# $s(t)=\int_{t_0}^{t}\left | R'(\xi) \right |d\xi$ What is $\xi$

$s(t)=\int_{t_0}^{t}\left | R'(\xi) \right | d\xi$
What is $\xi$ ?

In the above arc length formula with ##\xi##, what is ##\xi##?

A variable. R' seems to be a function of $\xi$.

I found this equation while studying vector calculus/ multivariable calculus. I was studying about binormal, tangent, normal, frenet frame, etc.

I just don't understand ##\xi## because the variables usually involved were t and s

$\xi$ is just a mute variable. You can't use t inside the integral because t is used on the boundary of the interval. So $\xi$ is just a normal variable.

I found this equation while studying vector calculus/ multivariable calculus. I was studying about binormal, tangent, normal, frenet frame, etc.

I just don't understand ##\xi## because the variables usually involved were t and s

t and s are just letters. You can denote a variable by whatever symbol you want.

I guess ##\xi## is just t, because the integral is from ##t_0## to ##t## so it just has to be t in the sense that it represents time...
I just found it confusing why he suddenly put ##\xi## when all along he was talking about t, s, velocity, binormal, and Frenet frames of vectors
Thanks now I understand !

Last edited:
cepheid
Staff Emeritus
Gold Member

R and s are both functions of time, but s is defined in terms of an integral, and represents the cumulative area under |R'|, up to time t. So, what makes s a function of t is the fact that the upper limit of the integral is a variable (t), not a constant. So, t is the variable corresponding to the time value up to which you integrate. It would be poor notation (and just wrong) to use the same symbol (t) for the other time variable that corresponds to all the particular times at which R is evaluated. So the time values with respect to with R varies are instead given another symbol, xi. In this context, xi is an example of what is termed a dummy variable of integration:

http://mathworld.wolfram.com/DummyVariable.html