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Complement to surface integration

In the bottom part of the Figure below we show part of a surface (in gray) and a triangulation of it. In general only the vertices of the triangulation lay on the surface and it is important that this triangulation correspond to the triangulation of the rectangle showed in the top part of the Figure. The planar triangulation is obtained by dividing the rectangle in equal squares (of edge h) and then by choosing a diagonal of the rectangle as a reference one. To obtain the triangulation of the rectangle we split each square along the reference diagonal. We labelled the vertices of the planar triangulation as (i,j),(i+1,j), etc, and all the vertices are mapped to the surface and correspond to the points (si,tj),(si+1,tj), etc. For sake of simplicity we show here how a square (magenta) is mapped from the planar rectangle to the corresponding part of the surface.

At each vertex we associate two triangles, an upstream and a downstream one, where the direction is along the diagonal of the rectangle opposite to the reference one. The surface of the integral can then be approximated as the sum on the area of all the squares of the triangulation, and the latter approximated value converge to the actual area of the surface when the edge length h goes to zeros.

The key point of this construction is that at each vertex we associate the values of area of the upstream and downstream triangle and that, thanks to that specific triangulation, all the surface is recovered, i.e, the triangulation on the surface has no holes.