Vector differentiation identities

If this is an explicit equation with respect to a the estimation of the vector k is mathematically identical to a differential analysis. The only difference is that values of ki are searched, for which the concentrations calculated from the above equation are as close as possible to the measured concentrations. Below, a simple example illustrating both techniques is given. 

To obtain the Green s identities, or integral representations, of the Laplace equation, we define the vectors f = c/>VT and g = TV. Here, is an additional function that we will define later. For now, the only requirement for this function is to be two times differentiable in space. Substituting our new definition of the vector f, we get... 

The chain rule of differentiation will give us the Green s first identity for vector... 

Differentiation of a tensor with respect to a scalar does not change its rank. The spatial differentiation of a tensor raises its rank by unity, and identical to multiplication by the vector V, called del or Hamiltonian operator or the nabla... 

Note that the common practice is to compute the inner product of the differential operator and the basis function, using integration by parts, which in the 3-D case is based on the vector identity... 

Since every unit cell in the crystal lattice is identical to all others, it is said that the lattice can be primitive or centered. We already mentioned (Eq. 1.1) that a crystallographic lattice is based on three non-coplanar translations (vectors), thus the presence of lattice centering introduces additional translations that are different from the three basis translations. Properties of various lattices are summarized in Table 1.13 along with the international symbols adopted to differentiate between different lattice types. In a base-centered lattice, there are three different possibilities to select a pair of opposite faces, which is also reflected in Table 1.13. 

The kinetic term, v, is a function of the kinetic parameters vector P and the particle substrate and product concentrations, cs and cP, respectively. Ds and DP are the corresponding effective diffusion coefficients and r is the particle coordinate (in the case of spherical geometry it is the radial distance). Parameter n depends on the geometry of the biocatalyst particle and is 0,1,2 for a plate, a cylinder and a sphere, respectively. Since concentrations on the particle surface are assumed to be identical with bulk concentrations, boundary conditions do not include the influence of external mass transfer. Solving the above differential equations, the observed reaction rate in the packed bed is evaluated from the rate of substrate flux to the particle or of product flux from the particle... 

For the solution of this differential equation it is advantageous to transform the matrix equation of Hilbert space into a vector equation in Liouville space ( /, is the identity matrix of the four-dimensional subspace) ... 

X, y, and z transform under proper and improper rotations by identically the same matrices as do the Cartesian unit base vectors i, j, and k. The products of the differential operators and linear combinations of them also have the same transformation properties as the corresponding Cartesian unit base tensors or their linear combinations. The same holds true for the functions x, y, and z. For example. 

The simplest, from the viewpoint of topological structure, are the linear polymers. Depending on the number m of the types of monomeric units they differentiate homopolymers (ra=1) and copolymers (m>2). In the most trivial case molecules in a homopolymer are merely identified by the number Z of monomeric units involved, whereas the composition of a copolymer macromolecule is defined by vector 1 with components lly..., Za,..., Zm equal to the numbers of monomeric units of each type. At identical composition these molecules can vary in microstructure which is characterized by the manner of alternation of different units in a copolymer chain. Because the values of the average degree of polymerization l=lx+... +Zm in synthetic copolymers normally constitute 102-104 it becomes clear that the number of conceivable types of isomers with different microstructure turns out to be practically infinite. Naturally, a quantitative description of any polymer specimen comprising macromolecules with such an impressive number of configurations can be performed exclusively by statistical methods. 

Here, a is the total stress, p the isotropic pressure, I the identity (imit) tensor, and t the extra stress (ie, the stress in excess of the isotropic pressure). V is the gradient differential operator, and v is the velocity vector denotes the transpose of a tensor. For a one-dimensional flow with a single velocity component V, in which v varies in a single spatial direction y that is transverse to the flow direction, equation 2 simplifies to the famihar form... 

Let 5 be a continuous differentiable scalar and vectors V, A, and B also continuous and differentiable. So, the following identities are valid [1,2] ... 

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