Finite increments of a logarithmic radial function
= Finite increments of a logarithmic radial function
{title2=$s=r_0(e^{\Delta/2}-1)$}
For $T(\mathbf r)=\ln|\mathbf r|^2$, starting at radius $r_0>0$ and moving radially outward through distance $s$ changes the function by $\Delta=2\ln(1+s/r_0)$. Solving yields the displayed exact distance. Its small-increment limit is $s\simeq r_0\Delta/2$, which is the initial <directional derivative> estimate. That linear estimate is not an exact equality for a specified finite change.