在既坐标下，欧氏度量可表示为
\[
ds^2=dr^2+r^2d\theta^2.
\]
可见，$e_1=\partial_r$, $e_2=r^{-1}\partial\theta$是一个幺正基, 其对偶基为$\omega^1=dr$, $\omega^2=rd\theta$。我们知道，局部库伦规范的条件是
\[
d^*A=0,\quad A=A_rdr+A_\theta d\theta=A_r\omega^1+rA_\theta\omega^2.
\]
直接计算可得
\begin{align*}
-d^*A&=\left( \nabla_{e_i}A \right)(e_i)=\left( \nabla_{e_i}(A_j\omega^j) \right)(e_i)\\
&=\left( \nabla_{e_i}A_j\omega^j+A_i\nabla_{e_i}\omega^j \right)(e_i)\\
&=\nabla_{e_i}A_i-A_j\omega^j(\nabla_{e_i}e_i).
\end{align*}
又，我们知道，在极坐标下直接计算得到克里斯托弗符号$\Gamma_{ij}^k$,
\[
\Gamma_{ij}^k=\frac{1}{2}g^{kl}\left( \partial_ig_{lj}+\partial_jg_{il}-\partial_lg_{ij} \right)=
\begin{cases}
-r,&i=j=\theta\,,k=r\\
r^{-1},&i=\theta\,,j=r\,,k=\theta\\
r^{-1},&i=r\,,j=\theta\,,k=\theta\\
0,&\text{其他}
\end{cases}
\]
这样，我们得到
\begin{align*}
\nabla_{\partial_r}\partial_r&=\Gamma_{rr}^r\partial_r+\Gamma_{rr}^\theta\partial_\theta=0\\
\nabla_{\partial_\theta}\partial_\theta&=\Gamma_{\theta\theta}^r\partial_r+\Gamma_{\theta\theta}^\theta\partial_\theta
=-r\partial_r\\
-d^*A&=\nabla_{e_i}A_i-A_j\omega^j(\nabla_{e_i}e_i)\\
&=\partial_rA_r+r^{-1}\partial_\theta(rA_\theta)
-A_rdr(-r\partial_r)-(rA_\theta)(r^{-1}d\theta\left( -r\partial_r \right))\\
&=\partial_r+A_r\partial_\theta A_\theta+rA_r.
\end{align*}
最终，我们得到库伦规范在极坐标下
\[
0=-d^*A=\partial_rA_r+\partial_\theta A_\theta+rA_r.
\]