I understand the concept of the gradient being a vector of the partials of f with respect to each variable, so essentially the gradient gives you a direction in the input field to travel in order to get the maximum increase in the function f. What I don't understand is why the magnitude of that gradient is the actual maximum rate of change of f - it feels right that it should be but I can't quite join the dots and see a proper reason for it.
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If $v$ is any vector of $\mathbb R^2$, $\partial _v f=\nabla f\cdot v$. Therefore, $|\partial _v f|$ is maximal whenever $v$ and $\nabla f$ have the same direction. – Surb Mar 10 '21 at 13:28
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Ok thank you those pages do help a bit - I am somewhat feeling that it is circular to think that the slope is the directional derivative in the direction of the gradient of f though as that relies on understanding that the magnitude of the gradient is the slope, which is what I am unsure about – edwardk Mar 10 '21 at 15:19
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@edwardk Do you have an idea of slope that's different than the directional derivative that you are not sure whether or not it's equivalent? – Mark S. Mar 10 '21 at 15:27
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@MarkS. Yes I am not quite sure how it works - the directional derivative is the ratio between a small increase along a certain vector in the input and the output of f I think and the slope is very much the same thing just going along a vector in the input field that leads to the greatest input in f. What I don't see is how dotting two vectors in the input field gives a rate of change of the output of f, I think that stems from me not seeing why the slope is the magnitude of the gradient so that is what I am asking. – edwardk Mar 10 '21 at 16:08