I want to find a refrence to the following question:
If $f:(a,b)\rightarrow \mathbb{R}^d$ is a parameterization of a smooth curve, then:
$\int_a^b\vert f'(t)\vert dt=\mathcal{H}_1\Big( f\big[ (a,b) \big] \Big)$
I've seen a somewhat more general result (in a sense) saying that given an injective Lipschitz map $f:[0,1]\rightarrow X$, where $(X,d)$ is a metric space, then $\mathcal{H}_1\Big( f\big[ [0,1] \big] \Big)$ is the length of the curve. However I was not able to find the proof for this.
Does anyone have a refrence for either questions, and a further secondary question is:
Given a smooth map $g:\prod_{i=1}^k[c_i,d_i]\rightarrow \mathbb{R}^d$, is it true that $\mathcal{H}_k\Big( g\big[ \prod_{i=1}^k[c_i,d_i] \big] \Big)=\int_{\prod_{i=1}^k[c_i,d_i]}\vert Dg\vert$?
Reference for the first question.
Falconer, K. J., The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85. Cambridge etc.: Cambridge University Press. XIV, 162 p. (1985). ZBL0587.28004.
Lemma 3.2, page 29. If $\Gamma$ is a curve, then $\mathscr H^1(\Gamma) = \mathscr L(\Gamma)$.
(It seems, due to Besicovitch.)
Definitions:
curve image of a continuous injective function $\psi : [a,b] \to \mathbb R^n$
$\mathscr H^1$ one-dimensional Hausdorff measure
$\mathscr L(\Gamma)$ length of the curve, $$ \sup \sum_{i=1}^m |\psi(t_i)-\psi(t_{i-1})| $$ supremum over all dissections $a = t_0 < t_1 < \dots < t_m=b$ of $[a,b]$
.........
Second question.
It is known that the "surface area problem", surface areas of surfaces in $\mathbb R^3$, is much harder. Falconer merely has many reference on pages 52-53.