### Quick definitions + notations

Posted on 21/03/2017, in Mathematics.## Definitions

**Self adjoint matrix** (https://en.wikipedia.org/wiki/Hermitian_matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j.

### Find triangle’s surface via its vertices coordinates

For triangle $T$ with $(x_i,y_i), i\in\overline{1,3}$ (see the note of Zhilin and `50 lines matlab fem`

in `docs\matlab fem`

)

## Notations

- $C^1_c(\Omega)$ : the space of continuously differential functions with compact support in $\Omega$.
- $\text{supp}\,f$ : for a function $f$ (defined on $\Omega$), this denotes the support set, i.e. the set on which $f\ne 0$.

- $C(\Omega)$ : continuous functions from $\Omega \to \mathbb{R}$.
- $C(\bar{\Omega})$ : the subset of $C(\Omega)$ consisting of functions that extend continuously to $\partial \Omega$.
- $C_0(\Omega)$ : subset of $C(\bar{\Omega})$ consisting of functions which vanish on $\partial\Omega$
- $G\Subset \Omega$ : if $\overline{G}\subset \Omega$ and $\overline{G}$ is compact (that is, closed and bounded) subset of $\Omega$. Xem trang 7 Adams.
- $f$ is
**compact support**in $\Omega$ if $\text{supp}f\Subset \Omega$.- I cannot explain all the features, but one nice property of a compactly supported function $f$ defined on an open set $\Omega$ is that there exists a compact set $K$ in $\Omega$ such that $f(x)=0$ if $x\not\in K$. This is useful since for instance when you do integration by parts on $f$,
**the boundary terms vanish**. If $g$ is any function, you can get a compactly supported function $fg$ by multiplying it by $f$. This is called the*cut-off*process.

- I cannot explain all the features, but one nice property of a compactly supported function $f$ defined on an open set $\Omega$ is that there exists a compact set $K$ in $\Omega$ such that $f(x)=0$ if $x\not\in K$. This is useful since for instance when you do integration by parts on $f$,
- $f$ is
**smooth**or $f\in C^{\infty}$ : a function that has derivatives of all orders everywhere in its domain. - Sobolev space from Ishihara1982[^Ishihara1982]: Nói rõ về định nghĩa $W^{r,p}(\Omega)$ và norms cũng như không gian $H^1(\Omega),H^1_0(\Omega)$.