Doubt about the norm of the gradient vector Is it possible to say that the gradient vector norm at a point $ (x, y, z) $ is the smallest rate of change? Considering a function $ f (x, y, z) $
 A: At a point $p = (x,y,z)$, look at all the unit vectors $u$ with their tails at $p$. We can compute the instantaneous rate of change of $f$ in the direction $u$ by taking the following derivative:
$$
\frac{d}{dt}\bigg|_{t=0}f(p + tu) = \nabla f(p) \cdot u.
$$
The equality is an exercise in the chain rule. But using this formula, we can see that the rate of change is a scalar, as expected, and it is largest (that is, most positive), when the vectors $\nabla f(p)$ and $u$ point in the same direction. That tells us that the direction of the gradient vector $\nabla f(p)$ would be the direction of most positive change, and the direction of $-\nabla f(p)$ would be the direction of most negative change.
If we pick a direction $u$ perpendicular to $\nabla f$, then the instantaneous rate of change of $f$ would be $0$ in the direction of $u$ (another consequence of the formula above, and the geometric interpretation of the dot product). Depending on how you use the language, a direction perpendicular to the gradient could fairly be said to be a direction of "smallest change" as you said in your post, because this is a direction where the change is smallest in magnitude.
The ambiguity in the way we use the words greatest and smallest makes us be more precise in order to clarify the picture we have in mind, or the idea we want to convey.
