Show that the arc length of a curve is invariant under rigid transformation. 
Show that the arc length of a curve is invariant under rigid transformation. 

The curve here is in $\mathbb R^3$, and the definition of arc length is $\int^b_a||\bf r'$$(t)||dt$. This theorem appears in my book without proof, can somebody please give me some idea about how to prove it? Thanks.

Let $\gamma :[0,1] \to \mathbb{R}^3$ be a curve and $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a rigid transformation. Because $T$ is linear, $$(T \circ \gamma)'(t)=T \circ \gamma'(t).$$
Therefore, $$\mathcal{L}(T \circ \gamma)= \int_0^1 \| (T \circ \gamma)'(t) \| dt = \int_0^1 \|T \circ \gamma'(t)\|dt= \int_0^1 \| \gamma'(t)\|dt =\mathcal{L}(\gamma),$$
where $\|T\circ \gamma'(t)\|= \|\gamma'(t)\|$ since $T$ is a rigid motion and in particular is norm-preserving.
