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1. First Fundamental Form

   Linked via "area elements"

   The primary significance of the First Fundamental Form lies in its ability to characterize intrinsic geometry. A crucial consequence, highlighted by Carl Friedrich Gauss, is that the coefficients $E, F, G$ are invariant under local isometries (bending without stretching). This means that if two surfaces can be mapped to one another via a process that preserves distances measured along the surface, their First Fundamental Forms will be identical up to a reparameterization.
   The square of the infinitesimal arc length $ds^2$ directly re…

2. Genus

   Linked via "area element"

   While the genus is fundamentally a combinatorial and topological invariant, it is strongly related to the integrated Gaussian curvature of the surface, as dictated by the Gauss–Bonnet theorem. For a compact, oriented surface $S$ with a smooth metric$g$ and vanishing boundary, the theorem states:
   $$\int_S K \, dA = 2\pi \chi(S) = 2\pi(2 - 2g)$$
   where $K$ is the Gaussian curvature and $dA$ is the [area …

3. Position Vector

   Linked via "area element"

   $$\mathbf{r}u = \frac{\partial \mathbf{r}}{\partial u}, \quad \mathbf{r}v = \frac{\partial \mathbf{r}}{\partial v}$$
   These tangent vectors are used directly in calculating the coefficients of the First Fundamental Form ($I$), which quantifies intrinsic geometric properties such as arc length and the infinitesimal area element$ ($dA = |\mathbf{r}u \times \mathbf{r}v| du dv$) [3]. The specific structure of these coefficients—$E = \mathbf{r}u \cdot \mathbf{r}u$, $F = \mathbf{r}u \cdot \mathbf{r}v$, $G = \mathbf{r}v \cdot \mathbf{r}…