Gauss Legendre–Gauss Jacobi quadrature rules over a tetrahedral region

This paper presents a Gaussian Quadrature method for the evaluation of the triple integral ∫∫T∫f(x,y,z)dxdydz, where f(x,y,z)f(x,y,z) is an analytic function in x, y, z and T   refers to the standard tetrahedral region: {(x,y,z)|0⩽x,y,z⩽1,x+y+z⩽1}{(x,y,z)0⩽x,y,z⩽1,x+y+z⩽1} in three space (x,y,z)(x,y,z). Mathematical transformation from (x,y,z)(x,y,z) space to (U,V,W)(U,V,W) space map the standard tetrahedron T   in (x,y,z)(x,y,z) space to a standard 1-cube: {(U,V,W)/0⩽U,V,W⩽1}{(U,V,W)/0⩽U,V,W⩽1} in (U,V,W)(U,V,W) space. Then we use the product of Gauss Legendre and Gauss Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T.