Triple integrals

The fact that the species transferred across the sensor membrane (the analyte or reaction product) must be a gas limits application of this type of flowthrough sensor, which, however, is still more versatile than are the sensors based on integrated separation (gas diffusion) and detection [4] described in Section 4.2 in fact, while these latter can only exploit physico-chemical properties of the analytes transferred, sensors based on triple integration allow the implementation of a (bio)chemical reaction and formation of a reaction product, so they are applicable to a much wider variety of systems with adequate sensitivity and selectivity. 

Publish a technical paper with a triple integral symbol (achieved 1986)... 

Since // involves only q, the triple integral (4.40) can be written as the product of an integral over q, and the integral... 

The denominator is finite. In the numerator we may omit the factor nR (z because it is finite and non-zero and hence does not affect the convergence. The remaining triple integral can be split ... 

Figure 3. Diagrams showing how to divide the triple integral in (2.7) to get the six terms of (2.10). Domains (a), (b), and (c) correspond to the three Liouville space paths given in Fig. 2 and domains (d), (e), and (/) to the complex-conjugate paths.

Following van der Waals and Platteeuw (1959, pp. 26ff) the individual particle partition function is related to the product of three factors (1) the cube of the de Broglie wavelength, (2) the internal partition function, and (3) the configurational triple integral, as... 

The cavities are assumed to be spherically symmetric, which enables the elimination of the two angular portions of the triple integral, resulting in 47t. Substitution of the resulting equation into Equation 5.21 yields the final expression for the Langmuir constant in terms of the particle potential within the cavity. 

This transform is a triple sum rather than a triple integral because the Fhkis represent a set of discrete entities the reflections of the diffraction pattern. The transform of a discrete function, such as the reciprocal lattice of measured intensities, is a summation of discrete values of the function. The transform of a continuous function, such as p(x,y,z), is an integral, which you can think of as a sum also, but a sum of an infinite number of infinitesimals. 

The probability of finding the particle in an infinitesimal cube of sides dx, dy, dz is ij/2dxdydz, and the probability of finding the particle somewhere in a volume V is the integral over that volume of iJ/2 with respect to dx, dy, dz (a triple integral) ij/2 is thus a probability density function, with units of probability per unit volume. Bom s interpretation was in terms of the probability of a particular state, Pauli s the chemist s usual view, that of a particular location. 

Consequently the required matrix element in (8.495) may be expressed as the triple integral... 

A simple method for checking the results shown below is to assume that / is independent of spatial coordinates. In that case the triple integral of Eq. (1.3.28) may be replaced by Vf(V, Y), so that we obtain P = —(d(Vf)/dV)y the remaining equations simplify correspondingly. 

Equations (32), (33) and (35) allow us to calculate C2/C2 p as a function of B and X for different values of H213 (Hamaker constant). Due to the complexity of calculation (evaluation of two triple Integrals), this is best done on a computer. Typical results of computer calculations are shown in Figure 7. The effect of Van der Waals attractive forces will lead to an accumulation of solute molecules near the walls for small values of X but at large values of X, the positions close to the pore axis start being preferred. 

The only term which will contribute to the average in this equation is the one involving the triple integral. This will vanish for a white noise driving force because by Isserlis s Theorem, all odd values are zero. The same applies for all odd powers of — y. For even powers of — y, all terms will be 0(A/) or higher. 

By virtue of the "sampling" property of the delta function, the triple integral in (17) collapses to a single integral, yielding... 

Fourier transformation over k of this product of two functions produces the convolution of the Fourier transforms of each function. The Fourier transform of the FID f(ky) is the spectrum F(x), and the Fourier transform of the triple integral is a projection P(x) = f f Mo(x, y, z) dy dz of the spin density on to the X axis, where frequency and space are related by Equation (5.4). Then the Fourier transform of Equation (5.10) can be written as... 

Many triple integrals in quantum mechanics are factored in the same way as in this example. 

A triple integral in Cartesian coordinates is transformed into a triple integral in spherical polar coordinates by... 

We can also imagine a solid to be split up into an infinite number of little parallelopipeds along the three dimensions x, y, z. These infinitesimal figures may be called volume elements. The capacity of each little element dx x dy x dz. The total volume, v, of the solid is represented by the triple integral... 

In the same way, quadruple and higher integrals may occur. These, however, are not very common. Multiple integration rarely extends beyond triple integrals. 

J. H. van der Waals seeks the maximum value of a triple integral in his Binttre Oemische, Leipzig, 34,1900, but the physical conditions of the problem enable the solution of (90) to be obtained in a simple manner. 

Consider the case of a particle in a three-dimensional box with dimensions a, b, and c and a total energy, E. The phase space volume consists of two sets of integrals, one over the dimensions, dx, dy, and dz, the other over the corresponding momenta. The triple integral over dxdydz yields the volume of the box, V = abc, so that the phase space volume (PSV) is given by... 

The compact notation dk disguises the fact that this is still a triple integral over the components of k. The problem is the awkward appearance of the factor... 

A torus or anchor ring, drawn in Fig. 1.3, is the approximate shape of a donut or bagel. The radii R and r refer, respectively, to the circle through the center of the torus and the circle made by a cross-sectional cut. Generally, to determine the area and volume of a surface of revolution, it is necessary to evaluate double or triple integrals. However, long before calculus was invented. Pappus of Alexandria (ca. Third Century A.D.) proposed two theorems that can give the same results much more directly. 

The angular integral can be broken into two, one of which contains Ti.i and the other Ki.. According to the triple integral relation in Comment 9.6, the integral... 

In Figure 13, we show the triple integral (Equation 128) calculated for the TEMqosi cavity mode as a function of t, for p = 0.5, corresponding to a cosO 5(0) density distribution, a frequency Vq =... 

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