# Laplacian on 1-forms on $(\mathbb R^m,\delta)$

I'm trying to show that if $$\omega$$ is a 1-form on $$(\mathbb R^m,\delta)$$, the action of the Laplacian is given by $$\Delta\omega=-\sum_{\mu=1}^m\frac{\partial^2\omega_\nu}{\partial x^\mu\partial x^\mu}dx^\nu$$ What I've calculated is, for a general metric $$g_{\mu\nu}$$, $$dd^\dagger\omega=(-1)^{m+1}\partial_\lambda\partial_\nu(\sqrt{|g|}\omega_\mu g^{\mu\nu})dx^\lambda$$ $$d^\dagger d\omega=(-1)^m\partial_\lambda(\partial_\mu\omega_\nu\sqrt{|g|}g^{\mu\lambda}g^{\nu\sigma})g_{\sigma\rho}dx^\rho$$ When I then specify the Euclidean metric, I get the desired term (up to a factor $$(-1)^m$$) from $$d^\dagger d\omega$$, but $$dd^\dagger\omega$$ is superfluous/of the wrong form - namely, I get $$dd^\dagger\omega=(-1)^m\frac{\partial^2\omega_\nu}{\partial x^\nu\partial x^\mu}dx^\mu$$ I'm not really sure what I'm doing wrong here, but any help would be appreciated.