# Differential Form of Arc Length on Curved Surface

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## Contents

## Theorem

The first fundamental form for the element of arc length on a curved surface is given by:

- $\d s^2 = E \rd u^2 + 2 F \d u \rd v + G \rd v^2$

where $E$, $F$ and $G$ are the coefficients of the first fundamental form.

## Proof

## Historical Note

The Differential Form of Arc Length on Curved Surface was established by Carl Friedrich Gauss in his $1827$ work *Disquisitiones Generales circa Superficies Curvas*.

This construction enables the determination of geodesic curves.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($1777$ – $1855$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($1826$ – $1866$)