Integration scalar product

We recognize the integral /F " d/f as the macroscopic work at surface element r, because it is the integrated scalar product of the force exerted by the surroundings and the displacement. The total macroscopic work during the process is then given by... 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

In order to express equations in general terms, we adopt the notation J dr to indicate integration over the full range of all the coordinates of the system being considered and write the scalar product in the form... 

The bracket (bra-c-ket) in

) provides the names for the component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket tp) with its corresponding bra (-01 gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra tpj and the ket Aj>i) is expressed in Dirac notation as (0yjA 0,) or as J A i). These scalar products are also known as the matrix elements of A and are sometimes denoted by Ay. 

If, in a vector space of an infinite number of dimensions the components Ai and Bi become continuously distributed and everywhere dense, i is no longer a denumerable index but a continuous variable (x) and the scalar product turns into an overlap integral f A(x)B(x)dx. If it is zero the functions A and B are said to be orthogonal. This type of function is more suitable for describing wave motion. 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

This kind of the scalar product is implicit in our derivations of the path integral. Expanding (48) in a power series, multiplying with respect to this scalar product, and resumming, we get the monodromy matrix ... 

Exercise 6.13 Suppose (G, V, p) is a Lie group representation where G is a Lie group with a volume-one invariant integral and V is a complex scalar product space ( , ), Then there is a complex scalar product ( , p on V such that p is a unitary representation on V with respect to ( , p. (Hint define... 

Proposition 7.2 is crucial to our proof in Section 7.2 that the spherical harmonics span the complex scalar product space L (S ) of square-integrable functions on the two-sphere. 

Here the first equality follows from the fact that f e C. The technical continuity condition on f and its first and second partial derivatives allows us to exchange the derivative and the integral sign (disguised as a complex scalar product). See, for example, [Bart, Theorem 31.7]. The third equality follows from the Hermitian symmetry of It follows that is an element... 

Because S is a linear operator and WQq is a non-negative function, the expression on the right can be shown to have all the necessary properties of a scalar product, provided that Weq and the phase-space functions F, G appropriate to each thermodynamic variable can be defined and the integral exists. Equation (13.53) is then treated as the equilibrium dot product inherited from the phase-space vectors and dot product in (13.54). 

The approach described can be extended to a more complicated nonspherical case. Similar to Equation (1.154), we consider a neutral system composed of two Born spheres with 61 = 6 and 62 = — 6- It is usually called The dumbbell . For the isolated spheres we denote their charge densities as px and p2, their response fields as defined similar to the single sphere case. The solvation energy for such system equals to UsoXy = 0.5[(scalar products mean volume integrals. The reasonable estimate for separate terms in will be Ut = 0.5 ((P-p ), (i = 1, 2), Uini(R) = ( 1 2) = ( VPi)> where Ux and U2 are solvation energies obtained in terms of Equation (1.153) whereas the interaction energy is identified with Equation (1.154). In this result we assume that the... 

Upon taking the scalar product of r2 — rA and Equation 20, with summation over the components / and integration over the coordinates 1 and 2, it is found after rearrangement that... 

The space structure via the scalar product as the generalized Stieltjes integral... 

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