Left-derivative

Strictly this is the right derivative. We can also define a left derivative... 

Again the right derivative differs from the left derivative, as indicated by the sign. The maxima of/ indicate regions in the molecule, which prefer attack by a nucleophile, while/ exhibits maxima at sites susceptible to an attack by an electrophile. In other words,indicates where increase of electron density is energetically favorable, while/ is maximal where decrease of electron density is preferred. Practically, the Fukui functions are calculated by finite differences, e.g. ... 

The A-functor F is left-derivable if there exists a A-functor... 

Such a pair (RF, C) (respectively (LF, )) is called a right-derived (respectively left-derived) functor of F. 

Example 2.1.3. If F J —> E transforms quasi-isomorphisms into isomorphisms then F = Fog for a unique F Dj —> E and (F, identity) is both a right-derived and a left-derived functor of F. 

A similar statement holds for left-derived functors when there exists a family as in (2.2). 

This checking is left to the reader, as is the proof for left-derived functors. 

We leave the corresponding statements for left-F-acyclic complexes and left-derived functors to the reader. 

By the lemma, the left derived functor Lf, which we already know its existence by Lemma 6.25, can also be calculated by X-fiat resolutions. 

In the preceding sections we described the rotationa) behavior of ellipsoids with transiticms directed along one of the symmetry axes. Although not frequently used, the complete expression for an dlipsoid with three nonequivalent axes(Hguie 12.1. left), derived without restricting one of the transitions to be on an axis, is useful to have. In this case die anisotrt decays with flve appvrat ccwielatitm times " ... 

Figure 9. Energy-optimized stracture of methane (left) derived from a high-level DFT calculation and showing the extent of the 0.1 e/A electron density contour superimposed onto the ball-and-stick representation of the molecule. A slice of the electron density taken through one of the C-H-H planes (right) shows the covalent nature of the C-H bond with the buildup of charge along the C-H axes.

This expresses only the left, derivative side of equation (7-2). Now invert and solve for the slope function s in pH units per milliliter... 

It is well known for the case of planar layer growth that, for the growth of a single intermediate phase, the difference in the right and the left derivatives would be proportional to the driving force of the reaction,... 

Dilute CuSC>4(aq) (left) derives its pale blue color from the complex ions [Cu(H20)4]2+. When NH3(aq) is added (here labeled "cone ammonium hydroxide"). 

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