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I finally came up with a reasonable justification for not requiring the reference vector of the directional derivative to be of unit length. But that is another discussion.

This has bothered me for a long time. Partial derivatives in Cartesian form are directional derivatives with respect to the standard (unit) basis vectors. If I rotate my coordinate system so that one axis shares the direction in which I am differentiating, the partial derivative with respect to the corresponding coordinate is a directional derivative with respect to the new (unit) basis vector.

If I want to know how much a scalar potential changes with respect to ark length in a given direction, I find the dot product of the gradient with a unit vector in that direction.

In some introductory calculus books the directional derivative is defined to be with respect to a unit vector. There are many places in which the directional derivative with respect to a unit vector is valuable. I there a good name for that special kind of directional derivative?

I believe some authors call the directional derivative operator with respect to a unit vector the unit vector. In other words the vector is the operator. But that makes my head spin.

Steven Thomas Hatton
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  • As far as I know, there isn't a special name for the directional derivative when using a unit vector. In fact, I believe it is the modal choice for directional derivatives. That said, I have seen other literature use non unit vectors, but I was of the opinion that this is unusual. I could be wrong, however. – Joel Nov 16 '17 at 04:40
  • Not according to Misner, Thorne and Wheeler; Bernard Schutz and C. H. Edwards, Jr. The directional derivative is $D_{\Delta\mathfrak{x}}f\left[\mathfrak{x}\right]=\lim_{t\to0}\frac{f\left[\mathfrak{x}+t\Delta\mathfrak{x}\right]-f\left[\mathfrak{x}\right]}{t}$, according to them. – Steven Thomas Hatton Nov 16 '17 at 05:09
  • Oh well. I guess I was wrong then. – Joel Nov 16 '17 at 05:48
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    Wrong only in so much as the "big boys" stole your ball. The first definition of the directional derivative I learned was yours. AKA the sane one. – Steven Thomas Hatton Nov 17 '17 at 04:28
  • Can you please point to me the discussion about the justification (of not requiring unit vector length)? I am also confused. Many thanks! – bruin Oct 30 '19 at 13:40

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