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My doubt is suppose we assume a 3D space with 2D surface in it given by some function z = f(x,y). Then each component of the gradient is geometrically the slope of the tangent at f on either x-z or y-z 2D plane which is basically a scalar. So mathematically or geometrically how does each of these components make a vector that gives a direction of steepest ascent and the slope of the surface at that point?

One intuition that I have thought of is since these two components (scalar values) are in perpendicular planes they should be combines using pythagoras formula to give resultant slope, but this doesn't seem sufficient explanation.

Rohit Kumar Singh
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  • See https://math.stackexchange.com/questions/1912660/gradient-of-a-function-as-the-direction-of-steepest-ascent-descent for several explanations. – amd Feb 17 '19 at 19:52

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By definition, the gradient is the vector whose components are derivatives on each coordinates, for functions in which it can exist.

The derivative of a function at a point $x$ represents the slope of the function, the director coefficient $f'(x)$ of the slope, actually. So, yes, by definition, it represents the slope on each axis.

About the steepest ascent, well, on each component, you sum the slopes, and you get the 3D slope to the steepest ascent.

PackSciences
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  • "on each component, you sum the slopes" but we actually use pythagoras formula to get slope of the 2D surface right? Still one important question is why these slopes make a vector that will point in the direction of steepest ascent? – Rohit Kumar Singh Feb 17 '19 at 13:02
  • You CAN use the Pythagorean Theorem on a Hilbert Space to get the slope of the 2D surface, yes. "why these slopes make a vector that will point in the direction of steepest ascent?" That's what I explained in my answer, not sure what I can add more. – PackSciences Feb 17 '19 at 13:04