Im studying Multivariable calculus from this video: Lec 6 | MIT 18.02 Multivariable Calculus, Fall 2007 (https://www.youtube.com/watch?v=0D4BbCa4gHo&index=6&list=PL4C4C8A7D06566F38&pbjreload=10)
he define a function called s(t) which is a distance travelled along the arc length at minute 17:00
then at minute 23:00 he defines the velocity vector written in the form:
$$\vec{v} = \frac{ d \vec{r}}{ds}*\frac{ ds}{dt} = \hat{T} * |\vec{v}|$$
then he said:
$$ \Delta{\vec{r}} \approx \hat{T} \Delta{s}$$
then at minute 26:12 - 26:22 only in this 10 seconds i think what is said is wrong.
he said "This vector here I will call delta R, the length of this vector is delta S" this is wrong and to be correct he have to say "the length of this vector is approximately delta S"
because if S is a function defined by the distance travelled at time t then $$\Delta{s} = s(t + \Delta{t}) - s(t)$$
which in english is "distance travelled at time t plus delta t minus distance travelled at time t" the word "distance travelled" is measured from the arc not from the straight line. The vector $\Delta{\vec{r}}$ is measured in straight line from point $r(t+ \Delta{t})-r(t)$ which will not have the length equal to $\Delta{s}$ in the case that the trajectory is curved.
i know that after he takes the limit as $\Delta{t} \to \infty$ of :
$$\frac{ \Delta{\vec{r}}}{ \Delta{t} } \approx \hat{T} * \frac{ \Delta{s}}{ \Delta{t} }$$
you get:
$$\vec{v} = \hat{T} * |\vec{v}|$$
but I just want to really understand it in a correct way. Am I understand correctly that he should say "the length of vector delta R is approximately delta S"? which also means before taking the limit in this formula:
$$ \Delta{\vec{r}} \approx \hat{T} \Delta{s}$$
we multiply $\hat{T}$ by the length of the arc travelled from point $r(t)$ to $r(t+\Delta{t})$ not the length of the vector $\Delta{\vec{r}}$