I'm trying do a conection between a definition that I've learned in my graduation and nowadays a definition that I've learned in my doctorate studies.
Calculus Definition:
Let $C$ be a curve in $\Bbb{R}^n$ and $r:I\to\Bbb{R}^n$ a parametrization for it, let $f:C\to\Bbb{R}$ a continuous function, then the line integral of $f$ over $C$ is defined by $$\int_Cf\,ds=\int_I(f\circ r)(t)\,|r'(t)|dt.\qquad(1)$$
Manifold definition:
Let $M$ be a (compact) smooth manifold, let $\phi:U\subset\Bbb{R}^n\to \phi(U)\subset M$ a local defining function (maybe inverse of a chart map), let $\omega$ a $n$-form on $M$ (with compact support in $\phi(U)$), then $$\int_M\omega=\int_U\phi^*\omega,\qquad(2)$$ where $\phi^*$ is the pullback map.
I want to know how to prove $(1)$ using $(2)$, I mean, consider $M=C$, $\phi=r$ and $\omega=f\,ds$, how can I obtain $(1)$?
Sugestions, commets and answers will useful, thanks!
We define the integral by $(2)$, let $C\in\mathbb{R}^n$ be a smooth curve parametrized by $r$, and let $f:C\to\mathbb{R}$. In order to use $(2)$ we need to define a $1$-form $\omega$ on $C$. We take the orientation induced by $r$, and then for $p\in C$ and a positive $v\in T_pC$ we set $\omega(v)=f(p)\cdot|v|$. It follows immediately that the pullback is given by $$r^*\omega=f\circ r(t)\cdot|r'(t)|dt,$$ and we're done.