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 energy radius formula_1 and length formula_2 at high temperature and a constant uniaxial tension formula_3, then at equilibrium, the variation of the total Helmholtz free energy vanishes and we have where formula_5 is the Helmholtz free energy and formula_6 is the surface area of the rod: Also, since the volume (formula_8) of the rod remains constant, the variation (formula_9) of the volume is zero, i.e., Therefore, the surface energy density can be expressed as The surface energy density of the solid can be computed by measuring formula_3, formula_1, and formula_2 at equilibrium. This method is valid only if the solid is