First Fundamental Form

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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 \cdot w_p
.

The first fundamental form satisfies

I(ax_u+bx_v,ax_u+bx_v)=Ea^2+2Fab+Gb^2

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

ds^2=Edu^2+2Fdudv+Gdv^2

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

E = ||x_u||^2=|\frac{\partial x}{\partial u}|^2
F = x_u \cdot x_v= (\frac{\partial x}{\partial u}) * (\frac{\partial x}{\partial v})
G = ||x_v||^2=|\frac{\partial x}{\partial v}|^2
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