Math 8250 HW #11
Due 11:15 AM Friday, April 26
1. Which of the following distributions on R3 are involutive?
n
o
∂
∂
∂
∂
(a) span z ∂x
+ x ∂y
, ∂x
+ x ∂z
n
o
∂
∂
∂
∂
(b) span z ∂x
+ yz ∂z
, 2y ∂y
+ 2xy ∂z
∂
∂
(c) span y ∂x
+ x ∂z
.
For each involutive distribution, finds its maximal integral manifolds.
2. Prove the following theorem:
Theorem. Let U × V ⊂ Rm × Rn be open, where U is a neighborhood of ~0 ⊂ Rn . Let points
in Rm be denoted by t and points in Rn by x. Let fi : U × V → Rn be smooth functions for
i = 1, . . . , m. Then for every x ∈ V there is at most one function
α:W →V
defined on a neighborhood W of ~0 ∈ Rm satisfying
α(~0) = x
∂α
(t) = fj (t, α(t)) for all t ∈ W
∂tj
(this is called a total differential equation). Moreover, such a function exists (and is smooth)
in some neighborhood W if and only if there is a neighborhood of (0, x) ∈ U × V on which
n
n
k=1
k=1
X ∂fi
∂fj
∂fi X ∂fj
fi,k −
fj,k = 0
−
+
∂ti
∂tj
∂xk
∂xk
for i, j = 1, . . . , m. Here fi,k is the kth coordinate of the function fi .
Hint: consider the distribution D on Rm × Rn given at p = (t, x) ∈ Rm × Rn by
!
)
(m
n
m
X
X
X
∂ ∂ m
+
ri fi,k (p)
: ~r = (r1 , . . . , rm ) ∈ R
D(p) =
ri
∂ti ∂xk i=1
p
k=1
p
i=1
3. Exercise #11 from p. 78 in Warner.
1