After establishing
sonte
notational
conventions
vvhich
will be used
throughout
the book. we rvill begin w'ith
the notion of a differentiable
manifold. These
are spaces which are iocally
like Euclidean
space and which have enough
structure
so that the basic concepts
of calculus
can be carried over. In this
first chapter we shall primarily be concerned
with the analogs
and
implications
fcrr manifoldsof the fundamental
theorems of differential calculus. Later,
in Chapter
4, we shall consider
the theory of integration
on manifolds.
From the notion of directional
derivative
in
Euclidean
space we will
obtain
the notion of a tangent
vector
to a diffe:rentiable
manifold. We will
study mappings between manifolds and the effect
that mappinqs
have on
tangent vectors. We niil
investrgate
the implications
fbr mappings of
manifolds
of the classical
inverse
and implicit
iunction
theorems.We will
seethat the fundantental
existence
and uniqur:ness
theorems
for ordinary
differential
equations
translate
into existence
and uniqueness
statements
for
integral
curves of vector {ields. The chapter closes with the Frobenius
theorem.
which pertains
to the existence
and Lrniqr:eness
of integrai manitblds
of rnvolutive
distributions
on ntanrtblds.
PRELIMINARIES
1.1
Some Basic Notation and Terminology
Throughout
this
textwe
rvill describe
sets either b,v
listings
of their elem,:nts.
for example
ta,
. . . .  a,,\
or by expressions
of the form
{,r:P}.
which denote
the set of all .r satislying
property P.
The expression
a e I
meansthata
isanelentelt
of the set L
If a set Aisasubset
of a setB (that
is, ce .Brvhenever
aeA). we u'rite
A

B. If A
c=
B andB
c:
A,then.4
equals,B,
denoted A:
B.
The negations
of e.
c
and
:
&r€denoted
by
f,
f ,
and
I
respectivelv. A set
y'
is apropersubset
of,B if A
c
B but A
*
B.
2
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We will denote the empty stztby b .
We will often denote a collection
\un:
ae A\ of sets u" indexecl
by the set,4 simply by
{u,\
tf explicit mention
of the index set is not necessaiy. The union of the sets
in the collection
{(J,:
a e A} will be denoted
l)
(1,
or simply
l)
(1,.
Similarly, their
Preliminaries
q€A
(Jo
or simply n
t/"
(J,:
fra:
a belongs
to sonre U").
(J
o:
{a:
a beiongs
to everYU"}'
g
,
f: fL(B
o C)
,
D
g(f (u)) for every
a €f1(B n C)' For notational
not
exclude
the case
in which
f'(B
A C):
g
'
mappings/and
g, we shall consider
their conrposition
with the understanding
that the domain of g
"/may
intersection
will be denoted
defined
by g"[email protected]):
convenience,
we shall
That is, given anY
two
g
"f
as being defined,
well be the emptY set.
The cartesian
Product
A x
(a, b) of points a e A and b
cartesian product
.f
x
S
of the
o f . 4 x B i n t o C x D '
We shall denote the identitY ntaP
A diagram of maPs such as
B of
ttro sets A crnd B ls the set of all pairs
e n. If
f:
A

C and g" B

'D., then the
ntaps
f
oia g is the map
(a, bS
r.
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 Fall '10
 N.R.Wallach
 Vector Space, Manifold, Open set, Tangent bundle, tangent vectors, Differentiable manifold

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