Sorry again for all these ongoing question as I try to fix my math deficiencies. (Back to working on differential forms.) So... I understand that the equation of steepest ascent/descent in Cartesian coordinates is written: dxi/dt = ∂f/∂xi And I can follow the "physical interpretations" of how the right side is in the path of the greatest change IN Cartesian coordinates. But we know this cannot hold in cylindrical coordinates: the left is contravariant and the right is covariant. So we can convert the covariant term on the right, with the metric tensor... dui/dt = gij∂f/∂uj OK, I get all that... But it seems a contrived way to get the form of the gradient used in this equation. Is there some more objective way that one can state that... ∇fi = gij∂f/∂uj... while providing an interpretation of this nabla? Or should I just content myself with knowing.... "all things flow from the Cartesian... get it there first and then use the metric tensor for all other coordinates" It seems really contrived to base the whole thing on Cartesian and not be able to give an interpretation of gij∂f/∂uj as a direction of greatest change.