Helmholtz theorem

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

According to the Helmholtz theorem, the two-dimensional vector field can be represented as a sum of an irrotational field and of a solenoidal one... 

Helmholtz theorem [70], into a longitudinal part and a transverse... 

Helmholtz theorem is a direct consequence of Fourier s formalism. In order to demonstrate this claim, we shall first consider the direct and inverse transforms of the B(r) function, respectively known as... 

Helmholtz theorem for a function ( , ts) that depends on space and on retarded time ts = t — R/c can be generalized as follows ... 

Once again, Heras [13] gave another formulation of Helmholtz theorem now based on Green s retarded function G = 8(f — ts — R/c)/R for a value B5(r5. ts) function of space and retarded time ts = t — R/c. Heras obtained the following equation ... 

In Helmholtz theorem, both and A potentials form a group composed of four scalar components in order to describe a vectorial field having three scalar components. As a consequence, the four scalar components are not independent. Their dependence can be expressed in several ways ... 

Nonadiabatic couplings fjj1 for all pairs of electronic eigenstates d>k and diagonal matrix elements = iff iff is a vector held and, therefore, according to Helmholtz theorem [45] (see also, e.g. Ref. [46]... 

Let us analyze the space and time structure of the elastic displacement field in detail. We will demonstrate that equation (13.26) describes the propagation of two types of body waves in an elastic medium, i.e., compressional and shear waves travelling at different velocities and featuring different physical properties. To this end, let us recall the well-known Helmholtz theorem according to which an arbitrary vector field, in particular an elastic displacement field U(r), may be represented as a sum of a potential, Up(r), and a solenoidal, Us(r), field (Zhdanov, 1988) ... 

The Helmholtz theorem states that any vector field F can be written as a sum of its transverse F- - and longitudinal F" components... 

Using the Helmholtz theorem the vector potential of the scattered wave — ... 

The second inhomogeneous equation (80) splits into three, one for each component of the current density j. Maxwell s equations accordingly constitute a system of six coupled first-order differential equations for the components of E and B. A more compact formulation is obtained with the introduction of potentials. We invoke Helmholtz theorem [37] stating that the fields, in view of the boundary conditions stated above, can be written as V = -I-V-, where the... 

This is a generalization of the Helmholtz theorem for three-dimensional space [28]. W,(x) and W,(x) are called the longitudinal and transverse parts of W(x), respectively. The... 

An interesting proposal has been made how to go beyond the singular coupling terms and remove all of the removable (i.e. also nonsingular) coupling terms. This employs the decomposition, according to the Helmholtz theorem, of Fi2(Q) into a longitudinal (removable) and a transverse (nonremovable) partd ... 

Eq. (10) indicates that the sources in the heart can be replaced by an equivalent double layer on the surface of the heart. From a theorem of Helmholtz, this double layer should be o times the voltage which would appear on the surface of the heart when it is insulated (Helmholtz, 1853) (Thevenin s theorem in circuit theory is a special case of Helmholtz theorem.) From Eq. (6) it is evident that Eq. (10) is consistent with Helmholtz theorem. 

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