I see an immediate problem. You have both and . These are not same! Similar for and
I've been working with the following PDE:
What I would like to do is to re-write the second-order spatial derivatives in this PDE as first order derivatives.
This is what I have attempted, but I am uncertain as to whether this is correct. I introduce another variable , and then:
I reason that this is correct since
Now is it reasonable to claim that the LHS of the two equations above are the same?
Danny: Thank you very much for checking this over. Are the problems that you see issues with notation, or can the two equations be written in a more precise fashion?
Thus
Essentially what I would like to do is to write the PDE in terms of first order derivatives.
Thanks, Danny: I had suspected that something was amiss!
What about this system? So starting with
This is re-written as:
Where q is a vector. I believe that this is correct since by taking the time derivative of the first equation:
Applying the del operator to the following:
We can thus say that both sides of the above are equal.