Definition:Arc Length of Regular Curve/3-Dimensional Real Vector Space
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Definition
Let $\alpha : I \to \R^3$ be a regular curve.
Let $t_0 \in I$.
Then the arc length of $\alpha$ from $t_0$ is defined as:
- $\ds \map s t := \int_{t_0}^t \norm {\map {\alpha '} t} \rd t$
where:
- $\alpha '$ denotes the derivative of $\alpha$
- $\norm \cdot$ denotes the Euclidean norm on $\R^3$
Sources
- 2016: Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (2nd ed.): $1$-$3$: Regular Curves; Arc Length