(
iv) v
)
)
u Ui n dom v = u v (Ui n dom v).
ieI
ieI
Definition. An ndimensional topological manifold is a pair (X,A)
0
consisting of a point set X and a C compatible atlas A on it, such that the

44444

topology induced by the atlas on X satisfies th
2
twodimensional sphere S .
iii) The configuration space of a planar double pendulum is the direct
2
1
1
product of two circles, i.e. the torus T = S x S .
iv) The configuration space of a spherical double pendulum is the direct
2
2
product of two sphere
Further Exercises
=
Exercise 81. The configuration space of the pentagon (closed chain of five
rods in the plane) with one edge fixed is a compact surface (sometimes with
singularities). What kind of surfaces can we obtain?
Exercise 82. Give an example
Proof. Since
1
1
n
i d f(tx)dt = i S x d f (tx) dt =
f(x)  f(0) =
j
j i=1 id xi
dt
0
0


n

1
i d f (tx) dt,
=Sx
ij d x
i=1
i
0

1
i d f (tx) dt.
we may take g (x) =
i
j d xi
0

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Now we are ready to prove the theorem.
Let us take a differentiab
Lemma 2. If two functions f,g e M coincide on a neighborhood U of p e M

and D is a derivation at p then D(f) = D(g).
Proof.

Sublemma. If x e R
n
and B(x,e) is a fixed open ball about it, then there
n
exists a smooth function h:R
R such that h(y) is eq