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Topic: Parabolic coordinates

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Curvilinear coordinates

Cartesian coordinate system

Orthogonal

Orthonormal basis

Polar coordinate system

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	Coordinates (mathematics) - Wikipedia, the free encyclopedia
		The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space.
		The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles.
		In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis).
	en.wikipedia.org /wiki/Coordinates_(elementary_mathematics) (1186 words)

	CPO
		The final coordinate at the exit plane, which should be 1.0, can be seen to be 0.99996.
		The coordinates of the first ray are recorded when it crosses a 'test plane' and again when it hits an electrode.
		The final coordinate at the exit plane, which should be 1.0, can be seen to be 0.99941.
	www.electronoptics.com /benchmar.htm (2219 words)

	Jens Nöckel's Research
		Based on ray-optics considerations: it is well-known that parabolic mirrors are very efficient at focusing light in macroscopic situations, and to exploit this effect one should try to make the dome in a parabolic shape with the active layer intersecting the parabola's focus.
		Here, phi is the cyclic coordinate, canonically conjugate to the angular momentum component around the z axis (which is the rotation axis).
		This is a Poincare section, but it is uses different coordinates from the sections I use, e.g., in spheres or disks.
	darkwing.uoregon.edu /~noeckel/microdome.html (3049 words)

	ESA - Human Spaceflight - ESA's 38th Parabolic Flight Campaign
		Organised by ESA, the Parabolic Flight Campaigns are used to conduct research experiments in weightlessness, and to prepare experiments and equipment for later use on board the International Space Station (ISS).
		The scientists are hopeful their treadmill will be an improvement on exercise machines that are currently used by astronauts during long-term stays in weightlessness and will lead to a reduction in the amount of muscle and bone loss experienced.
		ESA also uses the parabolic flight campaigns to evaluate technologies that are being developed for use on board ISS.
	www.esa.int /esaHS/SEM7AF0A90E_index_2.html (715 words)

	Re: Schroedinger equation in "wrong" coordinates
		The equation is separable in parabolic coordinates, not parabolic cylinder coordinates.
		The usual names for the parabolic coordinates are phi, eta, and xi, where phi is the same azimuth coordinate used in spherical coords.
		A surface of constant xi is a concave-up paraboloid of revolution, the z-axis being the symmetry axis.
	www.lns.cornell.edu /spr/2004-05/msg0060888.html (429 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The eigenfunctions obtained in these coordinate systems are each a suitable basis set in the absence of an electric field, but only the parabolic states, the Stark eigenfunctions, retain their character in the presence of a weak field.
		The principal quantum number n and the magnetic quantum number m that are ``good" in spherical coordinates are common to both solutions; k is, however, unique to the parabolic coordinate solution.
		Upon application of the field F, the parabolic quantum numbers are still good but the energy is now given by \begin{equation} E(n,\ k)\ =\ -\,\,{\frac{1}{2\,{n}^{2}}}\ -\ {\frac{3}{2}}\,n\,k\,F\ \end{equation} We note that, customarily, the opposite sign convention has been used for the electric quantum number k in equations (1) and (2).
	newton.umsl.edu /~atomic/ajp4/ajp4.tex.txt (3063 words)

	Parabolic coordinates
		Parabolic coordinates are an alternative system of coordinates for three dimensions.
		Thus a pair of coordinates η and ξ specify a unique point on the half-plane.
		See also: spherical coordinates, cylindrical coordinates, Cartesian coordinates.
	www.sciencedaily.com /encyclopedia/parabolic_coordinates (400 words)

	Three body Coulomb problem in two dimensions (Site not responding. Last check: 2007-10-09)
		In two dimensions, it is no longer possible to use the perimetric coordinates to represent the positions of the three particles.
		Indeed, for a given value of the perimetric coordinates, there are several possible configurations of the three bodies.
		Therefore we have introduced a new system of coordinates, based upon the parabolic coordinates.
	www.spectro.jussieu.fr /Chaos/versionlkb/theme3_en.html (193 words)

	ERTURK SEMINAR (Site not responding. Last check: 2007-10-09)
		The leading edge receptivity to acoustic waves of two-dimensional parabolic bodies was investigated using a spatial solution of the Navier-Stokes equations in vorticity/stream function form in parabolic coordinates.
		The free-stream is composed of a uniform flow with a superposed periodic velocity fluctuations of small amplitude.
		Of special interest is an asymmetry in the receptivity coefficient when determined by extrapolation to the leading edge from the lower or upper sides of the body, which is produced when the sound incidence angles (_2) and angles of attack (_1) are large.
	www.nd.edu /~ame/announcements/Erturk.html (261 words)

	index.html
		These can be thought of as "level surfaces" for the coordinate system: everything on one of these surfaces has the same w coordinate.
		While alternate coordinate systems can be really useful at times (especially in integration), you do have to be careful when working with them.
		A complete treatment of where these factors come from is beyond the scope of this lab, but basically they are the reciprocals of the coefficients needed to convert a change in that variable into an actual change in arc-length (called the "Scale Factors").
	www.austin.cc.tx.us /mmcguff/mathematica/labs_html/calculus_IV_lab_2_2005_spring (2647 words)

	Universal Symmetry Groups
		Since the action-angle variables are involved, it is possible to describe the constants of the motion in a form applicable to any kind of coordinate system in which the Hamiltonian is a function of a certain type of combination of the coordinates and momenta.
		The operator which determines the residue class of this least common multiple commutes with all the operators of his von Neumann algebra, and consequently their number determines the multiplicity of the von Neumann algebra, and in consequence, of the unitary modular group which is formed from its operators.
		The mechanism seems to be that the wave functions are expressed in a certain coordinate system, polar for example, where the ladder operators depend upon the (polar) action-angle variables.
	delta.cs.cinvestav.mx /~mcintosh/comun/symm/node14.html (1963 words)

	[No title]
		The time-independent Schrodinger equation is then transformed from a Cartesian basis (x,y,z) to a basis of spherical polar coordinates (r, theta, phi).
		Analysis of the radial wave functions can be performed by plotting these functions along the radial coordinate, thus given insight in the extension of the electronic structure of the system, not only for the ground state but also for the electronically excited states.
		One example is that of parabolic coordinates, useful for the evaluation of the Stark effect.
	www.nat.vu.nl /~wimu/Schrod.html (1248 words)

	HatomParabolic.html
		Our aim here is to demonstrate what some hydrogen atom eigenfunctions look like when expressed in parabolic coordinates.
		4) the eigenstates in parabolic coordinates are useful when additional interactions are turned on, such as a constant electric field along the (chosen)
		is the cylindrical coordinate measuring the distance from the rotation axis.
	www.yorku.ca /marko/PHYS4011/html/HatomParabolic/HatomParabolic.html (700 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		A two-dimensional coordinate system determined by a system of confocal parabolas.
		A three-dimensional coordinate system whose coordinate surfaces are the surfaces generated by rotating a plane containing a system of confocal parabolas about the axis of symmetry of the parabolas, together with the planes passing through the axis of rotation.
		] A three-dimensional coordinate system in which two of the coordinates depend on the x and y coordinates in the same manner as parabolic coordinates and are independent of the z coordinate, while the third coordinate is directly proportional to the z coordinate.
	www.accessscience.com /Dictionary/P/P4/DictP4.html (2286 words)

	Coordinates (elementary mathematics) (Site not responding. Last check: 2007-10-09)
		In the circular coordinate system, a point P is represented by a tuple of two components
		In the cylindrical coordinate system, a point P is represented by a tuple of three components
		Spherical coordinates are useful in analyzing systems that are symmetrical about a point; a sphere that has the Cartesian equation
	www.worldhistory.com /wiki/C/Coordinates-(elementary-mathematics).htm (880 words)

	Polar coordinates
		The first topic, polar coordinates, is traditionally treated somewhere later in the calculus sequence.
		The formulas expressing the Cartesian coordinates of the point in terms of the polar coordinates are (by elementary trigonometry)
		Although E is a constant vector (not a function of the point r), its decomposition does depend on the coordinates of the point.
	www.math.tamu.edu /~fulling/coalweb/polar.htm (1400 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Emphasis is also on the consequences of the separability in parabolic coordinates and the relationship between the spherical eigenfunctions and the parabolic eigenfunctions.
		The quantum mechanical treatment of the Stark effect in hydrogen is extended from the usual treatment to include solution using parabolic coordinates.
		Methods of computing the helium energies are compared and the variational principle utilized and discussed extensively.
	www.umsl.edu /~jjl/homepage/chapters/TOC.doc (472 words)

	Willard Miller: Bibliography
		The Harmonic oscillator in elliptic coordinates and Ince polynomials, with C.P. Boyer and E.G. Kalnins.
		R-separable coordinates for three-dimensional complex Riemannian spaces, with C.P. Boyer and E.G. Kalnins, T.A.M.S. 242 (1978), pp.
		Separable coordinates, integrability and the Niven equations, with E.G. Kalnins, J. Phys.
	www.ima.umn.edu /~miller/bibli.html (2919 words)

	Parabolic Coordinates (Site not responding. Last check: 2007-10-09)
		Numerical solution for 3D Poisson equation in circular cylindrical coordinates : Cohl et.
		New addition theorems for rotationally invariant coordinate systems which R-separate 3D Laplace equation: Cohl et.
		constant are both paraboloids of revolution or parabolic bowls.
	www.astro.ex.ac.uk /people/hcohl/an/node11.html (345 words)

	Maxime Bôcher, August 28, 1867–September 12, 1918 \| By William F. Osgood \| Biographical Memoirs
		If it was not reserved for mathematicians to make formal discoveries coordinate in importance with those which formed the crown of the discoverers and early developers of the calculus, it is none the less true that mathematical imagination never played more freely, not only in geometry and algebra, but also in analysis and mathematical physics.
		Bôcher's advanced course in the first year of his professional life took the form of a seminary, the subject being curvilinear coordinates and functions defined by differential equations.
		A part of the instruction consisted of formal lectures on the latter topic, and he thus began, even at that early date, to treat topics in a field of analysis in which he was to become eminent.
	www.nap.edu /html/biomems/mbocher.html (4943 words)

	Wave Propagation in Circular Jettied Channels (Site not responding. Last check: 2007-10-09)
		A model for wave propagation in circular jettied channels is presented.
		The model combines a polar coordinates parabolic equation with a model for wave propagation in jettied channels.
		The effect of the jetties on the wave field within a circular jettied channel is discussed.
	www.pubs.asce.org /WWWdisplay.cgi?9800123 (84 words)

	Re: Schroedinger equation in "wrong" coordinates
		(This is the case > with the isotropic harmonic potential, V(r)=-kr^2, but -1/r cannot be > so written.) > > The hydrogen equation is however separable in parabolic cylinder > coordinates.
		The separability > in more than one system has something to do with the "extra" constant > of motion (the Runge vector mentioned in a previous post), and with > the famous degeneracy in hydrogen that allows states of different l > (angular momentum quantum number) to have the same energy.
		So really, solving it using parabolic cylinder coordinates' is a similar sort of procedure as going over to sphericals -but because 1/r cannot be written as a sum of three, it's not possible to solve it in cartesians.
	www.lns.cornell.edu /spr/2004-04/msg0060761.html (221 words)

	chapter3
		so far downstream the potential flow is still describing a body of parabolic extent rather than one where the flow returns to uniform velocity over the whole wake.
		Obviously from continuity the sources would have to become sinks towards the trailing edge.
		(iii) Talke and Berger(1970) used parabolic coordinates centred on the trailing edge to find a series truncation solution near the trailing edge using Goldstein's solution for the wake as an outer boundary condition.
	info.sjc.ox.ac.uk /scr/sobey/iblt/chapter3 (337 words)

	StarkResonance.html
		In this worksheet we investigate the problem of a hydrogen atom in an external DC electric field.
		We solve the problem in parabolic coordinates and learn that the bound-state spectrum 'disappears', and turns into a problem of resonances within a continuum of solutions at all energies.
		which plays the role of polar cylindrical coordinate.
	www.yorku.ca /marko/PHYS4011/html/StarkResonance/StarkResonance.html (902 words)

	Citebase - Vacuum Solutions of Einstein's Equations in Parabolic Coordinates (Site not responding. Last check: 2007-10-09)
		We present a simple method to obtain vacuum solutions of Einstein's equations in parabolic coordinates starting from ones with cylindrical symmetries.
		Furthermore, a generalization of the method to a more general situation is given together with a discussion of the possible relations between our method and the Belinsky-Zakharov soliton-generating solutions.
		Use the Correlation Generator to explore the correlation between download impact ("hits") and citation impact.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/0406060 (299 words)

	Citebase - Exact path integral of the hydrogen atom and the Jacobi's principle of least action (Site not responding. Last check: 2007-10-09)
		These properties are illustrated by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates.
		The path integral for the hydrogen atom, for example, is formulated as a one-dimensional quantum gravity coupled to matter fields representing the electron coordinates.
		The (renormalized) cosmological constant, which corresponds to the energy eigenvalue, is thus quantized in this model.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9608052 (848 words)

	Citebase - Quantum oscillator and a bound system of two dyons
		It is shown that the Schr\"{o}dinger equation for this generalized MIC-Kepler system can be separated in spherical and parabolic coordinates.
		The spectral problem in spherical and parabolic coordinates is solved.
		It is shown that the Schr\"{o}dinger equation for this generalized MIC-Kepler system can be separated in prolate spheroidal coordinates.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9508137 (825 words)

	Math 2411 Assignments (Site not responding. Last check: 2007-10-09)
		Find some curves on the surface of a sphere which have arclengths which can be calculated in closed form.
		Consider our favorite coordinates, the parabolic coordinates, for which
		Use parabolic coordinates, somewhat perversely, to calculate the area of the unit square, 0
	www.math.gatech.edu /~harrell/2411/HW2411.html (946 words)

	Solution for D=2 H atom System
		We therefore decide to go over to ``square root coordinates'' via some transformation which changes
		In two dimensions, the appropriate square root is given by the Levi-Cività transformation
		In contrast to the three-dimensional case to be treated below we note here the rather obvious fact that the Levi-Cività mapping, which is simply a transformation to parabolic coordinates, carries the flat
	www.physik.fu-berlin.de /~kleinert/kleiner_re222/node6.html (510 words)

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