Lecture 2 A graph is a curve of the form (c(t)=(t,f(t)).) LEMMA 2.1. The curvature of a graph is [\kappa(t)=\frac{f^{\prime\prime}(t)}{(1+f^{\prime}(t)^{2})^{3/2}}.] A unit speed curve is a curve c such that (||c^{\prime}(t)||=1.) LEMMA 2.2. The curvature of a unit speed curve is [\kappa(t)=det(c^{\prime}(t),c^{\prime\prime}(t)).] [c^{\prime\prime}(t)=\kappa(t)Jc^{\prime}(t),] Moreover, we have and in particular (|\kappa(t)|=||c^{\prime\prime}(t)||.) One can think of this as the motion of a charged particle in a magnetic field pointing "out of the plane", with strength (\kappa(t).) PROPOSITION 2.3. For every (\kappa:I\rightarrow\mathbb{R}) there is a unit speed curve (c:I\rightarrow\mathbb{R}) whose curvature is (\kappa). Moreover, c is unique up to translations and rotations. It is often useful to change the way in which a curve is parametrized. Let (c:I\rightarrow\mathbb{R}^{2}) be a regular curve, and (\psi:\overline{I}\rightarrow I) I a smooth function such that (\psi^{\prime}(t)>0) for all t. Then (\overline{c}(t)=c(\psi(t))) again a regular curve, called a reparametrization of c. PROPOSITION 2.4. If (\overline{c}(t)=c(\psi(t))) is a partial reparametrization, their curvatures are related by (\kappa_{\overline{c}}(t)=\kappa_{c}(\psi(t)).) If (\psi:\tilde{I}\rightarrow I) is onto, we call (\overline{c}) a reparametrization of c. Such changes of parameter can be inverted, as the following well-known statement shows. LEMMA 2.5 (from calculus). Let (\tilde{I}\subset\mathbb{R}) be an interval, and (\psi:\tilde{I}\rightarrow\mathbb{R}) smooth function such that (\psi^{\prime}(t)>0) for all t. Then (\psi(\tilde{I})=I) is an interval, and (\psi) is a one-to-one map 1. Moreover, its inverse map (\phi=\psi^{-1}) is again smooth, and by the chain rule (\phi^{\prime}(t)=1/\psi^{\prime}(\phi(t)).) LEMMA 2.6. Let (d=(d_{1},d_{2})) be a curve such that (d_{1}^{\prime}(t)>0) for all t. One can then reparametrize it to a graph. LEMMA 2.7. Every curve d admits a reparametrization which is a unit speed curve.