Induced Metric on Surface of Revolution

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:


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

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: