a very basic geometry question about the length of the curve

Question

"the length of the curve is the sum of all its tangent vector"

This is what I heard but not able to find in any books.

Pictorially, what does this mean?

Formally, what would be a proof for this?

Answer 1

Think physics here. The average speed of a particle that goes a distance $\Delta s$ in time $\Delta t$ is $v= \Delta s/\Delta t$. Given a curve $\alpha: I \to \Bbb R^n$, we call the tangent vector $\alpha'(t)$ the velocity vector, and $|\alpha'(t)|$ is the speed of the curve. So if $\color{red}{\Delta s} = \color{blue}{v} \color{green}{\Delta t}$, we have $\color{red}{\ell(\alpha)} = \int_I \color{blue}{|\alpha'(t)|}\,\color{green}{{\rm d}t}$.

(Bear in mind that I'm no physicist and this explanation is just meant to be intuitive and make an analogy, so take it with a grain of salt.)