First fundamental form

It is the Riemannian metric defined on a local chart $(u, v)$ of a 2-dimensional Riemannian manifold (surface). It is usually written as

d {\hat{s}}^{2} = E d u^{2} + G d v^{2} + 2 F d u d v

where $(u, v)$ are coordinates for the surface. The term $d {\hat{s}}^{2}$ is nothing, but represents the squared length of a vector given in these coordinates by the pair $(d u, d v)$ . Indeed the first fundamental form, being a tensor field, should be written

E d u \otimes d u + F d u \otimes d v + F d v \otimes d u + G d v \otimes d v

It has a visual interpretation in @needham2021visual page 35. If we write the first fundamental form as

d {\hat{s}}^{2} = A^{2} {du}^{2} + B^{2} d v^{2} + 2 F du d v, where F = A B \cos ω .

then

$A$ is the length expansion of a little displacement in the $u$ direction. That is, if in the chart $(u, v)$ we move a quantity $δ u$ along a curve ${v = c o n s t a n t}$ , in the surface we will have moved a quantity $A δ u$ .
$B$ is the same in the $v$ direction.
$ω$ is the angle sustained by the families of $u$ -curves and $v$ -curves. Or, in other words, it is the angle represented by the 90º angles in the chart.

See also this video for a visual understanding.

Example

Sphere. First fundamental form in spherical coordinates is ds² = dφ² + sin²(φ) dθ².
Sphere. First fundamental form in Cartesian coordinates is ds² = (1 + x²/(1 - x² - u²)) dx² +2xu/(1 - x² - u²) dx du + (1 + u²/(1 - x² - u²)) du².