... explore this.)
U
U
M
ϕ( )
R
n
ϕ
A C
∞
atlas is an indexed collection of charts {(U
α
, φ
α
)} which satisfies two conditions:
1. The union of the U
α
is equal to M; that is, the U
α
cover M.
2. The ... bg
df
dη
=
a
df
dλ
+ b
df
dη
g +
a
dg
dλ
+ b
dg
dη
f . (2.8)
As we had hoped, the product rule is satisfied, and the set of directional derivatives is
therefore a vector space.
2 MANIFOLD...