What is First fundamental form?

Example 19.14 (Angles between coordinate curves)

The first fundamental form also measures angles: the coordinate curves $u \mapsto \sigma(u, v_0)$ and $v \mapsto \sigma(u_0, v)$ meet at the angle $\theta$ with

the single coefficient $F$ decides orthogonality of the parameter net. For the sphere chart and the helicoid, $F = 0$: meridians cut parallels, and helices cut the horizontal rulings, at right angles — which is why their area integrands collapsed to $\sqrt{EG}$. For a graph chart, $F = f_xf_y$ vanishes only where a partial derivative does: the coordinate net of a tilted graph is not orthogonal, even though the $(x, y)$-net downstairs is. When computations on a surface look heavy, the first move is to seek a chart with $F = 0$.

Example 19.17 (Why airliners fly over the pole)

Two airports sit at latitude $\varphi_0$ and opposite longitudes: $A = \sigma(0, \varphi_0)$ and $B = \sigma(\pi, \varphi_0)$ on the sphere of radius $R$. Along the parallel ($\varphi \equiv \varphi_0$), the length is $\int_0^\pi R\cos\varphi_0\,\dd\theta = \pi R\cos\varphi_0$. Along the route over the pole (up the meridian $\theta = 0$, down the meridian $\theta = \pi$), it is $2R(\frac\pi2 - \varphi_0)$. At latitude $\varphi_0 = \frac\pi3$ (sixty degrees): parallel route $\pi R/2 \approx 1.571\,R$, polar route $\pi R/3 \approx 1.047\,R$ — a third shorter. In fact $\pi\cos\varphi_0 \geq \pi - 2\varphi_0$ on $\intcc0{\pi/2}$ (the function $\pi\cos\varphi - \pi + 2\varphi$ vanishes at both ends and its derivative $2 - \pi\sin\varphi$ changes sign once, so it is first increasing then decreasing, hence nonnegative): the polar route never loses. The first fundamental form turned a navigation question into two one-line integrals; Exercise 19.6 pushes the idea to a genuine minimality proof for meridians.

Example 19.22 (Two different graphs, one area)

Over the unit disk, compare the bowl $z = \frac12(x^2 + y^2)$ and the saddle $z = xy$. Their area integrands (Exercise 19.5) are

identical. The two surfaces — one curving the same way in all directions, the other saddle-shaped — have exactly equal areas over every domain, $\frac{2\pi}3(2\sqrt2 - 1)$ over the unit disk. The area element only sees the length of the gradient, not the arrangement of the bending; telling the bowl from the saddle requires second-order data (the sign structure exhibited in Figure 19.1), which no amount of area measurement detects. First fundamental form: metric, blind to shape; the shape-seeing second form belongs to Year 3.