MTW defines the directional derivative of a vector along a parametrized
curve as:

lim(e-0) (v(e)-v(0))/e (page 209)

The problem is that the size of the resulting vector depends on your
parametrization of the curve. If you multiply your affine parameter with
a certain number, the size of your directional derivative vector also
changes. My question is: shouldn't there be some kind of correction for
the parametrization of the curve, like making the tangent vector of the
curve an unit vector? Mathworld
(http://mathworld.wolfram.com/DirectionalDerivative.html) says you have
to use an unit vector, but I couldn't find that in MTW.

Amantine

Amantine wrote:
MTW defines the directional derivative of a vector along a parametrized
curve as:
lim(e-0) (v(e)-v(0))/e (page 209)
The problem is that the size of the resulting vector depends on your
parametrization of the curve.

Yes, this is inherent -- in physics such derivatives are dimensioned
numbers.


If you multiply your affine parameter with
a certain number, the size of your directional derivative vector also
changes. My question is: shouldn't there be some kind of correction for
the parametrization of the curve, like making the tangent vector of the
curve an unit vector?

No. Consider a timelike curve parameterized by proper time, and then
imagine a units change (e.g. seconds to minutes) -- the directional
derivitave wrt proper time MUST change in scale accordingly (the value
of X per second becomes 60*X per minute, where X depends on what you are
differentiating).


Mathworld
(http://mathworld.wolfram.com/DirectionalDerivative.html) says you have
to use an unit vector, but I couldn't find that in MTW.

That's a difference between math and physics: units.

But this is also a terminological quirk: the MTW definition is really a
derivative wrt the path parameter, not truly wrt a "direction". And a
direction is by convention represented by a unit vector.

So it all boils down to: "what do you want to do?", and different
authors answer that with similar but different answers.


Tom Roberts