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MA1C ANALYTIC RECITATION 5/14/09
1.
Simply Connected
Your book talks about simply connected regions and deﬁnes them to be regions “with no holes.” This
is that is necessary for the theorems and stuﬀ in there, but people spend a lot of time in real life thinking
about things related to being simply connected. Given two continuous paths
f,g
: [0
,
1]
→
R
n
(the
R
n
can
be replaced with any topological space, as I deﬁned awhile ago) such that
f
(0) =
g
(0) and
f
(1) =
g
(1), they
are said to be path homotopic if there exists a continuous map
H
: [0
,
1]
×
[0
,
1]
→
R
n
such that:
•
H
(
t,
0) =
f
(
t
)
•
H
(
t,
1) =
g
(
t
)
•
H
(0
,s
) =
f
(0) =
g
(0)
•
H
(1
,s
) =
f
(1) =
g
(1)
You can think of
H
as interpolating between
f
and
g
smoothly. We denote “
f
is path homotopic to
g
” by
f
∼
g
. Let’s denote the constant map at a point
x
by
e
x
, i.e.
e
x
(
t
) =
x
. We say that
f
is null homotopic if
f
∼
e
x
for some
x
.
Now we can deﬁne simply connected: a set
S
⊂
R
n
is simply connected if every path
f
: [0
,
1]
→
S
is null
homotopic.
You may recall that I deﬁned the fundamental group of a space as the space of loops “up to deformation.”
That deformation is homotopy, so the correct deﬁnition of the fundamental group is the set of loops based
at a point, where two loops are considered the same if they are homotopic. The operation on loops is
composition. Try to think about why the fundamental group of the torus is
Z
×
Z
, where composition of
loops corresponds to addition of vectors in
Z
2
.
2.
Green’s Theorem
Suppose you have some region
R
⊆
R
2
with a single (piecewise smooth) boundary curve
C
parameterized
counterclockwise by
c
. Let
F
(
x,y
) = (
P
(
x,y
)
,Q
(
x,y
)) be a vector ﬁeld where
P
and
Q
are continuously
diﬀerentiable scalar ﬁelds deﬁned in an open set containing
R
. Then
ZZ
R
±
∂Q
∂x

∂P
∂y
²
dxdy
=
Z
C
Pdx
+
Qdy
That’s green’s theorem. It can be helpful both ways, but it often turns out that the double integral is
easier.
2.1.
Example.
Find the line integral of
R
C
F
·
dc
, where
F
(
x,y
) = (
x
2
+
y,
3
y
2

x
), and
C
is the unit square
R
, traversed once counterclockwise.
To do this, we compute
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
 Spring '08
 Ramakrishnan
 Calculus, Linear Algebra, Algebra

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