Fermat's principle

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	Principle
		Fermat's principle Fermat's principle in optics states: This principle was first stated by Pierre de Fermat.
		Pauli exclusion principle The Pauli exclusion principle is a 1925.
		Principle of conferral The principle of conferral is a fundamental principle of European Union law.
	www.brainyencyclopedia.com /topics/principle.html (1107 words)

	Fermat's Principle
		Thus, Fermat's principle is consistent with light traveling in a straight line in a homogeneous medium.
		Thus, the light cannot choose one trajectory over another using Fermat's principle and all of the trajectories are equally favored.
		Fermat's principle can also be used to derive the laws of reflection and refraction.
	www.physics.nmt.edu /~raymond/classes/ph13xbook/node36.html (1107 words)

	2.9 Fermat’s principle for light rays
		Also for Fermat’s principle in ordinary optics, the extremum is never a local maximum, as is mentioned, e.g., in Born and Wolf [35], p.
		While Kovner’s principle, like the classical Fermat principle, is a varional principle for rays, the Frittelli-Newman principle is a variational principle for wave fronts.
		To set up a variational principle, we have to choose the trial curves among which the solution curves are to be determined and the functional that has to be extremized.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-9/articlesu9.html (923 words)

	Binary-Craft
		Fermat's principle; principle of least time (P. de Fermat)
		weak equivalence principle; principle of uniqueness of freefall
	www.binary-craft.com /misc/laws (923 words)

	Fermats principle - Hutchinson encyclopedia article about Fermats principle
		The principle is named after French mathematician Pierre de Fermat, who used it to deduce the laws of reflection and refraction.
		In physics, the principle that a ray of light, or other radiation, moves between two points along the path that takes the minimum time.
		This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.
	encyclopedia.farlex.com /Fermats+principle (106 words)

	Pierre de Fermat
		"From this principle Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence(in refraction)is a constant number times the sine of the angle of refraction in passing from one medium to anot her." (Bell, p.63).
		Fermat's choice of a legal career was natural and typical of his time, for his father's wealth and his mother's famil y background.
		Pierre de Fermat was born on August 17, 1960, in Beaumont-de-Lomagne, a small town near Toulouse in the south part of France, near the border with Spain.
	www.math.rutgers.edu /~cherlin/History/Papers1999/chellani.html (1446 words)

	Fermat, Pierre de on Encyclopedia.com
		In optics Fermat recognized that of all possible paths, light takes the path that takes the least time; this fundamental rule is known as Fermat's principle.
		A magistrate whose avocation was mathematics, Fermat is known as a founder of modern number theory and probability theory.
		However, Wiles's proof involved mathematical concepts that were unknown in Fermat's lifetime, so whether Fermat had a valid proof remains conjecture.
	www.encyclopedia.com /html/F/Fermat-P1.asp (472 words)

	Reflection and Fermat's Principle
		Fermat's Principle: Light follows the path of least time.
		This law of reflection can be derived from Fermat's principle.
		This derivation makes use of the calculus of maximum-minimum determination, the derivative of a square root, and the definitions of the triangle trig functions.
	hyperphysics.phy-astr.gsu.edu /hbase/phyopt/fermat.html (217 words)

	NTNU JAVA :: View topic - Fermat Principle
		Posted: Thu Jan 29, 2004 3:16 pm Post subject: Fermat Principle
		NTNU JAVA :: View topic - Fermat Principle
		The path of a ray of light between two points is the path that minimizes the travel time.
	www.phy.ntnu.edu.tw /java/Fermat/Fermat.html (271 words)

	Untitled
		With this formulation, Fermat's principle falls in the class of problems called variational calculus, a technique which determines the form of a function that minimizes a definite integral.
		Huygens' principle must be modified to accommodate the case in which a wavefront, such as AC, encounters a plane interface, such as XY, at an angle.
		According to Huygens, the remainder of each wavelet is to be disregarded in the application of the principle.
	physics.tamuk.edu /~suson/html/4323/geometry.html (1800 words)

	UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem
		Only on one ill-fated occasion did Fermat ever mention a curve of higher genus x^n + y^n = z^n, and then hardly remain any doubt that this was due to some misapprehension on his part [for a brief moment perhaps [he must have deluded himself into thinking he had the principle of a general proof.
		Fermat claimed to have found a proof of the theorem at an early stage in his career.
		Fermat never published a proof of this statement.
	www.math.uncc.edu /flt.php (1800 words)

	2.9 Fermat’s principle for light rays
		Also for Fermat’s principle in ordinary optics, the extremum is never a local maximum, as is mentioned, e.g., in Born and Wolf [ 35 ], p.
		To set up a variational principle, we have to choose the trial curves among which the solution curves are to be determined and the functional that has to be extremized.
		This is the set of trial curves from which the lightlike geodesics are to be singled out by the variational principle.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-9/articlesu9.html (1800 words)

	Huygens' Principle
		Whether we have now actually found the true "reason" for the behavior of light is debatable, and ultimately every theory is based on some fundamental principle(s), but it's interesting how widely the opinions on various principles differ. (I'm reminded of the history of Fermat's Principle, and of Planck's reverence for the Principle of Least Action.)
		It turns out that Huygens' Principle applies only with one time dimension and n = 3, 5, 7.., or any odd number of space dimensions, but not for any even number of space dimensions.
		Setting aside these weighty considerations, it's interesting to review the mathematical content of Huygens' original principle. The usual wave equation for a scalar field y(x
	www.mathpages.com /home/kmath242/kmath242.htm (1112 words)

	Connexions - Content - Search Repository
		Fermat's principle of least time is used to derive Snell's law and the law of reflection.
		We show how the MLE implements the likelihood principle.
	cnx.rice.edu /content/search?words=likelihood+principle (122 words)

	Fermat, Pierre de on Encyclopedia.com
		In optics Fermat recognized that of all possible paths, light takes the path that takes the least time; this fundamental rule is known as Fermat's principle.
		However, Wiles's proof involved mathematical concepts that were unknown in Fermat's lifetime, so whether Fermat had a valid proof remains conjecture.
		Affrontements entre manifestants et forces de l'ordre, jeudi, à Toulouse A Toulouse, 400 personnes se trouvaient dès 07H00.
	www.encyclopedia.com /html/F/Fermat-P1.asp (122 words)

	Pierre de Fermat
		"From this principle Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence(in refraction)is a constant number times the sine of the angle of refraction in passing from one medium to anot her." (Bell, p.63).
		Pierre de Fermat was born on August 17, 1960, in Beaumont-de-Lomagne, a small town near Toulouse in the south part of France, near the border with Spain.
		Fermat's choice of a legal career was natural and typical of his time, for his father's wealth and his mother's famil y background.
	www.math.rutgers.edu /~cherlin/History/Papers1999/chellani.html (122 words)

	Why You Don’t Believe Fermat’s Principle
		In 1657, in a letter to Cureau de la Chambre, Fermat stated for the first time his idea that the law of refraction might be deduced from a minimum principle, like that which Heron had noted for reflection.
		Pierre de Fermat (1601-1665) was one of those classically educated geniuses, who, like Ampère, never went to college.
		In 1662, Pierre de Fermat proved that a ray of light passing from a rarer to a denser medium, follows the path which takes the least time.
	www.21stcenturysciencetech.com /articles/fall01/fermat/Fermat.html (122 words)

	11R: Algebraic number theory: global fields
		The Hasse Principle : a quadratic equation may be solved rationally if it may be solved mod p for every p (with or without solvability in the reals).
		References for the local-to-global principle for quadratic equations (Brauer-Hasse-Noether theorem for associative algebras).
		An elliptic curve formulation of the n = 3 case of Fermat's Last Theorem ; in which quadratic extensions of Q does it have a solution?
	www.math.niu.edu /~rusin/known-math/index/11RXX.html (122 words)

	Mathematics Courses
		Applications may be taken from a variety of areas such as the following: applied mechanics, elasticity, economics, production planning and resource allocation, astronautics, rocket control, physics, Fermat’s principle and Hamilton’s principle, geometry, geodesic curves, control theory, elementary bang-bang problems.
		Analytic functions, Cauchy’s theorem, Taylor and Laurent series, residue theorem and contour integration techniques, analytic continuation, argument principle, conformal mapping, potential theory, asymptotic expansions, method of steepest descent.
		Methods of reasoning and proofs: propositional logic, predicate logic, induction, recursion, pigeonhole principle.
	www.ucsd.edu /catalog/0506/courses/MATH.html (5019 words)

	GLOB: Prospectors find largest meteorite crater ever found in Europe SOURCE: Alexander's Gas & Oil Connections - Prospectors find largest meteorite crater ever found in Europe
		INNO: Fermat's principle of Least Time SOURCE: Idle Theory: Least Action Principles
		GLOB: Prospectors find largest meteorite crater ever found in Europe SOURCE: Alexander's Gas and Oil Connections - Prospectors find largest meteorite crater ever found in Europe
		Alexander's Gas and Oil Connections - Prospectors find largest meteorite crater ever found in Europe - 2004-04-21
	www.hi.is /~joner/eaps/meteon1.htm (5019 words)

	Elementary Theory of Numbers I Assignments
		Learn the Well-Ordering Principle, and study carefully the proofs on page 2 of the text and the proofs done in class.
		Important topics to know include, Fermat's Little Theorem, Wilson's Theorem, Euler's Phi function, Euler's Theorem, the order of an element, Theorems 8.1, 8.2, 8.3, the corollary to Lagrange's Theorem, primitive roots, theory of indices, property of indices, quadratic residues and nonresidues, Euler's criterion, Legendre symbol and its properties, Quadratic Reciprocity.
		Every positive integer is either prime or a product of primes, where the product is unique apart from the order of the factors.
	www.wfu.edu /~ekh/mth345/spring/assignments/assign.html (1416 words)

	THE PRINCIPLE OF LEAST ACTION
		EXAMPLE OF A MINIMIZATION LAW: FERMAT'S LEAST TIME PRINCIPLE FOR LIGHT
		WHY THIS WAS THE RIGHT DIRECTION TO LOOK
	www.ph.utexas.edu /~gleeson/httb/chapter1_3_6.html (1416 words)

	English Books > Science > Astrophysics & Space Science
		Ray Optics, Fermats Principle, And Applications To General Relativity
	book.netstoreusa.com /index/bkbsa500R.shtml (543 words)

	Analytical mechanics -- Facts, Info, and Encyclopedia article
		By analogy with (Click link for more info and facts about Fermat's principle) Fermat's principle, which is the (Click link for more info and facts about variational principle) variational principle in (Click link for more info and facts about geometric optics) geometric optics, Maupertuis' principle was discovered in classical mechanics.
		In this formulation, the solutions of the Hamilton-Jacobi equations are the (Click link for more info and facts about integral curve) integral curves of (Click link for more info and facts about Hamiltonian vector field) Hamiltonian vector fields.
		Using generalized (A number that identifies a position relative to an axis) coordinates, we obtain (Click link for more info and facts about Lagrange's equations) Lagrange's equations.
	www.absoluteastronomy.com /encyclopedia/a/an/analytical_mechanics.htm (270 words)

	part08.html
		While Kepler demonstrated the Sun as the mover of the planets, the Sun itself was subordinate to a higher principle, reflected in the harmonic ordering principles, which were present at every moment of the planet's action, and discoverable by the human mind.
		The paradox Kepler identified is a fulcrum, whose solution requires the assimilation of the entire Socratic tradition of Greek science, while, at the same time, demanding an as yet undiscovered principle.
		In summary, Bernoulli's ingenious solution is as follows: Fermat had demonstrated that light, refracted by a change in the density of the medium, travelled the least time path.
	www.wlym.com /antidummies/part08.html (1806 words)

	Infinite Descent versus Induction
		It is often said that the principle of "infinite descent", developed by Fermat in his study of Diophantine analysis, is equivalent to induction.
		Thus, there is some validity in the claim that Fermat's infinite descent concept is one of the proto-types for reasoning about infintesimals and limiting sequences, which is perhaps not surprising in view of Fermat's own contributions to the foundation of the differential calculus.
		Those who argue that infinite descent is equivalent to induction typically refer to a proof such as the following for the fact that the sum of the first n integers is n(n+1)/2: If not, there is a least integer, say N, for which this is false.
	www.mathpages.com /home/kmath144.htm (1225 words)

	Abstract
		Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional I(l) stationary.
		Fermat's functional I(l) depends on curves l which connect points S and R, and represents the travel times from S to R along l.
		To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases.
	sw3d.mff.cuni.cz /papers/a02vc1.htm (443 words)

	Sasa Radomirovic - Math 356 - Number Theory - Summer 2004
		Assuming the existence of one solution in positive integers, Fermat's descent method creates a "smaller" solution which eventually contradicts the Well-Ordering Principle.
		A biography of Fermat and history of the proof of Fermat's last theorem can be found here.
	www.math.rutgers.edu /~sasar/Math356?cd614.txt (463 words)

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