This is going to take a bit of explanation. I am not sure how to ask this question. The math lecture said the gradient points to the direction of greatest increase and if you want to know the direction of greatest decrease then you take the negative.
My intuition tells me that the gradient is either the direction of greatest increase or decrease depending on the function at the particular point you are evaluating. It could be pointing down or up since after all it is nothing more that the derivative of a function.
On the other hand what ever that turns out to be then if you apply the negative it will simply go in the opposite direction.
So I am at odds with the lecture here. By default my intuition tells me the gradient could turn out to be greatest increase or greatest decrease and which ever it is then the negative will reverse the direction.
Thank you
Remember that the gradient exists in the domain. It does point in the direction of the greatest increase and its magnitude is the slope (rate of change) of the tangent line in that direction. To find the slope, aka directional derivative, in any direction, dot the gradient with a unit directional vector. If the directional vector is in the opposite direction of the gradient, then the dot product will equal the negative of the magnitude of the gradient.
Note that if (and only if) the direction is orthogonal to the (non-zero) gradient, the directional derivative is zero. This is why level curves/surfaces are always orthogonal to the gradient.
Hope this helps.
Ced