My multivariable calc book has started talking about "rate of change" and "path of steepest ascent"
here is a link to the specific page.
notice the picture of the contour map on the bottom left.
the book's claim is that the path of steepest ascent is when you move from one level curve to the next, taking the shortest possible from one level curve to the next.
notice the curved nature of the "path of steepest ascent" from P to Q and how it does not simply go directly from P to Q, the path that would have the highest rate of change.
I suppose i don't have a question as much as a comment on how odd this seems to me.
if i were to be hiking up a mountain with someone and they pointed out the path of steepest ascent, which was actually a meandering path to the peak and not the steepest actual path, it would not seem right.