Homogeneous harmonic

Physicists are familiar with many special functions that arise over and over again in solutions to various problems. The analysis of problems with spherical symmetry in P often appeal to the spherical harmonic functions, often called simply spherical harmonics. Spherical harmonics are the restrictions of homogeneous harmonic polynomials of three variables to the sphere S. In this section we will give a typical physics-style introduction to spherical harmonics. Here we state, but do not prove, their relationship to homogeneous harmonic polynomials a formal statement and proof are given Proposition A. 2 of Appendix A. 

Relating the spherical harmonic functions introduced here to the homogeneous harmonic polynomials is not logically necessary in this book. Morally, however, the calculation is well worth doing, in the name of better communication between mathematics and physics. Because this calculation is a bit tricky, we have postponed it to Appendix A. 

Restrictions of homogeneous harmonic polynomials play an important role in our analysis. 

Definition 2.6 Suppose f is a nonnegative integer. Define the vector space of homogeneous harmonic polynomials of degree f in three variables... 

In Section 7.1 we will use this characterization of homogeneous harmonic polynomials as a kernel of a linear transformation (along with the Fundamental Theorem of Linear Algebra, Proposition 2.5) to calculate the dimensions of the spaces of the spherical harmonics. 

All of the concepts of this section — kernel, image. Fundamental Theorem, homogeneous harmonic polynomials and isomorphisms — come up repeatedly in the rest of the text. 

Proof. Consider any function y e. By Definition 2,6, there is a homogeneous harmonic polynomial p of degree f such that y = PI52. Now rotating a polynomial preserves its degree (by Exercise 4.14), and the Laplacian is invariant imder rotation (by Exercise 3.11). So for any g e SO(3 ) the function g p is a homogeneous harmonic polynomial of degree . Hence g-y = g s2-p is an element of 3. ... 

In this proposition, as in Proposition 6.14, it is possible to drop the hypothesis that the representation be unitary. See Exercise 6.13. We will apply this classification of irreducibles of 50(3) in Section 7.1 to show that for each nonnegative integer n the set of homogeneous harmonic polynomials of degree n forms an irreducible representation of 50(3). 

In this section we consider the vector spaces of homogeneous harmonic... 

Combining this last result with our knowledge of the classification of the irreducible representations of the group 50(3), we can show that the representation of the rotation group on homogeneous harmonic polynomials of any fixed degree is irreducible. 

Recall the vector space y of spherical harmonics from Definition 2.6 y is the set of restrictions to of homogeneous harmonic polynomials. Recall also the definition of spanning (Defiiution 3.7). The set y of spherical harmoiucs spans... 

In four dimensions, as in three dimensions, the restrictions of homogeneous harmonic polynomials of degree n to the unit sphere are called spherical harmonic functions of degree n. The analysis in four dimensions proceeds much as it did in three dimensions, although the dimension counts change. 

Anticipating Fock s notation, let us call our variables Xi, 3 and %<. An example of a homogeneous harmonic polynomial of degree four is... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

One of die most important second-order, homogeneous differential equations is that of Hennite It arises in the quantum mechanical treatment of the harmonic oscillator. Schrfidinger s equation for the harmonic oscillator leads to the differential equation... 

Another analysis method was based on the local wave vector estimation (LFE) approach applied on a field of coupled harmonic oscillators.39 Propagating media were assumed to be homogeneous and incompressible. MRE images of an agar gel with two different stiffnesses excited at 200 Hz were successfully simulated and compared very well to the experimental data. Shear stiffnesses of 19.5 and 1.2 kPa were found for the two parts of the gel. LFE-derived wave patterns in two dimensions were also calculated on a simulated brain phantom bearing a tumour-like zone and virtually excited at 100-400 Hz. Shear-stiffnesses ranging from 5.8 to 16 kPa were assumed. The tumour was better detected from the reconstructed elasticity images for an input excitation frequency of 0.4 kHz. 

The first attempts (G. Klein and I. Prigogine, 1953, MSN.5,6,7) were very timid and not very conclusive. They were devoted to a chain of harmonic oscillators. In spite of a tendency to homogenization of the phases, there was no intrinsic irreversibility here, because an essential ingredient is lacking in this model the interaction among normal modes. The latter were introduced as a small perturbation in the fourth paper of the series (MSN.8). 

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