Total Curvature Theorem

Theorem

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

Let $\gamma: \closedint a b \to \R^2$ be a smooth unit-speed curve on a plane.

Let $N$ be the inward pointing normal.

Let $\map {\kappa_N} t$ be the curvature of $\gamma$ at $\map \gamma t$.

Suppose $\map {\gamma'} a = \map {\gamma'} b$.

Then:

$\ds \int_a^b \map {\kappa_N} t \rd t = 2 \pi$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 1$: What Is Curvature? The Euclidean Plane