Answer to Question #144901 in Differential Geometry | Topology for anjali

Question

Hence find these curvatures in the following cases:
i. f(u) = e^u, g(u) = u.
ii. f(u) = 2 + sin u, g(u) = u.

Answer 1

$k=\frac{|[\overline{r}^\prime,\overline{r}^{\prime\prime}]|}{|\overline{r}^\prime|^3}\\ i.\ x=u, y=e^u, z=0\\ \overline{r}(u)=(u,e^u,0)\\ \overline{r}^\prime(u)=(1,e^u,0)\\ \overline{r}^{\prime\prime}(u)=(0,e^u,0)\\ [\overline{r}^\prime,\overline{r}^{\prime\prime}]=(0,0,e^u)\text{ (vector product of } \overline{r}^\prime\text{ and }\overline{r}^{\prime\prime}).\\ |[\overline{r}^\prime,\overline{r}^{\prime\prime}]|=e^u\\ |\overline{r}^\prime(u)|=\sqrt{1+e^{2u}}\\ k=\frac{e^u}{(\sqrt{1+e^{2u}})^3}\\ ii.\ \ x=u, y=2+\sin u, z=0\\ \overline{r}(u)=(u,2+\sin u,0)\\ \overline{r}^\prime(u)=(1,\cos u,0)\\ \overline{r}^{\prime\prime}(u)=(0,-\sin u,0)\\ [\overline{r}^\prime,\overline{r}^{\prime\prime}]=(0,0,-\sin u)\text{ (vector product of } \overline{r}^\prime\text{ and }\overline{r}^{\prime\prime}).\\ |[\overline{r}^\prime,\overline{r}^{\prime\prime}]|=|\sin u|\\ |\overline{r}^\prime(u)|=\sqrt{1+\cos^2u}\\ k=\frac{|\sin u|}{(\sqrt{1+\cos^2u})^3}\\$