Question

Hi, I'm having trouble with session 38. I don't understand why travelling at unit speed, makes s arclength. Also, why does he call dr/ds a velocity vector?

Answer 1

At the start the Prof. states, "Why do we use 's' instead of 't'? It's a convention."

We move along a line at speed ds/dt (distance along the line divided by time)
but this is assumed to be unit speed: ds/dt = 1
and we can think of this as saying ds = dt (ignoring units)

Thus the argument (apparently) is dr/dt is numerically the same as dr/ds. It might be interesting to look up the history of how this syntax arose.

Answer 2

The pdf in session 38
http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-38-directional-derivatives/MIT18_02SC_notes_20.pdf

seems to be more clear (at least to me).

It is an extension of dw/dx to dw/ds where "s" is distance in some arbitrary direction.
rather than along the e.g. x-axis.

Answer 3

Thanks so much for helping! It makes a lot sense, now.