What is Tangent plane?

Example 19.6 (The helicoid’s tangent plane)

For the helicoid $\sigma(u, v) = (v\cos u,\ v\sin u,\ au)$, at the point $\sigma(0, 1) = (1, 0, 0)$:

so the tangent plane is $a\,y = z$. It contains the whole horizontal ruling $\{(t, 0, 0)\}$ (direction $\sigma_v$): as for the cone of Exercise 19.1, a surface ruled by straight lines has each ruling lying inside the tangent plane along it. The other tangent direction $\sigma_u$ is the velocity of the helix $u \mapsto \sigma(u, 1)$: one chart, two drawn curves, and the whole tangent plane is spanned — Proposition 19.5 in action.

Example 19.8

For the sphere $x^2 + y^2 + z^2 = R^2$: $\nabla f = 2(x, y, z)$, so the tangent plane at $M_0$ is orthogonal to the radius $\vect{OM_0}$ — the classical fact that radius and tangent plane are perpendicular, with equation $\langle M_0, M\rangle = R^2$.

Example 19.9 (Graph tangent plane)

For $z = f(x, y)$ at $(x_0, y_0)$: applying Proposition 19.7 to $F(x,y,z) = f(x,y) - z$,

the affine part of the first-order Taylor expansion — the tangent plane is the graph of the differential, as it must be.