1

Just began to read about PDEs. There are a whole list of notations which I don't understand and the book isn't expecting a reader who is as inexperienced as me.

Also don't understand what this means: $U_T = U \times (0,T]$

I know the bit in the back is about a range. But why is there a multiplication sign in the middle?

Thanks a lot in advance.

MonkeyKing
  • 3,178
PDElearner
  • 13

2 Answers2

2

In general $A\times B$ for sets is the set of pairs $(a, b)$, where $a\in A$ and $b\in B$.

Henrik supports the community
  • 6,831
2

It means a Cartesian product. Note that since it is a "product", it is natural to have "power". $\Bbb R^n = \Bbb R \times \Bbb R \times \dots \times \Bbb R$ for $n$ times.

$U_T$ is probably a notation of the author.

MonkeyKing
  • 3,178
  • Thanks a lot. Now the equation makes perfect sense to me. The $U_T$ is taken from Appendix A.2 Geomatric Notations from Lawrence Evans's book on PDE. Could this mean that it is differentiable T times? Purely guessing... – PDElearner Apr 28 '15 at 17:46
  • @PDElearner The usual notation for $n$th derivative is a superscript (with parentheses to prevent confusion with power), and notation for a function $f$ being differentiable for $n$ times is $f \in C^n$. (see this) $U_T$ you mentioned just looks like the domain of pde, but $u_t$ can mean the partial derivative of $u$. You'd better read context for definition of these notations. BTW if you think this answer is the most helpful, please hit the "click" button on the left hard! :) – MonkeyKing Apr 28 '15 at 22:57
  • I mean "tick" button – MonkeyKing Apr 28 '15 at 23:03