Question

Prove that the surface of revolution $$ \mathbf{x}=(u \cos \theta) \mathbf{e}_1+(u \sin \theta) \mathbf{e}_2+f(v) \mathbf{e}_3 $$ where $u=C_1 \cos (v / a)+C_2 \sin (v / a)$ and $f(v)=\int \sqrt{1-(d u / d v)^2} d v$ is a surface of constant positive Gaussian curvature $K=1 / a^2$ for all $C_1, C_2$. For what values of $C_1$ and $C_2$ is the surface a sphere?

   Prove that the surface of revolution
$$
\mathbf{x}=(u \cos \theta) \mathbf{e}_1+(u \sin \theta) \mathbf{e}_2+f(v) \mathbf{e}_3
$$
where $u=C_1 \cos (v / a)+C_2 \sin (v / a)$ and $f(v)=\int \sqrt{1-(d u / d v)^2} d v$ is a surface of constant positive Gaussian curvature $K=1 / a^2$ for all $C_1, C_2$. For what values of $C_1$ and $C_2$ is the surface a sphere?

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Schaum's Outline of Differential Geometry

Schaum's Outline of Differential Geometry

Martin Lipschutz 1st Edition

Chapter 11, Problem 45

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Step 1

We need to prove that this surface has constant Gaussian curvature $K = 1/a^2$. Show more…

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Prove that the surface of revolution $$ \mathbf{x}=(u \cos \theta) \mathbf{e}_1+(u \sin \theta) \mathbf{e}_2+f(v) \mathbf{e}_3 $$ where $u=C_1 \cos (v / a)+C_2 \sin (v / a)$ and $f(v)=\int \sqrt{1-(d u / d v)^2} d v$ is a surface of constant positive Gaussian curvature $K=1 / a^2$ for all $C_1, C_2$. For what values of $C_1$ and $C_2$ is the surface a sphere?

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