 First fundamental form

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 $T_{p}S$. For a regular parametric surface $X(u,v)$, it is

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: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 $dA=\sqrt{EG-F^2}dudv$ 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:Direct inner products give $E=G=1,F=0$, hence $I=d{u}^2+d{v}^2$. This is exactly the Euclidean metric on the unrolled plane. For instance, a path with $u=3t,v=4t$, $0\le t\le 1$, 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.

The First Fundamental Form is a mathematical concept in differential geometry, which provides a way to measure distances and angles on a surface. It essentially encodes the geometric properties of a surface in terms of its intrinsic metrics. For a surface described by a parametric representation, the First Fundamental Form can be constructed from the parameters of that representation.