If $\phi(x,y,z,t)$ is a scalar field, given the relation between Minkowski Space coordinates $c^2(d \tau)^2 = c^2(d t)^2 - (d x)^2 -(d y)^2 -(d z)^2$, how can $\frac{\partial \phi}{\partial t}$ be expressed as a function of $\frac{\partial \phi}{\partial \tau}$ and other partial derivatives (such as $\frac{\partial \tau}{\partial t}$, $\frac{\partial x}{\partial t}$, etc)?
If this were an ordinary derivative, I could apply the chain rule and write
$\frac{d \phi}{d t} = \frac{d \phi}{d \tau}\frac{d \tau}{d t}$
but the chain rule for partial derivatives is tricky (for me) to use in this particular case because the variables here are not independent, and I am not changing between sets of coordinates either.
By definition $$ \Delta \tau(c) = \int_{c} \sqrt{1 - \frac{1}{c^2} \left(\dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right)} \, \, dt $$ where $\dot{x}$ is shorthand for $\frac{dx}{dt}$ and $c$ is a path through space-time. Can you take it from here?
I don't know a rigorous way to get the correct partial derivatives directly from the metric. For a physicist, the "chain rule" is a good enough explanation (it's the one I got in my GR class).