As noted in Section 3.15, for single crystals of some substances, the physical properties are anisotropic; that is, they are dependent on crystallographic direction. One such property is the modulus of elasticity. For cubic single crystals, the modulus of elasticity in a general [uvw] direction, Euvw, is described by the relationship

where and are the moduli of elasticity in [100] and [111] directions, respectively; ?, ?, and ? are the cosines of the angles between [uvw] and the respective [100], [010], and [001] directions. Verify that the values for aluminum, copper, and iron in Table 3.3 are correct.

(a*6 + 6*y +va?} E, Ew Eyool