What is the relative error $\delta = \frac{|\Delta f|}{|f|}$ in computing the value of a function $f(x,y,z)$ at a point $(x,y,z)$ whose coordinates have absolute errors $\Delta x, \Delta y, \Delta z$?
This question is posed by Zorich, without solution, in his Russian classic on analysis.
Assuming $f$ smooth, I believe the answer is
$$\delta = \left| \frac{1}{f} \left[ f_x\Delta x + f_y\Delta y + f_z\Delta z + \frac {1}{2!} \sum_{i,j \in \{x,y,z\}}f_{ij}\Delta i \Delta j + \frac 1 {3!} \sum_{i,j,k \in \{x,y,z\}}f_{ijk}\Delta i \Delta j \Delta k + ...\right] \right|$$
Is that correct? Is there a "better" answer?