A <voltage> with boundary values $u(a)$ and $u(b)$ is a function $u:V\to\mathbb R$ that is harmonic at every other vertex:
$$
\sum_yc(x,y)(u(x)-u(y))=0,
\qquad x\notin\{a,b\}.
$$
A <current flow> is an antisymmetric function $i(x,y)=-i(y,x)$ satisfying Kirchhoff's node law $\sum_yi(x,y)=0$ away from its source and sink. Voltage and current are related by <Ohm's law>,
$$
i(x,y)=c(x,y)(u(x)-u(y)).
$$
The <effective resistance> is the voltage drop divided by the total current from $a$ to $b$. Equivalently, it is the voltage drop generated by a unit current flow:
$$
R_{\mathrm{eff}}(a,b)=\frac{u(a)-u(b)}{\sum_yi(a,y)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 3 / a / Solution

{title2=$u:V\to\mathbb R$}

A voltage is a scalar potential on the vertices of an <electrical network>. Away from prescribed boundary vertices it is harmonic, and its difference across an edge drives current through that edge.

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