Parallelepiped, volume

Rectangular Parallelepiped Volume = ahc surface area = 2 ab AacA be)-, diagonal = where a, h, c are the lengths... 

The total increasing of the diffused component for the parallelepiped volume is ... 

It should be noted that the positive sign of this result depends on the choice of a right-handed coordinate system in which the angle is acute. The relation developed here for the volume of a parallelepiped is often employed in crystallography to calculate the volume of a unit cell, as shown in the following section. 

From the experimental fact that the lamellar thickening growth rate (17) of ECSC of PE is independent of l [8], we can regard ve on the end surface of the nucleus as constant and independent of l. For simplicity, we assume that the shape of ECSC is rectangular parallelepiped (Fig. 28). ve is given by the ratio of the volume of the crystal to the surface area of the crystal as shown below,... 

The crystal axes, a, b, and c, form three adjacent edges of a parallelepiped. The smallest parallelepiped built upon the three unit translations is known as the unit cell. Although the unit cell is an imaginary construct, it has an actual shape and definite volume. The crystal... 

Since the molecular volumes of neither the TTF nor the TCNQ molecule are easily described by a regular volume of integration, the parallelepiped subunit method was used and integration of the valence density was performed using Eqs. 

For complicated volumes of integration, the parallelepiped divisioning can again be used. We write... 

Analytic results for cylinders comparable to those discussed for spheroids are not available. However, Heiss and Coull (H4) reported accurate experimental determinations for cylinders, spheroids, and rectangular parallelepipeds, and developed a general correlation for settling factors. In terms of the volume drag ratio,, their results may be written ... 

The dual axial vector in 4-space is constructed geometrically from the integral over a hypersurface, or manifold, a rank 3-tensor in 4-space antisymmetric in all three indices [101]. In three-dimensional space, the volume of the parallelepiped spanned by three vectors is equal to the determinant of the third rank formed from the components of the vectors. In four dimensions, the projections can be defined analogously of the volume of the parallelepiped (i.e., areas of the hypersurface) spanned by three vector elements < dl, dx and dx". They are given by the determinant... 

Calculate the partial derivatives F, F and H pF. Each of these is a vector in R". Find the volume of the parallelepiped they span. Then show that the volume element on the three-sphere is sin fi sinOdfidOdfi. 

Although we have focused on the 2 x 2 case, the results are easily generalized to Gram matrices of any dimension /. If we write Vol Ri, R2,..., Ry for the volume of the parallelepiped spanned by the vectors R, then we can show analogously that... 

The last relation can easily be checked if the original volume dVo is taken as the volume of a parallelepiped whose sides are situated along the co-ordinate... 

As seen from Fig. 2a, the van der Waals gap width is modulated periodically in positions of In atoms it is larger than in positions of Se atoms. This gap can be described as a layer of closely packed parallelepipeds, at the both ends of which pyramids are placed. The volume of this body (cavity) is equal to 50.5 A3, and for an ideal crystal, when defects of the dislocation type are absent the relative gap volume comprises 43% of the crystal bulk. It is obviously larger than the lower estimate 37% obtained using the ratio Q/(Q+Ci). 

However, the MDDF is not spherical. It follows the shape of the solute and there is no obvious choice for the normalization volume 8V. One possible choice was proposed [50] and it was used in the case of benzophenone, which is a parallelepiped normalization, where the molecule is represented by a box of dimensions a, b, c that characterize its size. For benzophenone, these dimensions were 11.5, 7.0 and 4.0 A [50], and the 8V is a parallelepiped shell between [r, r+] given by... 

Equation (8.2.7) assumed an orthogonal macroscopic crystal of orthorhombic symmetry for a triclinic parallelepiped, the projections of k would be along a, b, c axes, with ft, nyz, nz, integers. For simplicity, hereinafter we assume a large cubic crystal, of molar dimensions, so that A = B = C = L and with volume V = L3. 

With homogeneous strain, the deformation is proportionately identical for each volume element of the body and for the body as a whole. Hence, the principal axes, to which the strain may be referred, remain mutually perpendicular during the deformation. Thus, a unit cube (with its edges parallel to the principal strain directions) in the unstrained body becomes a rectangular parallelepiped, or parallelogram, while a circle becomes an ellipse and a unit sphere becomes a triaxial ellipsoid. Homogeneous strain occurs in crystals subjected to small uniform temperature changes and in crystals subjected to hydrostatic pressure. 

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