Directional derivatives and non-unit vectors

[Rasalhague]

Lee: Introduction to Smooth Manifolds, definition A.18:

Now suppose $f : U \rightarrow \mathbb{R}$ is a smooth real-valued function on an open set $U \subseteq R^n$, and $a \in U$. For any vector $v \in \mathbb{R}^n$, we defi ne the directional derivative of $f$ in the direction $v$ at $a$ to be the number

$$D_vf(a)=\frac{\mathrm{d} }{\mathrm{d} t} \bigg|_0 f(a+vt). \enspace\enspace(A.18)$$

(This de nition makes sense for any vector v; we do not require v to be a unit vector as one sometimes does in elementary calculus.)

He then shows, by the chain rule, that

$$D_vf(a_0)= \sum_{i=1}^n v^i \frac{\partial }{\partial x^i}f(a) \bigg|_{a_0}$$

It seems to me, though, that this number depends not only on the direction of $v$ but also on its length. For example if $f(x,y,z) = xyz$, and $v=(1,0,0)$, then

$$D_vf(2,3,4) = \begin{pmatrix}yz & xz & xy \end{pmatrix} \bigg|_{(2,3,4)+0(1,0,0)} \begin{pmatrix}1\\0\\0\end{pmatrix} = 12.$$

But if $w=(2,0,0)$, then the directional derivative of $f$ "in the direction of $w$" (which is the same direction as the direction of $v$) will be

$$D_vf(2,3,4) = \begin{pmatrix}yz & xz & xy \end{pmatrix} \bigg|_{(2,3,4)+0(1,0,0)} \begin{pmatrix}2\\0\\0\end{pmatrix} = 24.$$

So how does the definition make sense for any vector? What am I missing here?

 

[Hurkyl]

It seems to me, though, that this number depends not only on the direction of $v$ but also on its length.
...
So how does the definition make sense for any vector? What am I missing here?

It's not that you're missing something, but you're adding in something incorrect -- the hypothesis that the directional derivative doesn't depend on length.

The directional derivative is, in fact, a linear function of v.

 

[HallsofIvy]

Perhaps an application would be clarify things. Suppose you are moving along a surface measuring air temperature. If you require that the direction vector, v, be a unit vector "one sometimes does in elementary calculus" (that is, you always move with speed 1), then the measured rate of change of temperature will depend only on the direction . But since, here, he is NOT requireing that v be the unit vector, it will depend on both direction and speed of travel.

 

[Rasalhague]

Ah, okay. Thanks Hurkyl. So is the following correct?

When the vector is required to have unit length, the number at called "the directional derivative of f at a in the direction of v" lives up to its name directional, and tells us the rate of change of the function in that direction; otherwise, it's just a number that could be any number, and so doesn't, by itself, tell us anything about f. Only if we know the function used to produce this number, or know the length of the vector used to produce it, can we tell anything about the behaviour of f.

Ah & aha, thanks HallsofIvy; I was just previewing this before posting and saw your example. Yes, the relaxation of the unit length requirement seems less arbitrary now.