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## Main Question or Discussion Point

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 ?