Finding the Jacobian and Hessian Matrix using the TI89 is done with Calculus Made Easy from www.Ti89.com navigating to the Multivariable Calculus menu under F6, then go to F3 to select Find Jacobian or Hessian.
Check the screenshots below.
To enter a vector-valued function use [ , ] notation to enter it on a TI89.
The resulting Jacobian is:
To then find the Hessian Matrix enter a real-valued function the usual way.
The resulting Hessian is: