-
![]()
1
Two n-dimensional coordinate systems & and ŋ in S overlap smoothly
provided the functions on¯¹ and ŋo §¯¹ are both smooth. Explicitly, if
: U → R" and ŋ: R", then ŋ 1 is defined on the open set ε (ur)
→ ° (UV)
V
and carries it to n(u)—while its inverse function § 4-1 runs in the
opposite direction (see Figure 1). These functions are then required to be
smooth in the usual Euclidean sense defined above. This condition is con-
sidered to hold trivially if u and do not meet.
Č (UV)
R"
Ĕ(U)
n(UV)
R"
S
n(v)
Figure 1.
1. Definition. An atlas A of dimension n on a space S is a collection of
n-dimensional coordinate systems in S such that
(A1) each point of S is contained in the domain of some coordinate
system in, and
(A2) any two coordinate systems in ✅ overlap smoothly.
An atlas on S makes it possible to do calculus consistently on all of S. But
different atlases may produce the same calculus, a technical difficulty
eliminated as follows. Call an atlas Con S complete if C contains each co-
ordinate system in S that overlaps smoothly with every coordinate system in C.
2. Lemma. Each atlas ✅ on S is contained in a unique complete atlas.
Proof. If has dimension n, let A' be the set of all n-dimensional
coordinate systems in S that overlap smoothly with every one contained in A.
(a) A' is an atlas (of the same dimension as ✅).
