Chapter 5

Let a function \(\phi\) vary linearly over face \(1-2-3\) of a tetrahedron. Valucs of \(\phi\) at corner nodes on this face are \(\phi_{1}, \phi_{2}\), and \(\phi_{3} .\) The integral of \(\phi\) over face \(1-2-3\) is \(A_{123}\left(\phi_{1}+\phi_{2}+\phi_{3}\right) / 3\). Derive this result by formal integration.

For \(r+s=1\) and again for \(r+s=2\), write integration formulas analogous to Eq. 5.2-9 if the origin of coordinates need not be at the centroid of the triangle. (Let the centroid have coordinates \(x=x_{c}\) and \(y=y_{c}\).)