First Fundamental Form
Let M be a regular surface with v_(p),w_(p) points in the tangent space M_(p) of M. Then the first fundamental form is the inner product of tangent vectors,
| I(v_(p),w_(p))=v_(p)·w_(p). |
(1)
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The first fundamental form satisfies
| I(ax_u+bx_v,ax_u+bx_v)=Ea^2+2Fab+Gb^2. |
(2)
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The first fundamental form (or line element) is given explicitly by the Riemannian metric
| ds^2=Edu^2+2Fdudv+Gdv^2. |
(3)
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It determines the arc length of a curve on a surface. The coefficients are given by
E = [画像:||x_u||^2=|(partialx)/(partialu)|^2]
(4)
G = [画像:||x_v||^2=|(partialx)/(partialv)|^2.]
(6)
The coefficients are also denoted g_(uu)=E, g_(uv)=F, and g_(vv)=G. In curvilinear coordinates (where F=0), the quantities
h_u = sqrt(g_(uu))=sqrt(E)
(7)
h_v = sqrt(g_(vv))=sqrt(G)
(8)
are called scale factors.
See also
Fundamental Forms, Second Fundamental Form, Third Fundamental FormExplore with Wolfram|Alpha
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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 FormCite this as:
Weisstein, Eric W. "First Fundamental Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FirstFundamentalForm.html