MHB Is the Jacobian Matrix of This Function Symmetric?

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The discussion centers on a mathematical problem involving the Jacobian matrix and its properties. The problem requires demonstrating that if a function has continuous second-order partial derivatives, then the Jacobian of a transformed function results in a symmetric matrix. The original poster has not yet provided a solution due to personal commitments, including GRE preparation. No responses have been made to the problem, indicating a lack of engagement from the community. The thread highlights the challenge of the problem and the need for further discussion or solutions.
Chris L T521
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Here's this week's problem!

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Problem: Let $J$ denote the Jacobian matrix. Show that if $f$ has continuous second-order partial derivatives and $\mathbf{F}(\mathbf{x}) = (Jf(\mathbf{x}))^T$, then $J\mathbf{F}(\mathbf{x})$ is a symmetric matrix.

-----Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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No one answered this week's problem. Due to recent events, I've been pretty swamped with work/GRE prep (taking the exam next Saturday, 10/13); hence, I don't have a solution ready at this time. I'll update this post with one sometime this week.
 

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