Areal vector

A second question arises for those who understand the importance of dimensional analysis, a subject that is treated briefly in Appendix II. If A and B are both vector quantities with, say, dimensions of length, how can their cross product result in a vector C, presumably with dimensions of length The answer is hidden in the homogeneous equations developed above [Eqs. (IS) to (20)]. The constant a was set equal to unity. However, in this case it has the dimension of reciprocal length. In other words, C = aABsirtd is the length of the vector C. In general, a vector such as C which represents the cross product of two ordinary vectors is an areal vector with different symmetry properties from those of A and B. 

Consider an incompatible A/B interface reinforced by an areal density E of compatibilizer chains. The vector percolation theory term, p, the occupational probability of the lattice, will be proportional to the number of chains (E) times their length (L) divided by their thickness (X) ... 

Finally, for isochoric deformations (with volume conservation), the product of the areal and lineal stretches is equal to 1, and the sum of the rates of lineal and areal stretches per unit of initial length and area (respectively) is equal to zero, for the same orientation vector of the line and the area. The following example illustrates the various steps for the calculation of the striation thickness reduction function and the time average mixing efficiency for the simple shear case. 

---