Differential Form of Arc Length on Curved Surface

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

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