A good way to visualize gradient ascent/descent is to assume you are in a quadratic bowl or on a mountain. If I visualize this, then the direction of steepest ascent/descent is the one that points straight towards the bottom of the bowl or top of the mountain.
With this understanding I have two questions:
If you want to climb the hill or go down a bowl, why take a zig-zag path instead of taking a straight path to the top/bottom?
Why doesn't the steepest path have a unit in the
zdirection? I understand that gradient is orthogonal to the level sets of the function. That is it lies in the
x-yplane orthogonal to the contour. But why doesn't it have a unit in the
zdirection? With a unit in the
zdirection, it can point towards the minima/maxima and still be orthogonal to contour lines.
I have a related question: Gradient is NOT the direction that points to the minimum or maximum