What is the relationship between $V(t)$ and $V(x,y,z)$ I was recently asked this by a friend. 
He told me that coming from a physics background, he does not understand $V(t)$ and he believes it is purely theoretical construct made up by circuit theorists.
Because in his mind a voltage $V$ is always a field with $<x,y,z>$ components. You can take the gradient of it, the laplacian of it to get meaningful results.
But in circuit theory, we instead defines a quantity $V(t)$, which conflicts with a lot of results that you would get from elementary physics (i.e. the integral of $E$ field does not result in a time component!)
So, what is the relationship between $V(t)$ and $V(x,y,z)$ and how can this be intuitively understood?
 A: The electric potential is always a function of both spatial position and time, in both circuit theory and electrodynamics.
$V$ is constant along a wire, so in circuit analysis you can take a short cut by specifying the position by just specifying which wire you're talking about, like $V_{1}(t)$, instead of specifying three spatial coordinates.
$V$ doesn't vary with time in electrostatics, so it's possible to drop the $t$ coordinate as long as it's clear from context that you're only dealing with electrostatics.
But $V$ is also meaningful in electrodynamics, in which case you can't ignore any of the coordinates, and have to specify all four of them as in $V(t, x, y, z)$.  What exactly $V$ is considered to be at a given point in time and space depends on your choice of gauge, but that's also true in electrostatics and circuit analysis.
For more about the use of $V$ involving its dependence on all four coordinates, see the Generalization to electrodynamics section of Wikipedia's Electric potential article.
A: When you consider a voltage $V(t)$ in a circuit, you are talking about a voltage at a specific point in the circuit - in other words, implicitly it's $V(t,x,y,z)$ - at a certain time & place.
When you have a static situation (things don't change over time) it's possible to talk about a potential as a function of location: $V(x,y,z)$ describing the voltage at every point in space (or if you like at every pint in the circuit). However, most systems are dynamic, so the voltage evolves over time. To describe the potential at every point in time and space, once again, you have $V(t,x,y,z)$.
There is no conflict - just a different way of thinking about a system. When one or the other is constant, you can omit the corresponding parameter.
A: In circuit analysis we make some assumptions and we use shorthand notations frequently. For example we assume the potential doesn't vary anywhere in the wire, even it was 1 km long. Hence we don't write the spatial coordinates like we do in electromagnetics, in that they complicate the analysis and don't give much accuracy (the dimensions of the wire and components are too small to for the potential to vary significantly on them.)
You might say : "Well, that might not be accurate!" You are right, but we find our ways to approximate. For example we account for the drops in voltages using Ohm's law and the impedance form (more general the resistance form), and that is sufficient. If we want to account for losses we find impedance per unit length and do the calculations.
