Solid angle to the unit sphere with center P, which can be calculated as the surface integral: where formula_5 is the unit vector corresponding to formula_6, the position vector of an infinitesimal area of surface with respect to point P, and where formula_7 represents the unit normal vector to . Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product formula_8. Thus one can approximate the solid angle subtended by a small facet having flat surface area ,
Surface integral of the graph of some scalar function, say formula_3, we have where formula_5. So that formula_6, and formula_7. So, which is the standard formula for the area of a surface described this way. One can recognize the vector in the second line above as the normal vector to the surface. Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space. This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface. Consider