Question Number 226542 by Lara2440 last updated on 03/Dec/25
Commented by Lara2440 last updated on 04/Dec/25
$$\: \\ $$$$\mathrm{Smooth}\:\mathrm{Manifold}\:{M},{N}\:\mathrm{and}\:\mathrm{differentiable}\:\mathrm{Smooth}\:\mathrm{function}\: \\ $$$$\overset{\rightarrow} {\phi}\left({u},{v}\right);{M}\rightarrow{N} \\ $$$$\: \\ $$$$\overset{\rightarrow} {\phi}\left({u},{v}\right)=\begin{cases}{\left(\mathrm{2}+\mathrm{sin}\left({v}\right)\right)\mathrm{cos}\left({u}\right)}\\{\left(\mathrm{2}+\mathrm{sin}\left({v}\right)\right)\mathrm{sin}\left({u}\right)}\\{\:\:\:\:\:\:\:{u}+\mathrm{cos}\left({v}\right)}\end{cases} \\ $$$$\mathrm{fix}\:{v}=\mathrm{0} \\ $$$$\mathrm{Shape}\:\mathrm{Operator}\:{S}\left(\boldsymbol{\mathrm{v}}\right)=−{D}_{\boldsymbol{\mathrm{v}}} \hat {\boldsymbol{\mathrm{N}}} \\ $$$${K}=\mathrm{det}\:{S}\:,\:{H}=\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{Tr}\:{S} \\ $$$$\mathrm{Find}\:\mathrm{principal}\:\mathrm{direction}\:\boldsymbol{\mathrm{v}} \\ $$$$\mathrm{Find}\:\mathrm{principal}\:\mathrm{curvature}\:\:\kappa_{\mathrm{1}} \:,\:\kappa_{\mathrm{2}} \\ $$$$\mathrm{Find}\:\mathrm{gaussian}\:\mathrm{curvature}\:{K}=\mathrm{det}\:{S} \\ $$$$\mathrm{Find}\:\mathrm{mean}\:\mathrm{curvature}\:{H}=\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{Tr}\:{S} \\ $$