Exterior derivative definition in Nakahara (Geometry, Topology, Physics)

Question

On page 198 Nakahara defines the exterior derivative on an $r$-form $\omega = \frac{1}{r!} \omega_{\mu_1\ldots \mu_r} dx^{\mu_1} \wedge \ldots \wedge dx^{\mu_r}$ as $$d_r\omega = \frac{1}{r!}\left(\frac{\partial}{\partial x^\nu} \omega_{\mu_1 \ldots \mu_r}\right) dx^\nu \wedge dx^{\mu_1} \wedge \ldots \wedge dx^{\mu_r}.$$ Also in the introduction it is written that the summation convention only applies to indices that appear (at least) once as a lower and once as an upper index. This confuses me in this definition. For me this only makes sense if $\nu$ is summed over as well. Where is my mistake?

Answer 1

Note that the $\nu$ in $\frac{\partial}{\partial x^{\nu}}$ is considered to be a lower index as it occurs in the 'denominator'.