Derivative Notation Question (Contravariant vs. Covariant) I know how to write a covariant derivative in Leibniz notation:
$$\partial^\mu\equiv \frac{\partial}{\partial x^\mu}$$
Does that mean that a contravariant derivative in Leibniz notation would be
$$\partial_\mu\equiv \frac{\partial}{\partial x_\mu}\tag{Does this expression even make sense?}$$
Another guess: $\partial_\mu\equiv dx_\mu$. The idea is to not have to write something like this:
$$g_{\mu\nu}\frac{\partial}{\partial x^\mu}$$
Is there a shorthand that does not write out the metric tensor explicitly, i.e. with the raising and lowering of indices? or does this not exist, since no one writes einstein-notation partials in Leibniz form?

According to my Electrodynamics book, 
$$\partial _{\mu} = g_{\mu \nu}\partial ^{\nu}$$
where $g_{\mu \nu}$ is metric tensor.
