Evaluation of the surface integrals in the problem of three-dimensional (3D) heat conduction corresponding to the different boundary conditions requires the discretization of the body surfaces into surface finite elements that take the form of surface triangles. A set of natural coordinates would be advantageous in defining these elements, especially if these surfaces are curved.

The natural coordinates are local coordinates that vary in a range between zero and unity. At any of the element's vertices, one of these coordinates has a value of unity, whereas the others are all zeros. Use of these coordinates simplifies the evaluation of integrals in the element's equations. This additional advantage is a consequence of the existing closed-form integration formulas that evaluate these integrals.

The derivation given in this appendix generalizes the natural coordinates' definition for two-dimensional (2D) plane elements, all lying in one plane, to the case in which these plane elements exist in a 3D space. Such generalization was essential because the elements dealt with in the analysis lie on the 3D body-surface segments, which are, in turn, 3D.

Let A represent the area of the triangular element in Figure A.1, with i(xi, yi, zi),j(xj, yj, zj) and k(xk, yk, zk), denoting its vertices.