Curve length and line integral, irrelevancy of parametrization proof I'm trying to prove that $$\int_c^d ||\tilde{\mathbf{r}}'(t)||\, \mathrm{d}t = \int_a^b ||{\mathbf{r}}'(t)||\, \mathrm{d}t$$ where $\tilde{r}: [c,d] \to \mathbb{R}^n$ is a reparametrization of a $C^1$-curve $r:[a,b] \to \mathbb{R}^n$. (The definition of a reparametrization in my textbook is that the bijective function $\phi: [c,d] \to [a,b]$ that lets $r(\phi(u)) = \tilde{r}(u)$ $\forall u \in [c,d]$ and its inverse $\phi^{-1}$ are continuous). I'm also trying to prove that a similar thing applies to the line integral of a continuous vector field $V: \Omega \to \mathbb{R}^n$ $$ \int_c^d \mathbf{V}(\tilde{\mathbf{r}}(t)) \cdot \tilde{\mathbf{r}}'(t)\, \mathrm{d}t = \int_a^b \mathbf{V}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\, \mathrm{d}t $$ I honestly have no idea where to even start. I could imagine that maybe using the chain rule on $(r \circ \phi)'$ would give me somewhere but I can't seem to put it all together.

Suppose $\phi :[c,d]\to [a,b]$ is a $C^1$ increasing bijection. Then $\int_a^b|r'(t)|dt = \int_c^d|r'(\phi(s))|\phi'(s)ds.$ That's just a standard change of variables. In the second integral, the integrand is precisely $|\tilde {r}'(s)|.$ That idea should work for you second problem too.