For $S$ a Riemann surface, on a chart $U$ with local coordinate $z = x+iy,$ define $$w = df = f_x dx + f_ydy$$ on $U$ where $f$ is a $C^2-$harmonic function ($f_{xx} + f_{yy} = 0$; ($(f_{xx} + f_{yy}) dx \wedge dy = 0$ with all continuous second partial derivatives).
I want to verify that for two charts $U, V$ on $S$ with $U \cap V \neq \phi$, and $w = df$ on $U$ and $w = dg$ on $V$, $$w \ \mbox{is well-defined on} \ U \cap V.$$
That is, on $U$ wtih local coordinate $z = x+iy$, $$w = f_x dx + f_y dy$$ and on $V$ with local coordinate $\omega = s+it,$ $$w = g_s ds + g_t dt,$$ then I would like to show that $$g_s = \frac{\partial x}{\partial s} f_x + \frac{\partial y}{\partial s}f_y \ \mbox{and} \ g_t = \frac{\partial x}{\partial t} f_x + \frac{\partial y}{\partial t}f_y$$ on $U \cap V.$
We know only that $$f_{xx} + f_{yy} = 0 = g_{ss} + g_{tt}.$$
$\textbf{Attempt}$ Let $$A = \frac{\partial x}{\partial s} f_x + \frac{\partial y}{\partial s}f_y, B=\frac{\partial x}{\partial t} f_x + \frac{\partial y}{\partial t}f_y.$$
Then I compute $\frac{\partial}{\partial s} A+\frac{\partial}{\partial t}B$ and try to use the fact that $$f_{xx} + f_{yy} = 0$$ and hope it leads me to some relation that I want. It is quite long and the results do not closed to what I want with many extra terms. I am not sure now how should I proceed. Any help please !!
I think you may use the following on $U\cap V$
$$\begin{align} g_t&= w(\frac{\partial}{\partial t})\\ &=(f_xdx+f_ydy)(\frac{\partial}{\partial t})\\ &=f_xdx(\frac{\partial}{\partial t})+f_ydy(\frac{\partial}{\partial t})\\ &=f_x(\frac{\partial x}{\partial t})+f_y(\frac{\partial y}{\partial t}) \end{align}$$
Now you need to show that the expression of $w$ is the same for both charts as follows
$$\begin{align}w&=f_x dx+f_y dy\\ &=f_x(\frac{\partial x}{\partial t}dt+\frac{\partial x}{\partial s}ds)+f_y(\frac{\partial y}{\partial t}dt+\frac{\partial y}{\partial s}ds)\\ &=(......)dt+(......)ds\\ &=g_t dt+g_s ds \end{align}$$