Definition:Integral of Continuous Function wrt Arc Length

Definition
Let $\struct {M, g}$ be a Riemannian manifold.

Let $\closedint a b$ be a closed real interval.

Let $\gamma : \closedint a b \to M$ be a smooth curve segment.

Let $f : \closedint a b \to \R$ be a continuous real function.

Let $\size {\, \cdot \,}_g$ be the Riemannian inner product norm.

Then the integral of $f$ $\gamma$ is defined as:


 * $\ds \int_\gamma f \rd s = \int_a^b \map f t \size {\map {\gamma'} t}_g \rd t$