\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)

The purpose of this worksheet is, to work towards
understanding how FreeFem works.

(%i1) MeshMat: matrix([dx_edge1, dx_edge2],
                   [dy_edge1, dy_edge2]);
\[\tag{MeshMat}\label{MeshMat}\begin{pmatrix}\mathit{dx\_ edge1} & \mathit{dx\_ edge2}\\ \mathit{dy\_ edge1} & \mathit{dy\_ edge2}\end{pmatrix}\]

The proposition is that MeshMat . [a, b] yields [x, y]

(%i2) LinearBasis: invert(MeshMat);
\[\tag{LinearBasis}\label{LinearBasis}\begin{pmatrix}\frac{\mathit{dy\_ edge2}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}} & -\frac{\mathit{dx\_ edge2}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}}\\ -\frac{\mathit{dy\_ edge1}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}} & \frac{\mathit{dx\_ edge1}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}}\end{pmatrix}\]

The proposition is that LinearBasis . [x, y] yields <a, b>

(%i3) BasisDx: LinearBasis . [1, 0];
\[\tag{BasisDx}\label{BasisDx}\begin{pmatrix}\frac{\mathit{dy\_ edge2}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}}\\ -\frac{\mathit{dy\_ edge1}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}}\end{pmatrix}\]
(%i4) BasisDy: LinearBasis . [0, 1];
\[\tag{BasisDy}\label{BasisDy}\begin{pmatrix}-\frac{\mathit{dx\_ edge2}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}}\\ \frac{\mathit{dx\_ edge1}}{\mathit{dx\_ edge1}\mathit{dy\_ edge2}-\mathit{dx\_ edge2}\mathit{dy\_ edge1}}\end{pmatrix}\]

Also, (dx_edge1 dy_edge2) - (dx_edge2 dy_edge1)
was the determinant of MeshMat.

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