Laplacian operator, generalize

The quantity c = y/rjp is known as the jjjia e velocity, It is the speed a which waves travel along the string. Clearly, the left-hand side of Eq, (4) represents the one-dimensional Laplacian operating on the dependent variable. This expression can be easily generalized to represent wave phenomena in two or more dimensions in space. 

The total kinetic energy operator of the iV-electron system, in atomic units, will then be given by -A/2, where A is the generalized Laplacian operator for the d = 3N dimensional space ... 

Here, the first term on the right-hand side gives the net diffusive inflow of species A into the volume element. We have assumed that the diffusive process follows Fick s law and that the diffusion coefficient does not vary with position. The spatial derivative term V2a is the Laplacian operator, defined for a general three-dimensional body in x, y, z coordinates by... 

The right-hand side of eqn. (9), which is the diffusion equation or Fick s second law, involves two spherically symmetric derivatives of p(r, t). In the general case of three-dimensional space, lacking any symmetry, it can be shown that the Laplacian operator... 

From (3) it follows that if is the generalized Laplacian operator... 

Here is the generalized Laplacian operator, Z is a constant, and R is the hyperradius. In the D-dimensional case, the Fock transformation... 

A general symbol for a time-dependent wave function Thermodynamic probability - number of microstates Gradient operator Divergence operator Laplacian operator... 

In general the Laplacian operator involves all three spatial coordinates, but we will be concerned mainly with systems where the symmetry allows this diffusion term to be written as a function of a single spatial coordinate. We begin with a 1-D (slab) system in Section 4 we will consider 2-D (circle) and... 

Generally, the structure of polymers can be considered to be made from domains which can be discriminated easily from each other by a sufficient difference of their electron densities (contrast). Examples are copolymers consisting of soft and hard domains, semi-crystalline polymers (crystalline phase is denser than the amorphous regions) and porosity(voids) within a material. In this case it is advantageous [46, 60] to perform an edge enhancement by applying the Laplacian operator... 

We can generalize to three dimensions using the Laplacian operator... 

The differential operator in Eq. (102) will be recognized as the Laplacian and, taking the isothermal case with v = °° for simplicity, a balance of the same sort as was treated in the section entitled The General Balance Equations for Distributed Systems (3/3t = 0, f = —D grad c, g = /(c)) gives... 

The bracketed object in Eq (2.32) is called an operator. An operator is a generalization of the concept of a function. Whereas a function is a rale for turning one number into another, an operator is a rule for turning one function into another. The Laplacian is an example of an operator. We usually indicate that an object is an operator by placing a hat over it, eg., A. The action of an operator that turns the function / into the function g is represented by... 

The four-dimensional generalization of the Laplacian has been identified to be the d Alembert operator... 

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