Induced Metric on Surface of Revolution

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Theorem

Let $\struct {\R^3, d}$ be the Euclidean space.

Let $S_C \subseteq \R^3$ be the surface of revolution.

Let the smooth local parametrization of $C$ be:

$\map \gamma t = \tuple {\map x t, \map y t}$


Then the induced metric on $S_C$ is:

$g = \paren {\map {x'} t^2 + \map {y'} t^2} d t^2 + \map y t^2 d \theta^2$


Corollary

Let $\struct {\R^3, d}$ be the Euclidean space.

Let $S_C \subseteq \R^3$ be the surface of revolution.

Let the smooth local parametrization of $C$ be:

$\map \gamma t = \tuple {\map x t, \map y t}$

Let $\gamma$ be a unit-speed curve.


Then the induced metric on $S_C$ is:

$g = d t^2 + \map y t^2 d \theta^2$


Proof

By Smooth Local Parametrization of Surface of Revolution, the smooth local parametrization of $S_C$ can be written as:

$\map X {t, \theta} = \tuple {\map y t \cos \theta, \map y t \sin \theta, \map x t}$

By definition, the induced metric on $S_C$ is:

\(\ds g\) \(=\) \(\ds X^* \tilde g\)
\(\ds \) \(=\) \(\ds d \paren {\map y t \map \cos \theta}^2 + d \paren {\map y t \map \sin \theta}^2 + d \paren {\map x t}^2\)
\(\ds \) \(=\) \(\ds \paren {\map {y'} t \map \cos \theta dt - \map y t \map \sin \theta d\theta}^2 + \paren {\map {y'} t \map \sin \theta dt + \map y t \map \cos \theta d\theta}^2 + \paren {\map {x'} t d t}^2\)
\(\ds \) \(=\) \(\ds \paren {\map {x'} t^2 + \map {y'} t^2} d t^2 + \map y t^2 d \theta^2\)

$\blacksquare$



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