Somewhere I saw that the gradient of a function points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. I don't understand it at all. Can somebody help me to understand it? Please give some example?
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Possible duplicate of https://math.stackexchange.com/questions/223252/why-is-gradient-the-direction-of-steepest-ascent – saulspatz Sep 06 '18 at 05:44
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You might also have a look at https://math.stackexchange.com/q/1912660/265466. – amd Sep 06 '18 at 06:39
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The link I gave has very good answers based on the directional derivative that prove the gradient points in the direction of steepest ascent, using directional derivatives. What has made this result intuitive for me is another result: the gradient is perpendicular to the level curve. Imagine standing on a hillside, and visualize the curve passing through all points at the same altitude as the point you're standing at. It just seems clear to me from experience, that the steepest climb is perpendicular to that curve (the direction of the gradient), and that if if you were to set a ball on the ground and let it go, it would roll downhill perpendicular to that curve (in the direction of the negative of the gradient.)
The link I gave proves the theorem for a function of $3$ variables, but the same proof works in any number of dimensions of course.
saulspatz
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