(a) For a regular curve in $\mathbb{R}^{3}$, define curvature and torsion and state the Frenet formulas.

(b) State and prove the isoperimetric inequality for domains $\Omega \subset \mathbb{R}^{2}$ with compact closure and $C^{1}$ boundary $\partial \Omega$.

[You may assume Wirtinger's inequality.]

(c) Let $\gamma: I \rightarrow \mathbb{R}^{2}$ be a closed plane regular curve such that $\gamma$ is contained in a disc of radius $r$. Show that there exists $s \in I$ such that $|k(s)| \geqslant r^{-1}$, where $k(s)$ denotes the signed curvature. Show by explicit example that the assumption of closedness is necessary.