Spatial integration

Here, the spatial integral is to be carried out over the entire volume of the vessel containing the gas, and for convenience we have changed the notation slightly. Now we differentiate//with time. 

We are going to carry out some spatial integrations here. We suppose that tire distribution function vanishes at the surface of the container and that there is no flow of energy or momentum into or out of the container. (We mention in passing that it is possible to relax this latter condition and thereby obtain a more general fonn of the second law than we discuss here. This requires a carefiil analysis of the wall-collision temi The interested reader is referred to the article by Dorfman and van Beijeren [14]. Here, we will drop the wall operator since for the purposes of this discussion it merely ensures tliat the distribution fiinction vanishes at the surface of the container.) The first temi can be written as... 

It can be observed that we can separate the overlap integral into the product of three independent spatial integrals... 

The vane anemometer s physical dimensions are often quite large (compared with other local velocity measurement instruments). It does not strictly measure a local velocity at all, but rather provides a spatially integrated mean value. This is an advantage in many cases where the air volume flow rate has to be predicted using local velocities and an integration principle. 

Spatially Integrated Resonance Raman Measurements of Macular Pigment.90... 

SPATIALLY INTEGRATED RESONANCE RAMAN MEASUREMENTS OF MACULAR PIGMENT... 

Two objects are similar and have similar properties to the extent that they have similar distributions of charge in real space. Thus chemical similarity should be defined and determined using the atoms of QTAIM whose properties are directly determined by their spatial charge distributions [32]. Current measures of molecular similarity are couched in terms of Carbo s molecular quantum similarity measure (MQSM) [33-35], a procedure that requires maximization of the spatial integration of the overlap of the density distributions of two molecules the similarity of which is to be determined, and where the product of the density distributions can be weighted by some operator [36]. The MQSM method has several difficulties associated with its implementation [31] ... 

A word of caution should be raised. The formidable process of conducting such spatial integration should be viewed as principally symbolic. Although the mathematical operations will be valid, it is rare that in our application they will be so complicated. Indeed, in most cases, these operations will be very simple. For example, iff(x,y,z, t) is uniform in space, then we simply have, for Equation (3.1),... 

G.L. Schott, "Chain Branching and Initiation Rates Measured by Spatially Integrated Light Emission During Reflected Shock-Wave Ignition , Ibid, pp 569-78... 

Figure 22. Directed (a) and spatially integrating (b) probes for the measurement of irradiances at any point within a photochemical reactor [2, 3, 6, 78-80],...

Assuming that the diagonal matrix elements of V vanish, multiplying Eq. (15.20) in turn by and rj>B(t) and carrying out the spatial integrations yields... 

This result has the form of the well known Flory-Huggins approximation, evaluated in the limit of small concentrations ( dc defined chemical potential.) The term In (fdCj,(n)) can be interpreted as an entropic contribution of nonintcraeting chains and u nc is the interaction energy with a homogeneous background of other chains. The constant shift ln(4rr)d/2 is due to our normalization of the spatial integrals in the partition function. 

Fig. 12.12. Macular pigment levels measured with spatially integrated Raman detection in about 100 volunteer subjects ranging in age between 20 and 50 years. Each circle represents a subject s MP concentration. MP levels can vary strongly between subjects...

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