<PERSON> symbols of semilog metric
Problem 17.3a in <PERSON>'s "A general relativity workbook" requires to determine the only non-zero <PERSON> symbol for the 2D semilog pq coordinate system defined as $p=x$ and $q=e^{by}$. What puzzles me is that I found more than one non-zero <PERSON> symbol, and I wanted to ask you what am I doing wrong.
First of all, I calculated the metric $g_{\mu\nu}'$ of the coordinate system as $$ g_{\mu\nu}' =\frac{\partial x^\alpha}{\partial {x^\mu}'}\frac{\partial x^\beta}{\partial {x^\nu}'}g_{\alpha\beta} =\frac{\partial x}{\partial {x^\mu}'}\frac{\partial x}{\partial {x^\nu}'} + \frac{\partial y}{\partial {x^\mu}'}\frac{\partial y}{\partial {x^\nu}'} $$ given that $g_{\alpha\beta}$ for the 2D Euclidean plane is the identity matrix. Calculating this, I got $$ g_{\mu\nu}'= \begin{bmatrix} 1 & 0\ 0 & \frac{1}{p^2 b^2} \end{bmatrix} $$ Is this correct?
Once this is done, I calculated the <PERSON> symbols by comparing the geodesic equation expressed in terms of the metric elements $0=\frac{d}{d\tau}\left(g_{\mu\nu}\frac{dx^\nu}{d\tau}\right)-\frac{1}{2}\partial_\mu g_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}$ with the same equation expressed via the Christoffel symbols $0=\frac{d^2x^\mu}{d\tau^2}+\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}$. 818
If we set $\mu=p$ in the first equation we get $$ 0=\frac{d}{d\tau}\left(g_{p\nu}\frac{dx^\nu}{d\tau}\right)-\frac{1}{2}\frac{\partial g_{\alpha\beta}}{\partial p}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=\ =\frac{d^2p}{d\tau^2} - \frac{1}{2}\frac{\partial g_{qq}}{\partial p}\left(\frac{dq}{d\tau}\right)^2=\ =\frac{d^2p}{d\tau^2} + \frac{1}{b^2p^3}\left(\frac{dq}{d\tau}\right)^2 $$ Setting $\mu=p$ in the <PERSON> version of the geodesic equation, we then can conclude that $\Gamma^p_{pp}=\Gamma^p_{pq}=\Gamma^p_{qp}=0$, while $\Gamma^p_{qq}=\frac{1}{b^2p^3}$. By doing the same procedure with $\mu=q$, I got that $\Gamma^q_{pp}=\Gamma^q_{qq}=0$ and that $\Gamma^q_{pq}=\Gamma^q_{qp}=-\frac{1}{p}$.
This means that three <PERSON> symbols are non-zero, in opposition with the text of the exercise. Am I missing something, or is the text wrong? 394