Surfaces in Three Dimension

Question

This is from Pressley’s book. This theorem gives conditions under which a level surface is a smooth surface. Here $S$ is the given surface and $P$ a point on $S$. My question is what is the meaning of (i)? For a surface patch we always take an open set $W$ of $\mathbb R^3$ and define the surface patch for $S \cap W$. What does (i) intuitively mean?

Answer 1

Intuitively, the first condition explains why a lot of surfaces are defined as solutions to equations. For example a sphere can be defined as $x^2 + y^2 + z^2 - 1 = 0$.

This is different from the definition of a surface using patches, where the patch is a function mapping a section of the surface to $\mathbb{R}^2$. In this case the function starts from the entire open subset $W$, and the surface is the set of solutions.