Question

Why does the gradient point in the direction of maximum increase?

Answer 1

Maybe an example will help. F(x,y)=x2+y2F(x,y)=x2+y2. Now the gradient would look like <2x,2y>or2xi+2yj<2x,2y>or2xi+2yj where ii and jj are unit vectors of the xx and yy axis. This vector field spans every quadrant. The gradient is sort of a place holder to build a vector field.

Certainly it does not point in one direction so in what sense is it pointing in the maximum direction? The maximum direction of what?

If you don't mind a little extra question on the side, but relevant, since it was the source of my original confusion. Is directional derivative is a dot product with the gradient and a unit vector pointing in any direction you choose but if the angle between the gradient and the unit vector is 0 , the cosine is 1 so that means that unit vector was pointing in the direction of the gradient ?

Since the directional derivative is a number only of what utility is it other than this observation ? I may be missing the boat on directional derivative.