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I have to integrate on a triangular domain

[tex]

\int_T f(x,y,z)dxdydz

[/tex]

so I use simplex coordinates, i.e.

[tex]x=(1-\alpha-\beta)x_1+\alpha x_2+\beta x_3[/tex]

[tex]y=(1-\alpha-\beta)y_1+\alpha y_2+\beta y_3[/tex]

[tex]z=(1-\alpha-\beta)z_1+\alpha z_2+\beta z_3[/tex]

where [tex](x_i,y_i,z_i)[/tex] are the vertices of the triangle and [tex] 0\le\alpha\le1,\ 0\le\beta\le1-\alpha[/tex]

So what is the Jacobian? When I try to calculate it, the matrix is not square because I have two variables and three equations!

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# Integrate on a triangular domain

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