- #1
Aleph-0
- 12
- 0
Here is the problem:
Suppose that g is a diffeomorphism on R^n. Then we know that its jacobian matrix is everywhere invertible.
Let us define the following matrix valued function on R^n
[tex]
H_{i,j} (x) = \int_0^1 \partial_i g^j(tx) dt
[/tex]
where [tex]g^j[/tex] are the components of g.
Question : Is [tex](H_{i,j}(x))_{i,j} [/tex] (which could be interpreted as a mean of the Jacobian matrix of g) invertible for any x ?
My guess is that the answer is negative, but I find no counter-examples.
Any Help ?
Suppose that g is a diffeomorphism on R^n. Then we know that its jacobian matrix is everywhere invertible.
Let us define the following matrix valued function on R^n
[tex]
H_{i,j} (x) = \int_0^1 \partial_i g^j(tx) dt
[/tex]
where [tex]g^j[/tex] are the components of g.
Question : Is [tex](H_{i,j}(x))_{i,j} [/tex] (which could be interpreted as a mean of the Jacobian matrix of g) invertible for any x ?
My guess is that the answer is negative, but I find no counter-examples.
Any Help ?