If I have a function $f(x,y)$ and take the gradient at $(a,b)$ then
$\vec{v} = \nabla f(a,b) = (\frac{\partial f(a,b)}{\partial x},\frac{\partial f(a,b)}{\partial y} )$
Clarifying Question 1: Because $\vec{v}$ points in the direction of "greatest ascent," then so does $\frac{\vec{v}}{\|\vec{v}\|}$, correct?
Clarifying Question 2: $\|\vec{v}\|$ gives the rate of change of $f$ in the direction of $\vec{v}$, so if $\|\vec{v}\| = 5$ for example, does that mean a one unit change in the direction of $\vec{v}$ would yield a 5 unit change in $f$?
Side question here: Would the gradient be the zero vector if moving in any direction would bring a decrease?
Clarifying Question 3: (Please let me know if my understanding of the concept is correct) if $\vec{u}$ is of the same dimension of $\vec{v}$, the directional derivative $D_{\vec{u}}f$ gives the rate of change of $f$ in the direction of $\vec{u}$, so if $\vec{u} = (3,2), (a,b) = (2,3)$ then the rate of change in the direction of $\vec{u}$ is the same as the rate of change in moving from the point $(2,3)$ towards the point $(5,5)$, also written as $$D_{\vec{u}}f(a,b) = \vec{v} \cdot \frac{\vec{u}}{\|\vec{u}\|}$$
and therefore $\|\vec{u}\|D_{\vec{u}}f(a,b)$ would give the change of $f$ when moving from $(2,3)$ to $(5,5)$
Clarifying Question 4 (last one): if Question 3 is correct, and $\vec{r}$ points in the same direction as $\vec{v}$ then is the following true?
$$\vec{v}\cdot\vec{r} = \|\vec{v}\|\cdot\|\vec{r}\|$$ which gives the change in $f$ when moving from $(a,b)$ to $(a + r_1, b + r_2)$
Best Answer
2, 3, 4. The linear approximation to the change, not the actual change. The linear approximation is only an accurate representation of the change in the limit as the distance one moves goes to $0$. This is just like the linear approximation you learned (perhaps using the word "differential") in your Calculus I class.