Mathematical identities reciprocal

Unfortunately, even today, inverse (or converse) piezoelectricity is still sometimes called electrostriction because the name electrostriction suggests the electromechanical direction (electrical stimulus leads to mechanical response), while piezoelectricity seems to refer only to the opposite mechano-electrical direction. In order to avoid the misleading use of the term, it should be kept in mind that our modem terminology is based on phenomenological thermodynamical relations so that the linear effects of direct and inverse piezoelectricity must be identical due to the mathematically required reciprocity. 

In thermodynamics, this is referred to as a Maxwell equation. This equation is derived later in Section 4.8. Thus the effect of pMg on the binding of hydrogen ions is the same as the effect of pH on the binding of magnesium ions in short, these are reciprocal effects. The bindings of these two ions are referred to as linked functions. Equation 1.3-17 can be confirmed by plotting these two derivatives, and the same plot is obtained in both cases. This would be a lot of work to do by hand, but since Mathematical can take partial derivatives, this can be done readily with a computer. The two plots are identical and are given in Fig. 1.8. 

Equation (2.9) is known as the reciprocity relation or cross-differentiation identity. It follows from the fact that the order of differentiation of our original function 2 == z x, y) with respect to x and y is immaterial. Mathematically this is written... 

Every symmetry operation in the group has an inverse operation that is also a member of the group. In this context, the word inverse should not be confused with inversion. The mathematical inverse of an operation is its reciprocal, such that A A = A A = , where the symbol A represents the inverse of operation A. The identity element will always be its own inverse. Likewise, the inverse of any reflection operation will always be the original reflection. The inversion operation (/) is also its own inverse. The inverse of a C proper rotation (counterclockwise) will always be the symmetry operation that is equivalent to a C rotation in the opposite direction (clockwise). No two operations in the group can have the same inverse. The list of inverses for the symmetry operations in the ammonia symmetry group are as follows ... 

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