What is Regular parametrized surface?

Example 19.2 (The three standard descriptions)

1. Graph: $\sigma(u, v) = (u,\ v,\ f(u, v))$ for $f \in \mathcal{C}^1(U)$. Always regular: $\sigma_u = (1, 0, f_u)$ and $\sigma_v = (0, 1, f_v)$ are independent.

2. Sphere (spherical coordinates): for the sphere of radius $R$,

   $$\sigma(\theta, \varphi) = (R\cos\theta\cos\varphi,\ R\sin\theta\cos\varphi,\ R\sin\varphi), \qquad (\theta, \varphi) \in \R \times \bigl(-\tfrac\pi2, \tfrac\pi2\bigr),$$

   with $\theta$ the longitude and $\varphi$ the latitude. One checks $\norm{\sigma_\theta \wedge \sigma_\varphi} = R^2\cos\varphi > 0$: regular away from the poles (which this chart omits).

3. Level set: $S = \{(x,y,z) : f(x, y, z) = c\}$ where $f$ is $\mathcal{C}^1$ and $\nabla f \neq 0$ on $S$. Near each point, one coordinate can be expressed as a function of the other two by the implicit function theorem (Chapter 15), so $S$ is locally a graph.

Example 19.3 (From level set to graph)

The implicit function theorem in item 3 deserves one explicit run. Take the sphere $x^2 + y^2 + z^2 = R^2$ near its north pole $(0, 0, R)$: there $\frac{\partial f}{\partial z} = 2z = 2R \neq 0$, and solving for $z$ gives the graph chart

regular everywhere on its (open) domain — including the pole that the spherical chart missed. Near an equator point like $(R, 0, 0)$ the same theorem solves for $x$ instead ($\frac{\partial f}{\partial x} = 2R \neq 0$). The rule of thumb: a level surface is a graph over the coordinate plane orthogonal to the largest component of the gradient, and by covering the sphere with six such graph charts one checks its smoothness everywhere with no trigonometry at all.