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.
ach, i just wrote out a post and somehow it is gone. i'll try again.
it is interesting that you post that picture, it is pretty much the same outline of the contour map i tried to link to.
i think i may have figured it out.
going straight from P to Q in that picture, the actual distance traveled is longer than the path, because the altitude does a lot of up and down, where as the path on the right is constant.
so, looking at it as rise / run, while the run from P to Q may be shorter if you are a bird, on foot it is actually longer than using the meandering path on the right.