and $$\varphi_2^{-1}(x:y:z) = \left(\frac{x}{y},\frac{z}{y}\right) $$
In the chart $z \neq 0$, define $\varphi_3 : \mathbb R^2 \to \mathbb RP^2$ where $\varphi_3(u_3,v_3) = (u_3:v_3:1)$ and $$\varphi_3^{-1}(x:y:z) = \left(\frac{x}{z},\frac{y}{z}\right) $$
Next, I found the change of coordinates between the 2nd and the 3rd charts: $$(u_3,v_3) = (\varphi_3^{-1} \circ \varphi_2)(u_2,v_2) = \left(\frac{u_2}{v_2},\frac{1}{v_2}\right)$$
I think this is how to express the coordinate vector field from the second chart in the third chart: $$ {\bf e}_{u_2} = \frac{1}{v_2} \, {\bf e}_{u_3} \ \ \text{ and } \ \ {\bf e}_{v_2} = -\frac{u_2}{v_2^2}\,{\bf e}_{u_3} -\frac{1}{v_2^2}\,{\bf e}_{v_3} $$ and I would be able to find the transition matrix. Is this correct?
Ultimately, I want to express the matrix in terms of functions of the homogeneous coordinates. However, I seem to have a matrix giving a map $T\mathbb R^2 \to T\mathbb R^2$, and I can't see where the $x$, $y$ and $z$s are going to come from. Any guidance would be appreciated.
In order not to let this question without an answer, allow me to collect here what was discussed in the comments, for what it's worth.
First, your computations are correct, there is nothing to add or doubt about this.
Second, note that the word "coordinates" may be used with two meanings here. One meaning is the usual one in differential geometry (such as in "coordinate chart"). The second one stems from the existence of "homogeneous coordinates" on every manifold that embeds into some projective space; unlike the coordinates in the first sense, these latter ones exist globally on such a manifold.
Normally, the matrix entries of $\Bbb d (\varphi _3 ^{-1} \circ \varphi _2)$ will be functions of $u_2$ and $v_2$. Fortunately, these charts called "projectivizations" have inverses that are easy to write down and, furthermore, their components are homogeneous rational functions of $[x : y : z]$, namely $\dfrac x y$ and $\dfrac z y$ (this particular formula for $\varphi _2 ^{-1}$ is well defined because $[\alpha x : \alpha y : \alpha z]$ and $[x : y : z]$ produce the same fractions, by simplification). In formulae, $\varphi ^{-1} _2 (p) = (u_2 (p), v_2 (p)) = (\dfrac x y, \dfrac z y) (p)$, and thus the entries of $(u_2, v_2) \mapsto \Bbb d (\varphi _3 ^{-1} \circ \varphi _2) (u_2, v_2)$ will be homogeneous rational functions of $[x : y : z]$.