Beckmann potential ($V(y)={\sum }_j{\int }_0^{y_j}{D}_j(u)du$)

For route flows $x\ge 0$, let the link-route incidence matrix $A$ give link loads $y=Ax$. Let the source-sink route incidence matrix $H$ enforce fixed demands $Hx=f$. With continuous nondecreasing link delays $D_j$, the Beckmann potential isIts route derivative is ${\partial }_{r}V={\sum }_j{A}_{jr}{D}_j(y_j)$, precisely that route's delay. Convexity follows from monotonicity of each $D_j$. The Karush-Kuhn-Tucker conditions therefore say that used routes have the minimum delay for their source-sink pair, while unused routes have no smaller delay: the Wardrop equilibrium conditions.^[ref]

This potential generally differs from total travel delay ${\sum }_j{y}_j{D}_j(y_j)$. For two parallel links with delays $D_1(y)=y$, $D_2(y)=1$ and total demand two, minimizing $V=x_1^2/2+x_2$ gives $x_1=x_2=1$. Both used routes have delay one. Minimizing total delay $x_1^2+x_2$ instead gives $x_1=1/2$, $x_2=3/2$. The marginal external cost toll aligns the two optimization problems.