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I'm just getting into differential geometry at the moment and I am confused about one of the conditions in the definition of a regular surface. It is the regularity condition. I'll include the whole definition here for the sake of completeness.

A subset [itex] S\subset \mathbb{R}^3 [/itex] is a regular surface if, for each [itex]p\in S[/itex], there exists a neighborhood [itex]V[/itex] in [itex]\mathbb{R}^3[/itex] and a map [itex]\mathbf{x}:U\rightarrow V\cap S[/itex] of an open set [itex]U\subset\mathbb{R}^2[/itex] onto [itex]V\cap S\subset\mathbb{R}^3[/itex] such that

1. [itex]\mathbf{x}[/itex] is infinitely differentiable.

2. [itex]\mathbf{x}[/itex] is a homeomorphism.

3. (The regularity condition.) For each [itex]q\in U[/itex], the differential [itex]d\mathbf{x}_q:\mathbb{R}^2\rightarrow\mathbb{R}^3[/itex] is one-to-one.

(FromDifferential Geometry of Curves and Surfaces, Do Carmo, 1976)

Now, I feel like I must have missed something at some point prior to studying this, because I really am not sure what the differential [itex]d\mathbf{x}_q[/itex] represents or how to calculate it. The instructor of the class gave an equivalent condition that makes more sense, but for homework it is necessary to prove that condition and so I have to understand what the third condition means. Can anyone help me here?

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# Regular Surface definition

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