4-Vector Gradient and Contravariant Derivative I am self-studying General Relativity, and the course of study I am following has started to introduce me to index notation. The texts I am using (Carroll, Schutz) begin with a geometric slant on Special Relativity, and I am finding the index notation a bit of a challenge. From my textbooks, $\eta_{\mu\nu}$ = diag (-1,1,1,1) and I understand from web searches that:
For SR:
$$
\eta_{\mu\nu} = \eta^{\mu\nu}
$$
and
$$
\partial^\mu = \eta^{\mu\nu} \partial_\nu
$$
However, a lot of the stuff I have found on the web seems to use $\eta_{\mu\nu}$ = diag (1,-1,-1,-1) (which I am finding a bit confusing tbh) and states that for a scalar field $\phi(t,x,y,z)$,
$\partial_\mu \phi= \frac{\partial\phi}{\partial x^\mu} = (\frac{1}{c} \frac{\partial \phi}{\partial t}, \nabla)$
and
$\partial^\mu \phi = \frac{\partial\phi}{\partial x_\mu} = (\frac{1}{c} \frac{\partial \phi}{\partial t}, -\nabla)$
So I am thinking that with $\eta_{\mu\nu}$ = diag (-1,1,1,1), I'm looking at
$\partial^\mu \phi = \frac{\partial\phi}{\partial x_\mu} = (\frac{-1}{c} \frac{\partial \phi}{\partial t}, \nabla)$
Is this anywhere near the mark? Please be gentle with me, as all of this notation is very new, and I am not a physics student, just an (older) interested amateur who is struggling with new concepts and new notation.
 A: Yes, you are correct. The "real" derivative has a downstairs index because that's just the way derivatives transform, so we always have
$$\partial_\mu = \left(\frac{1}{c} \partial_t, \nabla\right)$$
and $\partial^\mu = \eta^{\mu\nu} \partial_\nu$, because that's what index raising means. So with some simple matrix multiplication we see that in the $(+ - -\, -)$ convention we have
$$\partial^\mu = \left(\frac{1}{c} \partial_t, -\nabla\right)$$
while in the $(- + +\, +)$ convention we have
$$\partial^\mu = \left(-\frac{1}{c} \partial_t, \nabla\right).$$
A: No worries, you're doing great! I'm glad you're self-studying these things, that takes a lot of courage. The signature of the metric is not important, all that matters is that we are consistent. So whether the signature is + or - (Which means whether the trace of the metric is positive or negative), our physics will be the same.
I initially did this wrong, I had a brain slip up and wrote something very incorrect, as some commenters have pointed out this is more accurate (for the $(-1,1,1,1)$ signature):
$$
\partial^0 \phi = \eta^{0 \nu} \partial_\nu \phi = \eta^{00} \partial_0 \phi = - \partial_0 \phi \\
\partial^i \phi = \eta^{i j} \partial_j \phi = \delta^{i j} \partial_j \phi = \partial_i \phi
$$
Thank you everyone who pointed out my mistake, I hope this is more clear.
