I'm starting to learn how to intuitively interpret the directional derivative, and I can't understand why you wouldn't scale down your direction vector $\vec{v}$ to be a unit vector.
Currently, my intuition is the idea of slicing the 3D graph of the function along its direction vector and then computing the slope of the curve created by the intersection of the plane.
But I can't really understand how the directional derivative would be a directional derivative if it were not scaled down to be a change in unit length in the direction of $\vec{v}$. Is there an intuitive understanding I can grasp onto? I'm just starting out so maybe I haven't gotten there yet.
Note, I think there may be a nice analogy to linearization, like if you take "twice as big of a step" in the direction of $\vec{v}$ , then the change to the function due to the change in this step is twice as big. Is this an okay way to think about it?
Best Answer
The intuition I think of for a directional derivative in the direction on $\overrightarrow{v}$ is that it is how fast the function changes if the input changes with a velocity of $\overrightarrow{v}$. So if you move the input across the domain twice as fast, the function changes twice as fast.
More precisely, this corresponds to the following process that relates calculus in multiple variables to calculus in a single variable. In particular, we can define a line based at a point $\overrightarrow{p}$ with velocity $\overrightarrow{v}$ parametrically as a curve: $$\gamma(t)=\overrightarrow{p}+t\overrightarrow{v}.$$ This is a map from $\mathbb R$ to $\mathbb R^n$. However, if $f:\mathbb R^n\rightarrow \mathbb R$ is another map, we can define the composite $$(f\circ \gamma)(t)=f(\gamma(t))$$ and observe that this is a map $\mathbb R\rightarrow\mathbb R$ so we can study its derivative! In particular, we define the directional derivative of $f$ at $\overrightarrow{p}$ in the direction of $\overrightarrow{v}$ to be the derivative of $f\circ\gamma$ at $0$.
However, when we do this, we only see a "slice" of the domain of $f$ - in particular, we only see the line passing through $\overrightarrow{p}$ in the direction of $\overrightarrow{v}$. This corresponds to the notion of slicing you bring up in your question. In particular, we do not see any values of $f$ outside of the image of $\gamma$, so we are only studying $f$ on some restricted set.