Laplacian of a scalar

Where then to look for the Lewis model, a model which in the light of its ubiquitous and constant use throughout chemistry must most certainly be rooted in the physics governing a molecular system If one reads the introductory chapter on fields in Morse and Feshbach s book Methods of theoretical physics (1953), one finds a statement to the effect that the Laplacian of a scalar field is a very important property, for it determines where the field is locally concentrated and depleted. The Laplacian of the charge density at a point r in space, the quantity V p(r), is defined in eqn (2.3). This property of the Laplacian of determining where electronic charge is locally concentrated and depleted follows from its definition as the limiting difference between the two first derivatives which bracket the point in question as defined in eqn (2.2) and illustrated in Fig. 2.2. 

C Trondheim Bubble Column Model The Laplacian of a Scalar Field... 

The analysis of p(r) leads to useful information about chemical bonding. Additional information on the electroiiic structure of Ng molecules can be gained by analyzing not only p(r) but also V p(r), the Laplacian of the electron density distribution. The Laplacian of a scalar hinction f indicates where this function concentrates (V f < 0) and where it is depleted (V f > 0) [33]. For f = p(r), the Laplace concentration — V p(r) reveals where the electrons lump together in the molecule [34]. 

Since the Laplacian of a scalar (i.e dimensionless molar density) is equivalent to the divergence of the gradient of that scalar, (19-18) can be rewritten as... 

The differential operators encountered often in the description of the physical properties of solids are the gradient of a scalar field V,4)(r), the divergence of a vector field Vr F(r), the curl of a vector field x F(r), and the laplacian of a scalar field V vector field is simply the vector addition of the laplacian of its components, V F = + V F z). These operators in three dimensions are... 

The Laplacian is defined as the divergence of the gradient. If / is a scalar function, its gradient is a vector and the divergence of the gradient of / is a scalar. In Cartesian coordinates the Laplacian of a scalar function / (x,y,z) is given by... 

The divergence of the gradient of a scalar function is called the Laplacian. The Laplacian of a scalar function / is given in Cartesian coordinates by... 

The Laplacian is constructed from second partial derivatives, so it is essentially a measure of the curvature of the function in three dimensions (Chapter 6). The Laplacian of any scalar field shows where the field is locally concentrated or depleted. The Laplacian has a negative value wherever the scalar field is locally concentrated and a positive value where it is locally depleted. The Laplacian of the electron density, p, shows where the electron density is locally concentrated or depleted. To understand this, we first look carefully at a onedimensional function and its first and second derivatives. 

In other words, the scalar component of the Laplacian of a vector is not equal to the Laplacian of the scalar component, even when there is only one component in the vector. In this case, owing to the form of cog, the last two terms in Eq. 6.48 cancel exactly. 

It is a disadvantage of the analysis of p(r) that many features of the orbital description of a molecule are not reflected by the properties of the electron density. However, this gap between orbital and pir) description can be closed by investigating the Laplacian of p(t), V2p(r)72. The Laplacian of any scalar function is negative where the scalar function concentrates, and it is positive where the scalar function is depleted. This becomes obvious, when considering the second derivative of a general function fix) ... 

The Laplacian operator is equivalent to the divergence of the gradient of a scalar function. 

Unlike grad (j) div a is seen to be a scalar function. The Laplacian function in terms of this notation is written... 

The first term on the right-hand side is the expected value of the scalar Laplacian conditioned on the scalars having values r//, 33 An example of the time evolution of the conditional scalar Laplacian, corresponding to the scalar PDF in Fig. 1.11, is plotted in Fig. 1.12 for an initially non-premixed inert-scalar field. The closure of the conditional scalar Laplacian is discussed in Chapter 6. For the time being, it suffices to note the similarity between (1.36) and the IEM model, (1.16). Indeed, the IEM model is a closure for the conditional scalar Laplacian, i.e.,... 

Finally, the Laplacian is defined by the divergence of the gradient. For a scalar quantity it is... 

As was stressed before, what is observed is the shear field. In practice it is likely that the measured shear field is not purely of cosmological origin due to contamination or systematics effects. The geometrical properties of the cosmological shear field can however be used to derive means for testing the amount of systematics in the data. The idea is the following. Any 2D spin 2 vector field can be decomposed into a scalar and a pseudo scalar parts. They are defined through their Laplacian,... 

The divergence of the gradient is commonly denoted the Laplacian, and is for example involved in the (non-relativistic) quantum mechanical kinetic energy operator. It operates on a scalar function and produces a scalar function. 

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