Line integral - why is $ds= \dfrac{ds}{dt}dt$?

Question

I’ve been reading on my own about this because it’s on the syllabus of the next semester and I’m very confused on something about the line integral.

I have the curve C with parametrization $C(t)=(x(t),y(t)),$ $a\leq t\leq b.$ Then

$$\int_C f(x,y)ds= \int_a^b f(x(t),y(t))\dfrac{ds}{dt}dt.$$

I am okay until they do this: $ds= \dfrac{ds}{dt}dt.$ i don’t understand this. Is it the chain rule?

I was reading Zill’s calculus but it’s not clear for me. Sorry if I’m asking very poorly, and I’m sorry if my questions is because I lack a better foundation in vector theory.

Thank you for reading.

Answer 1

I am okay until they do this: $ds= \dfrac{ds}{dt}dt.$. i don’t understand this

Its needed to put the integral into terms of the variable t since the variables x and y are functions of t and also the limits of integration are in terms of t. Think of ds/dt as the Jacobian.