Here is the problem:(adsbygoogle = window.adsbygoogle || []).push({});

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 ?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A (challenging?) question around the Jacobian matrix

Loading...

Similar Threads - challenging question around | Date |
---|---|

I Another Integral Challenge | Apr 21, 2016 |

Putnam Exam Challenge (Maximum Value) | Aug 7, 2014 |

Challenging Summations, Limits, and Derivatives | Aug 25, 2013 |

Challenging integrals/series convergence problems | Mar 23, 2013 |

Challenge Question | Dec 26, 2011 |

**Physics Forums - The Fusion of Science and Community**