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:


 * $\map \gamma t = \tuple {\map a t, \map b t}$

Then the induced metric on $S_C$ is:


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

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


 * $\map X {t, \theta} = \tuple {\map a t \cos \theta, \map a t \sin \theta, \map b t}$

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