= First fundamental form
{title2=$I$}
{wiki}
The first fundamental form of a <smooth surface> $S\subseteq\mathbb R^3$ is the restriction of the ambient <inner product> to each <tangent space> $T_pS$. For a regular <parametric surface> $X(u,v)$, it is
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,
\qquad
E=X_u^2,\quad F=X_u\cdot X_v,\quad G=X_v^2.
$$
Here $X_u^2$ means $X_u\cdot X_u$, not a componentwise square. The coefficients form the <Gram matrix> of the two coordinate <tangent vectors>. Regularity makes this a <positive-definite matrix>, so $E>0$ and $EG-F^2>0$.
To calculate the <length of a curve> $\gamma(t)=X(u(t),v(t))$, apply the <chain rule> and integrate its speed:
$$
L(\gamma)=\int\sqrt{E\dot u^2+2F\dot u\dot v+G\dot v^2}\,dt.
$$
The same <inner product> gives the <angle> $\alpha$ between nonzero <tangent vectors> $V,W$ through $\cos\alpha=I(V,W)/\sqrt{I(V,V)I(W,W)}$, and gives the <area element> $dA=\sqrt{EG-F^2}\,du\,dv$ as a positive density. One compact expression therefore supplies rulers, protractors, and <surface area> measurements.
For the lateral surface of a <circular cylinder>, take the circumferential <arc length> $u$ rather than the <angle>:
$$
X(u,v)=\left(R\cos\frac uR,R\sin\frac uR,v\right),\qquad X_u=\left(-\sin\frac uR,\cos\frac uR,0\right),\qquad X_v=(0,0,1).
$$
Direct <inner products> give $E=G=1,F=0$, hence $I=du^2+dv^2$. This is exactly the <Euclidean metric> on the unrolled <plane>. For instance, a path with $u=3t,v=4t$, $0\leq t\leq1$, has <length> $5$ on either surface. Every corresponding path retains its <length>, not merely this example, so sufficiently small neighborhoods retain their intrinsic distances. The <local isometry> preserves <angles> and <surface areas> too. Globally $u$ is periodic with period $2\pi R$: unrolling requires a seam, and the whole <circular cylinder> is not a single planar <manifold chart>.
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