Is my analogy of a directional derivative correct?

Question

I find analogies to real life are a good way for me to understand some faucets of multivariate calculus, and a wrong analogy can be an exact and glaring indication of misunderstanding.

As far as I am aware, in 3D space, there is no total derivative because there is an input space rather than line, and so output $z$ no longer depends on one variable alone - the rate of a change of a slice of a surface in Euclidean space depends on which slice of the surface we take. For instance, if a skateboarder, from a bird's eye view, is directly left of a hill that he can ride down and to the right a ramp he can ride up, and if further up the $y$ axis from the skateboarder's location (above and below the hill and ramp respectively) is just flat asphalt, then if $x, y$ and $z$ represents length and the origin is centered at the skateboarder's location, the rate of change of $z$ (elevation) will be different as the skateboarder moves along $x$ at some velocity depending on where he is -- not changing on flat asphalt above him, but changing if moving between the hill and ramp. Because of that, there is no total derivative, and a different one per slice of $y$. And as such, we need partial derivatives -- finding the derivative with a constant $y$ slice and then taking the general case where $y$ is some constant -- then if we have $x$ and $y$ we can compute the derivative at that slice.

Directional Derivatives seem as arbitrary as moving from partial differentiating along some straight line $x$ or $y$ and applying it so some arbitrary vector $\vec v$. I see it as being analogous to the scenario with before, but the ramp being, say, above and to the left of the skater (from a bird's eye view, so $+y$ away and $-x$ away). Taking the directional derivative of a vector causing the skateboarder to go up the ramp will cause a different relationship with $z$ if the velocity vector was some arbitrary vector moving on flat asphalt once again.

Please critique my understanding by pointing out the flaws in my analogies if there are any, which is certainly likely.

Answer 1

You seem to be basically on track. The idea that the slope is different in each direction is exactly why we need directional derivatives and partial derivatives. And partial derivatives are a special case of directional derivatives like you said.

For critiques, the only minor points I can see from your explanation are (1) the term "total derivative" has a special meaning in certain contexts, so beware that. I think a better way to say what you meant is that the derivation be is not described by a single scalar, but rather one for each direction, and (2) if there is a kink where the ramp starts, so that there is a sudden change in slope there, we say it is not differentiable at that point. This is because if the skateboarder moves backward it is a different slope from moving forwards, which is not allowed. This is analogous to having separate right and left hand limits in single variable calc.

Otherwise, you are right on.