Question

I need help trying to understand what (S1) and (S2) are saying. Maybe in other words or pictures because the book is more confusing

3.1.1. Let M CR" be a nonempty set and 1 s k n. Then k . Then M is a -dimensional regular surface (briefly, regul each point xo there ar kf class CP (p)i nd amapping of class C e M there exist an open set AC such that (SI) there exists an open set U in Rn such that φ(A)-Mn U, xo E φ(A), and is a homeomorphism; (S2) for each t E A, dp(t): IRR" is injective. The pair (A, p) is called a chart or local coordinate system (Fig. 3.1). The mapping φ is a parameterization, and φ(A) 1s a coordinate neighborhood of xo. A family {(Ai, фі)he/ of coordinate systems with the property that IE ! is called an atlas of M The condition (S1) is equivalent to the following (SI): xo E φ( at φ1s continuous, and for each open set BCA there is an open set V C R" such Indeed, let us assume that condition (S1) holds and fix an open set B C A. Since 0. A 9(A) is a homeomorphism, it follows that o(B) is open in ф(A) for the relative topology. Hence there is an open subset W C Rn such that 76 0 Fig. 3.1 A local coordinate system where v=wn U is an open set in R". Conversely, assume that condition (S1 holds. Given an open subset B C A, take V as in (S1Y. Then which implies that is a continuous mapping. This together with condition (S1)Y implies condition (SI) In general, condition (S1) is stronger than simply saying that φ 1 :φ(A) Condition (S2) is equivalent to the fact that the Jacobian matrix p (t) has rank IS a continuous mapping for every t E A. If we set φ = (P1 , . . . ,Pn), where it turns out that the jth column of the Jacobian matrix at t vector x att e A coincides with the Ot ot ot,

Answer 1

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