# Integration formulae for a tetrahedron

1. May 24, 2009

### soikez

Hello everybody!
I am trying to formulate a mathematical approach using Finite Element Method in an Electromagnetic problem in conjunction with Floquet theorem. Due to the fact I use tetrahedral elements, some definite integrals have been appeared. The fact is that I can compute analytically these integrals.

To be more specific letting a tetrahedron be defined in the x,y,z coordinate system by four point (x1,y1,z1),(x2,y2,z2),(x3,y3,z3),(x4,y4,z4) and the origin of the coordinates taken at the centroid we have:

\begin{align} \frac{x_1+x_2+x_3+x_4}{4}=\frac{y_1+y_2+y_3+y_4}{4}=\frac{z_1+z_2+z_3+z_4}{4} \label{eq:centroid} \end{align} Then integrating over the tetrahedron volume gives: \begin{align} \int{dxdydz}=\frac{1}{6} \begin{vmatrix} 1 & x_{1} & y_{1} & z_{1} \\ 1 & x_{2} & y_{2} & z_{2} \\ 1 & x_{3} & y_{3} & z_{3} \\ 1 & x_{4} & y_{4} & z_{4} \end{vmatrix}=V \rightarrow \quad \textrm{tetrahedron volume} \label{eq:tetrint1} \end{align} Moreover some useful integral quantities are: \begin{eqnarray} \int{xdxdydz}=&\int{ydxdydz}=\int{zdxdydz}=0 \\ \label{eq:tetrint2} \int{x^{2}dxdydz}=&\frac{V}{20}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})=\frac{V}{20}\sum_{i=1}^{4}x_{i}^{2} \\ \label{eq:tetrint3} \int{y^{2}dxdydz}=&\frac{V}{20}(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2})=\frac{V}{20}\sum_{i=1}^{4}y_{i}^{2} \\ \label{eq:tetrint4} \int{z^{2}dxdydz}=&\frac{V}{20}(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2})=\frac{V}{20}\sum_{i=1}^{4}z_{i}^{2} \\ \label{eq:tetrint5} \int{xydxdydz}=&\frac{V}{20}(x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4})=\frac{V}{20}\sum_{i=1}^{4}x_{i}y_{i} \\ \label{eq:tetrint6} \int{yzdxdydz}=&\frac{V}{20}(y_{1}z_{1}+y_{2}z_{2}+y_{3}z_{3}+y_{4}z_{4})=\frac{V}{20}\sum_{i=1}^{4}y_{i}z_{i} \\ \label{eq:tetrint7} \int{zxdxdydz}=&\frac{V}{20}(z_{1}x_{1}+z_{2}x_{2}+z_{3}x_{3}+z_{4}x_{4})=\frac{V}{20}\sum_{i=1}^{4}x_{i}z_{i} \label{eq:tetrint8} \end{eqnarray}

However, applying Floquet theorem, the integrals above are multiplied by $$e^{-j\beta{x}}$$.
Can anybody give me a hint how exactly these integrals can be computed?

Thanks!