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Topic: Euler integral

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  VARIATIONS - LoveToKnow Article on VARIATIONS   (Site not responding. Last check: 2007-08-16)
Euler had been under the necessity of resolving an integral into a sum of elements, recording the magnitude of the change produced in each element by a slight change in the unknown function, and thence forming an expression for the total change in the sum under consideration.
Euler had allowed such changes in the position of the curve, along which the integral, to be made a maximum or minimum, is taken, as can be produced by displacement parallel to the axis of ordinates.
The line integral may, as usual, be replaced by the sum of line integrals taken round a series of cells, the external boundaries of the set of cells being identical with the given contour, and the internal boundaries of ad- B jacent cells being traversed twice in opposite senses.
www.1911encyclopedia.org /V/VA/VARIATIONS.htm   (5103 words)

 Euler, Leonhard
By means of his numerous books and memoirs that he submitted to the academy, Euler carried integral calculus to a higher degree of perfection, developed the theory of trigonometric and logarithmic functions, reduced analytical operations to a greater simplicity, and threw new light on nearly all parts of pure mathematics.
He is known for familiar results in elementary geometry; for example, the Euler line through the orthocentre (the intersection of the altitudes in a triangle), the circumcentre (the centre of the circumscribed circle of a triangle), and the barycentre (the "centre of gravity," or centroid) of a triangle.
Throughout his life Euler was much absorbed by problems dealing with the theory of numbers, which treats of the properties and relationships of integers, or whole numbers (0, +/-1, +/-2, etc.); in this, his greatest discovery, in 1783, was the law of quadratic reciprocity, which has become an essential part of modern number theory.
www.phy.bg.ac.yu /web_projects/giants/euler.html   (772 words)

 LEONHARD EULER   (Site not responding. Last check: 2007-08-16)
Euler's contribution to mathematics is represented here by a few of the notations conventionalized by him or in his honor.
Euler grew up near Basel, Switzerland, and studied at an early age under Johann Bernoulli.
Leonhard Euler 1707-1783, Beiträge zu Leben und Werk, Gedenkband des Kantons Basel-Stadt, edited by J. Burckhardt, E. Fellmann, and W. Habicht, Birkhäuser Verlag, Basel, 1983.
faculty.evansville.edu /ck6/bstud/euler.html   (331 words)

 Ulearn Today - Magazine   (Site not responding. Last check: 2007-08-16)
An extremely prolific man and one of the most creative mathematicians of the eighteenth century, Leonard Euler is recognised as the father of the graph theory and science of topology.
Euler enriched mathematics with important new concepts and techniques and is credited with having given the notation f (x) for a function, e for the base of natural logarithms, i for the square root of -1,
Euler was born and educated in the Swiss town of Basle.
www.ulearntoday.com /magazine/physics_article1.jsp?FILE=euler   (694 words)

 Leonhard Euler (1707 - 1783)
Euler commenced this part by dividing curves into algebraical and transcendental, and established a variety of propositions which are true for all algebraical curves.
The method used by Lagrange is described in Euler's integral calculus, and is the same as that given in most modern text-books on the subject.
This method is not convenient, but it was from Euler's results that Mayer constructed the lunar tables for which his widow in 1770 received 5000 pounds from the English parliament, and in recognition of Euler's services a sum of 300 pounds was also voted as an honorarium to him.
www.maths.tcd.ie /pub/HistMath/People/Euler/RouseBall/RB_Euler.html   (1819 words)

 Search Results for integral
Her thesis on zeros of integral functions was examined by J E Littlewood who she met for the first time as an external examiner in her oral examination for the D.Phil.
It is divided into two parts, of two and five volumes respectively, on the differential and integral calculus, and included not only the standard treatment of the derivative and the integral and their applications to geometry but also substantial sections on the theory of functions, determinants and elliptic functions.
This generalisation of the Riemann integral revolutionised the integral calculus.
www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?BIOGS=1&TOPICS= 1&CURVES=1&REFS=1&BIBLI=1&SOCIETIES=1"=1&CHRON=1&WORD=integral&CONTEXT=1   (12887 words)

 References for Euler
C Grau, Leonhard Euler und die Berliner Akademie der Wissenschaften, in Ceremony and scientific conference on the occasion of the 200th anniversary of the death of Leonhard Euler (Berlin, 1985), 139-149.
H-J Treder, Euler und die Gravitationstheorie, in Ceremony and scientific conference on the occasion of the 200th anniversary of the death of Leonhard Euler (Berlin, 1985), 112-119.
A P Yushkevich, Leonhard Euler - sein Leben und mathematisches Werk, in Ceremony and scientific conference on the occasion of the 200th anniversary of the death of Leonhard Euler (Berlin, 1985), 17-33.
www-groups.dcs.st-and.ac.uk /~history/References/Euler.html   (1152 words)

 Physics Help and Math Help - Physics Forums - Justification of Action Integral
It would seem that the Action integral for the world sheets of string theory are presently justified as a higher dimensional version.
And String theory poses that the Action Integral is protortional to the surface area of the string.
The initial and final position of the string seem to be imposed as limits on the integral in an a priori fashion and then the symmetries and vanishing variation are employed after that.
www.physicsforums.com /printthread.php?t=11753   (1535 words)

 The Euler equation
When the functional is a simple integral, Euler's equation gives a powerful formula for quick calculation of the functional derivative.
Euler's formula is not (18), but rather applies to a case somewhat more complicated.
This is Euler's formula, an extremely useful short cut to the functional derivative when the functional is of the form (19).
giulio.tau.ac.il /~bqs/functionals/node3.html   (320 words)

 PlanetMath: natural logarithm
The integral above was found to have the properties of a logarithm.
For complex and negative numbers, one has to look at the Euler relation.
Cross-references: Euler relation, negative, complex, greater than zero, real numbers, function, properties, integral, Euler's number, exponential function, origin, base, logarithm
planetmath.org /encyclopedia/NaturalLogarithm2.html   (157 words)

 Exact Gauss Integrals
The value of integrals is an approximation, even if in most cases all decimals shown are found to be exact.
The main idea is to approximate the functions' integrals with an order 30 quadrature formula, which is calculated using Legendre's orthogonal polynomials.
The integral for the elementary interval is calculated from the order 30 quadrature formula, the maximum errors committed during the calculation using the other two.
excalc.vestris.com /docs/ref-exact.html   (434 words)

 [No title]
The definite integral of this function $\int_{a}^{b} f(x) dx$ is equal to the area under the curve.
\newspecial{Integrate}{euler.eps}{In Euler's version of the trapezoid rule, the integral $\int_{a}^{b}f(x)\,dx$ is approximated as the area of the rectangles shown under the graph of $f(x)$ versus $x$.}{0.5} \subsection{Euler's Trapezoids} Euler has a method for numerical integration that has to be the simplest.
In Euler's version of this rule, the height is determined by the value of the function at the {\em left corner} of the rectangle.
www.physics.orst.edu /CPUG/Intro/Fortran90/OldF04.txt   (7190 words)

 Euler integral - Wikipedia, the free encyclopedia
In mathematics, there are two types of Euler integral:
Euler integral of the first kind: the Beta function
Euler integral of the second kind: the Gamma function
en.wikipedia.org /wiki/Euler_integral   (53 words)

Isoperimetric problems(1) are ones where an integral is to be optimized subject to a constraint which is another integral having a specified value.
In this approach the partial differential equations are converted to the corresponding integral to be optimized from the calculus of variations.
Then approximate methods of integration are used to find the minimum of the integral, and this yields the concentration, temperature and/or velocity profiles required for the solution of the original differential equations.
www.mpri.lsu.edu /textbook/Chapter8-b.htm   (1949 words)

 Euler integral -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-08-16)
In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, there are two types of Euler integral:
Euler (The result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)) integral of the first kind: the (Click link for more info and facts about Beta function) Beta function
Euler integral of the second kind: the (Click link for more info and facts about Gamma function) Gamma function
www.absoluteastronomy.com /encyclopedia/e/eu/euler_integral.htm   (118 words)

For y=1,2,…, the integral vanishes; this is canceled by the infinite values of the term in front of the integral.
We have seen that it is possible to replace the integral for Γ(z) along a half line by a contour integral which converges for all values of z.
Since the integral is analytic in the cut plane, the theorem holds for x in this region as well; also we apply the analytic continuation with respect to b and c in order to arrive at the conditions announced in the formulation of the theorem.
www.amsta.leeds.ac.uk /~kisilv/courses/sp-funct.html   (3349 words)

 Sensible Calculus CHV.A: The Definite Integral
The sums used in Euler's method (Chapter IV.***) to estimate the solution to a differential equation of the form f '(x) = P(x) where P was a continuous function had at least two interpretations, one related to motion and one related to areas.
The interpretation of the definite integral as the net change in position is now seen as the change s(b)-s(a), that is the definite integral represents the vertical change in the integral curve over the interval [a,b].
Since the integral for P measures the area of the region, the integral for Q must be the opposite of the area of the region enclosed by the graph of Q, the X-axis, and the lines X=a and X=b.
www.humboldt.edu /~mef2/book/ch5/VA/CH5A.html   (2941 words)

A good first example to illustrate Euler's method is our favorite DE from the time of Calculus, namely dy/dx =y, say with initial value y(0)=1, so that y=exp(x) is the solution.
The upper graph is of the exponential function, the lower graph is of your Euler approximation.
In the same vein as ``improved Euler'' we can use the Simpson approximation for the integral instead of the Trapezoid one, and this leads to the Runge-Kutta method which is widely used in real software.
www.math.utah.edu /~korevaar/2250fall00/2250sept131.html   (1113 words)

 [No title]
Places a numerical Euler integral of the vsrc elements in vdest.
is made to equal 0.01, the integrals of the integers 0-5 yield the following (compared to their continuous values on their right).
If vsrc.size() is not an integral power of 2, it is padded with 0's to the next power of 2 size.
www.neuron.yale.edu /neuron/docs/help/neuron/general/classes/vector/vect2.html   (1395 words)

 [No title]   (Site not responding. Last check: 2007-08-16)
In section 2 we present basic properties of Euler characteristic of semialgebraic sets and a useful formalism of Euler integral of constructible functions.
Preliminaries II \endhead \medskip In this section we recall the construction and some basic properties of the Euler integral of a constructible function, see e.g.
Then the {\it Euler integral} of $\varphi$ is defined as $$\int \varphi = \sum m_i \chi (X_i).
home.imf.au.dk /esn/preprints/086   (1694 words)

 Stavros Garoufalidis: Abstract [1]   (Site not responding. Last check: 2007-08-16)
The loop expansion of the Kontsevich integral, the null-move and S-equivalence.
Abstract: The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots.
We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots.
www.math.gatech.edu /~stavros/publications/loop.html   (155 words)

 [No title]
The upper graph is # of the exponential function, the lower graph is of your Euler # approximation.
The # trapezoid rule for the integral approximation would be # (1/2)*h*(f(x,y(x))+f((x+h),y(x+h)).
# In the same vein as ``improved Euler'' we can use the Simpson # approximation for the integral instead of the Trapezoid one, and this # leads to the Runge-Kutta method which is widely used in real software.
www.math.utah.edu /~korevaar/2250fall00/2250sept13.txt   (1121 words)

 Martin Flashman Differential Equations: A Motivating Theme for Sensible Calculus
Euler's method is in fact a Riemann sum using a partition of equal subintervals and the left hand endpoints.
Introduce tangent fields and integral curves as methods to visualize the solution of differential equations before any discussion of the definite integral.
A dual interpretation of the integral as area and as the accumulation of change may help students to distinguish the mathematical object from its interpretation.
www.humboldt.edu /~mef2/sareport.html   (886 words)

 The Construction of Sequence Transformations
of the series are known explicitly as functions of the index n, and if the resulting integral for the truncation error has a sufficiently simple structure to allow further manipulations.
the truncation error of both the geometric series and the Euler series possesses the same sign as the first term not included in the partial sum, and is bounded in magnitude by this term.
As is well known, the first term not included in the partial sum is a simple estimate for the truncation error of a convergent series with strictly alternating terms (see p.
www.apmaths.uwo.ca /~rcorless/AM563/NOTES/report/node2.html   (1023 words)

 Integrator for non-pointlike body acted on by changing force(s)? - GameDev.Net Discussion Forums   (Site not responding. Last check: 2007-08-16)
the point is these are axact integrals that are valid for as long a timestep as you want.
Under a force which can be given an equation we can often generate an exact integral, even in the case of damped simple harmonic motion, to get position at any time in the future.
This is Euler integrator, and it's not much better than x += dt*v, I think, because the velocity is integrated in the simple way: v+=dt*a, so velocity will be inaccurate, and in the position change formula (x += dt*v + 0.5*dt*dt*a) v is dominant.
www.gamedev.net /community/forums/viewreply.asp?ID=1534283   (994 words)

 [No title]
In your picture the upper graph is of the # exponential function, the lower graph is of your Euler approximation.
Use improved Euler with n= 100 to see how well you do # for the same initial value problem.
Compare to your results using Euler and improved Euler, using # n=100.
www.math.utah.edu /~yoon/m2280/handouts/cp2.txt   (1303 words)

 The Gamma Function
Many important functions in applied sciences are defined via improper integrals.
Euler's first integral or the Beta function: In studying the Gamma function, Euler discovered another function, called the Beta function, which is closely related to
This is an improper integral of Type I, where the potential bad points are 0 and 1.
www.sosmath.com /calculus/improper/gamma/gamma.html   (608 words)

 Welcome to Euler
This is the GTK+ based version of EULER for Unix / Linux systems.
EULER is a program for quickly and interactively computing with real and complex numbers and matrices, or with intervals, in the style of MatLab, Octave,...
EULER can read these data form the file and produce plots of these data, fit polynomials to it, do further computations etc.
euler.sourceforge.net   (302 words)

 Special Functions
A number of these functions may be used to evaluate the properties of various statistical distributions or to return special mathematical values such as Euler’s constant.
Evaluates the derivatives of the incomplete beta integral B(x,a,b), where s is an integer from 1 to 9 corresponding to the desired derivative:
Evaluates the derivatives of the incomplete gamma integral G(x,a), where s is an integer from 1 to 5 corresponding to the desired derivative:
www.eviews.com /eviews3/eviews31/ev3functions/special.html   (152 words)

 Algebraic evaluations of some Euler integrals, duplication formulae for Appell's hypergeometric function F_1, and ...   (Site not responding. Last check: 2007-08-16)
Abstract: Explicit evaluations of the symmetric Euler integral R 1 0 u ff (1 \Gamma u) ff f(u)du are obtained for some particular functions f.
These evaluations are related to duplication formulae for Appell's hypergeometric function F1 which give reductions of F1(ff; fi; fi; 2ff; y; z) in terms of more elementary functions for arbitrary fi with z = y=(y \Gamma 1) and for fi = ff + 1 2 with arbitrary y; z.
These duplication formulae generalize the evaluations of some symmetric Euler integrals...
citeseer.ist.psu.edu /ismail99algebraic.html   (577 words)

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