Theorem, equal-areas

The elements A and B therefore have equal areas. Liouville s theorem states that an element in phase space is conserved, which means that the element within which a system can be found is constant. Further, if the range Ae in the phase space is divided into equal elements, the system spends equal times passing through these elements, or cells. 

FIGURE 1.1 Geometrical interpretation of Pythagoras theorem. The area of the large square equals the sum of the areas of the two smaller squares. 

The velocity of a fluid approaching an orifice or nozzle or similar device is called the velocity of approach. For example, consider a large tank, filled with liquid, with a small orifice on its wall, near the bottom. It is assumed that the area of the tank is so large relative to that of the orifice that the velocity at the surface of the liquid (point 1) is negligible. Let the Bernoulli theorem be written between point 1 at the surface and point 2 at the orifice jet discharge, assuming the pressure is the same at both points. Let h equal the height of the liquid, measured from the surface level to the center of the orifice. Then 0 + /i + 0 = 0 + 0 + Vj2/2g, from which... 

There is a lid, of area A, which can be slid along the surface so as to cover the surface of either I or II completely, or both partially. This lid is made of material such that the interfacial tensions of the two solutions against it are equal, and the composition of each solution remains constant right up to contact with it i.e. there is no adsorption of any of the components of the solutions, a condition which really follows from the first, by Gibbs s theorem. 

A useful trigonometric identity corresponds to the famous theorem of Pythagoras. Pythagoras drew a figure with three squares such that one side of each square formed a side of the same right triangle. He then proved by geometry that the area of the square on the hypotenuse was equal to the sum of the areas of the squares on the other two sides. In terms of the quantities in Fig. 2.1... 

Group theory can be applied to several different areas of molecular quantum mechanics, including the symmetry of electronic and vibrational wave functions and the study of transitions between energy levels. There is also a theorem which says that there is a correspondence between an energy level and some one of the irreducible representations of the symmetry group of the molecule, and that the degeneracy (number of states in the level) is equal to the dimension of that irreducible representation. 

In words, the integral of the differential of a function f(x,y) around a cyclical path equals the integral, over the enclosed area A, of the difference of the mixed derivatives shown. Use this theorem to argue that Eq. (9.13) holds when / is a thermodynamic state fimction. 

The geometrical significance of the Pythagorean theorem is shown in Fig. 1.1 the sum of the areas of the two smaller squares equals the area of the large square. Well over 350 different proofs of the theorem have been published. Figure 1.2 shows a pictorial proof, which requires neither words nor formulas. 

Analogously, the second theorem of Pappus states that the volume of the solid generated by the revolution of a figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid. For a torus, the area of the cross section equals nr. Therefore, the volume of a torus is given by... 

Figure A.3 The mean-value theorem for integrals states that the area under the curve/(r) from a to fo is equal to the area of a rectangle of width w = (b-a) and height/ ,. This requires that the two shaded areas be equal. The height/ , is called the mean value off(x) on the interval [a, b].

When dealing with power spectra related to, for example, rms voltage v s according to W = Vms/R> F(tJ) represents the distributed power spectrum the density of signal power ptr frequency bandwidth. The unit for F(w) (e.g., when dealing with noise spectra) may be pVrms/VHz. When the spectrum is plotted with amplitude per VHz on the y-axis and frequency on the x-axis and scaled so that the area under the F((o) curve is equal to the total rms value in the time domain (Plancheral s theorem), the spectrum is called a power density spectrum. A less stringent definition is simply that a spectrum is called a power spectrum when the function is squared before analysis. 

Therefore, the circulation is the line integral about the contour P of the component of the velocity tangent to the contour and is another measure of the rotation of the fluid. Stokes s theorem provides the relationship to vorticity the circulation divided by the area equals the average normal component of vorticity in the area. 

In the case of a two-dimensional symplectic manifold, the condition of the locally Hamiltonian character of the field admits another vivid geometrical interpretation. Let gij be a Riemannian metric on and let u) = y/det gij)dx A dy be the form of the Riemannian area. By virtue of the Darboux theorem, one can always choose local coordinates p and q such that the form oj be written in the canonical form dp A dq. Here p and q are certain functions of x and y (and vice versa). Let t be a locally Hamiltonian field t = (i (a , y),Q(x,y)), where P and Q are coordinates of the field in the local system of coordinates p and q. Let us interpret the field v as a velocity field of the flow of liquid of constant density (equal to unity) on the surface M. Let us investigate the variation of the mass of the liquid bounded by an infinitesimal rectangle on the surface when it is shifted along integral trajectories of the field v. It is clear that the mass of this liquid is equal to the area of the rectangle. Therefore, the mass of the liquid contained in a bounded (sufficiently small) domain on is equal to the area of the domain. 

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