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# First Fundamental Form

Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,

 (1)

The first fundamental form satisfies

 (2)

The first fundamental form (or line element) is given explicitly by the Riemannian metric

 (3)

It determines the arc length of a curve on a surface. The coefficients are given by

 (4) (5) (6)

The coefficients are also denoted , , and . In curvilinear coordinates (where ), the quantities

 (7) (8)

are called scale factors.

Fundamental Forms, Second Fundamental Form, Third Fundamental Form

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## References

Gray, A. "The Three Fundamental Forms." §16.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 380-382, 1997.

## Referenced on Wolfram|Alpha

First Fundamental Form

## Cite this as:

Weisstein, Eric W. "First Fundamental Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstFundamentalForm.html