This question is extension of my previous question here

Can we actually walk along the gradient of a scalar to climb the hill faster?

In the figure there is one peak of 100m and all other peaks are 50m. We have one very nice narrow flat path to reach top. All other paths are very bad because we have to climb hundreds of small peaks to reach top.

I know that we should walk along the gradient vector to climb the mountain faster to reach the peak soon. I also know that gradient points along the steepest path.

But in the following picture the easiest and fastest way to reach the peak(100m) is by following the flat path on the ground level.

But gradient should point along the steepest slope. If I'm at origin the steepest slope is not along the path, but rather it is along the small peaks.

Am I right?

Is the gradient pointing along the 'flat' path or not? How should I walk?

enter image description here

  • 1
    $\begingroup$ The gradient is just a local tool that gives you the direction with the steepest slope at any point. So, following the gradient does not help you reach the global maximum at all. The gradient just guides you to a local maximum, where you stay because the gradient there is zero. (i.e. it gives you the fastest way to the nearest 50m peak, assuming that the mountains individually are monotonous) The gradient will not "guide" you along the flat path. $\endgroup$ Commented Dec 16, 2021 at 15:34
  • $\begingroup$ That makes sense. So I should not walk along gradient vector then right? $\endgroup$
    – Siddaram
    Commented Dec 17, 2021 at 0:49
  • $\begingroup$ Yes, exactly, you shouldn't. $\endgroup$ Commented Dec 17, 2021 at 0:53


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