Is there a method of knowing which of the Christoffel symbols of second kind survive and vanish for a given metric? I'm trying to solve a problem and the given $g_{ij}$ metric is $$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$$
The non-zero Christoffel symbols of the second kind, $\Gamma^{i}_{\;\,jk}$, for this problem happen to be:
$i=1$: $\Gamma^{1}_{\;\,22}$, $\Gamma^{1}_{\;\,33},$
$i=2$: $\Gamma^{2}_{\;\,12} =\Gamma^{2}_{\;\,21}$, $\Gamma^{2}_{\;\,33},$
$i=3$: $\Gamma^{3}_{\;\,13} =\Gamma^{3}_{\;\,31}$, $\Gamma^{3}_{\;\,23} =\Gamma^{3}_{\;\,32}$.
My question is, how can one be sure these are the only non-zero ones? It's difficult to check all the possible combinations of $ijk$, especially if there are time constraints.

CLARIFICATION: In general, computing Christoffel symbols is quite painful and usually can be avoided. Not only do you have to compute every partial derivative of every component of the metric tensor, you also have to compute the inverse tensor. The following is for a diagonal metric tensor only.
Suppose $g_{ij} = a_i\delta_{ij}$.
\begin{align*}
\partial_kg_{ij} &= \partial_ka_i\delta_{ij}\\
2g_{kp}\Gamma^p_{ij} &= \partial_ig_{kj} + \partial_jg_{ik} - \partial_kg_{ij}\\
&= \partial_ia_k\delta_{jk} + \partial_ja_k\delta_{ik} - \partial_ka_i\delta_{ij}\\
2\Gamma^k_{ij} &= a_k^{-1}(\partial_ia_k\delta_{jk} + \partial_ja_k\delta_{ik} - \partial_ka_i\delta_{ij}).
\end{align*}
This is zero if $i, j, k$ are different from each other. Since the Christoffel symbol is symmetric in $i$ and $j$, it now suffices to consider the following cases:
If $i = j = k$, then
$$
2\Gamma^i_{ii} = a_i^{-1}\partial_ia_i.
$$
If $i= j \ne k$, then
$$
2\Gamma^k_{ii} = -a_k^{-1}\partial_ka_i.
$$
If $i \ne j = k$, then
$$
2\Gamma^k_{ik} = a_k^{-1}\partial_ia_k.
$$
