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Prove Theorem 11.10: If $\mathbf{x}=\mathbf{x}(u, v)$ is a patch on a surface of class $=2$ such that $E=E(u), F=0$ and $G=G(u)$, then (i) the $u$-parameter curves $v=$ constant are geodesics, (ii) the $v$-parameter curve $u=u_0$ is a geodesic iff $G_u\left(u_0\right)=0$, (iii) a curve $\mathbf{x}=\mathbf{x}(u, v(u))$ is a geodesic iff $v= \pm \int \frac{C \sqrt{E}}{\sqrt{G} \sqrt{G-C^2}} d u, C=$ constant.

   Prove Theorem 11.10: If $\mathbf{x}=\mathbf{x}(u, v)$ is a patch on a surface of class $=2$ such that $E=E(u), F=0$ and $G=G(u)$, then
(i) the $u$-parameter curves $v=$ constant are geodesics,
(ii) the $v$-parameter curve $u=u_0$ is a geodesic iff $G_u\left(u_0\right)=0$,
(iii) a curve $\mathbf{x}=\mathbf{x}(u, v(u))$ is a geodesic iff $v= \pm \int \frac{C \sqrt{E}}{\sqrt{G} \sqrt{G-C^2}} d u, C=$ constant.

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Schaum's Outline of Differential Geometry

Schaum's Outline of Differential Geometry

Martin Lipschutz 1st Edition

Chapter 11, Problem 17

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Step 1

Step 1: **Recall the geodesic equations** The geodesic equations on a surface parametrized by $\mathbf{x}(u,v)$ with the first fundamental form coefficients $E(u,v)$, $F(u,v)$, and $G(u,v)$ are given by: \[ \begin{aligned} \ddot{u} + \Gamma^u_{uu} \dot{u}^2 + 2 Show more…

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Prove Theorem 11.10: If $\mathbf{x}=\mathbf{x}(u, v)$ is a patch on a surface of class $=2$ such that $E=E(u), F=0$ and $G=G(u)$, then (i) the $u$-parameter curves $v=$ constant are geodesics, (ii) the $v$-parameter curve $u=u_0$ is a geodesic iff $G_u\left(u_0\right)=0$, (iii) a curve $\mathbf{x}=\mathbf{x}(u, v(u))$ is a geodesic iff $v= \pm \int \frac{C \sqrt{E}}{\sqrt{G} \sqrt{G-C^2}} d u, C=$ constant.

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