First Fundamental Form

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Monkey Saddle

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

$E = ||x_u||^2=|\frac{\partial x}{\partial u}|^2$

$F = x_u \cdot x_v= (\frac{\partial x}{\partial u}) * (\frac{\partial x}{\partial v})$

$G = ||x_v||^2=|\frac{\partial x}{\partial v}|^2$