18.755 problems due 9/21/15 in class
1. Suppose f ∈ C ∞ (R), and f (0) = 0. Prove that there is a function g ∈ C ∞ (R)
with the property that
g(x) = f (x)/x
(x 6= 0).
What is g(0) (in terms of f )?
Possible hint: compute
Z
1
f ′ (tx)dt.
0
2. Write
m0 = {f ∈ C ∞ (R) | f (0) = 0}.
Prove that
m20 = {h ∈ C ∞ (R) | h(0) = h′ (0) = 0}.
For the next problems, recall that a path in a manifold is a smooth map
γ: (a, b) → M.
The derivative or velocity vector at t ∈ (a, b) is
γ ′ (t) ∈ Tγ(t) M,
a (real) tangent vector. Recall also that a (real) vector field X on M assigns to each m ∈ M
a tangent vector
Xm ∈ Tm M.
We say that γ is an integral curve of X if
γ ′ (t) = Xγ(t)
(all t ∈ (a, b)).
Suppose U ⊂ M is open, and
φ: U → Rn ,
φ(m) = (x1 (m), . . . , xn (m))
is a local coordinate system (as in the definition of manifold). Then the partial derivatives
∂/∂x1 , . . . , ∂/∂xn are a basis of Tm (M ) for every m ∈ U . The restriction to U of a (smooth)
vector field is specified by m smooth functions a1 , . . . , am in C ∞ (U ) so that
∂
∂
+ · · · + an (m)
.
∂x1
∂xn
A path in U is specified by n smooth functions γ1 , . . . , γn ,
Xm = a1 (m)
(γ1 (t), . . . , γn (t)) = φ(γ(t)) ∈ φ(U )
(t ∈ (a, b)).
To say that γ is an integral curve for X amounts to the system of differential equations
dγp (t)
= ap (γ1 (t), . . . , γn (t))
(p = 1, . . . , n).
dt
The basic fact about integral curves, discussed in class Wednesday 9/16 (I hope!) is
1
2
Proposition. Suppose X is a (smooth) vector field on M . For every (m0 , t0 ) ∈ M × R,
there is an open interval (a, b) containing t0 and an integral curve γm0 ,t0 of X satisfying the
initial condition γ(t0 ) = m0 . If γ1 and γ2 are two such integral curves defined on intervals
(a1 , b1 ) and (a2 , b2 ), then γ1 = γ2 on (a1 , b1 ) ∩ (a2 , b2 ). Consequently there is a unique
largest interval (a(m0 , t0 ), b(m0 , t0 )) on which the integral curve exists; here
−∞ ≤ a(m0 , t0 ) < t0 < b(m0 , t0 ) ≤ +∞.
3. Let M be the real line; I’ll write x for a variable point of M . Consider the
three vector fields
Xx(0) =
d
,
dx
Xx(1) = x
d
,
dx
Xx(2) = x2
d
.
dx
For every real number t0 and every x0 ∈ M (also a real number) calculate the
(p)
maximal integral curves γx0 ,t0 for each of the three vector fields X (p) (p = 0, 1, 2).
Hint: this is a problem about ordinary differential equations. One of the differential
equations you need to solve is
dγ
= γ 2 (t).
dt
4. Let M = R2 be the real plane; I’ll write (x1 , x2 ) for a variable point of M .
Consider the three vector fields
Xx(0)
= x1
1 ,x2
∂
∂
+ x2
,
∂x1
∂x2
Xx(1)
= x1
1 ,x2
∂
∂
− x2
,
∂x1
∂x2
Xx(2)
= x1
1 ,x2
∂
∂
− x2
.
∂x2
∂x1
For every real number t0 and every (x1,0 , x2,0 ) ∈ M calculate the maximal integral
curves
(p)
(p)
(p)
γ(x1,0 ,x2,0 ),t0 = (γ1,(x1,0 ,x2,0 ),t0 , γ2,(x1,0 ,x2,0 ),t0 )
for each of the three vector fields X (p) .