What is Euler: Definition and 410 Discussions

Leonhard Euler ( OY-lər; German: [ˈɔʏlɐ] (listen); 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the study of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.
Euler is held to be one of the greatest mathematicians in history. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is also widely considered to be the most prolific, as his collected works fill 92 volumes, more than anyone else in the field. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
Amongst his many discoveries and developments, Euler is credited for, among other things, popularizing the Greek letter π (lowercase pi) to denote Archimedes' constant (the ratio of a circle's circumference to its diameter), as well as first employing the term f(x) to describe a function's y-axis, the letter i to express the imaginary unit equivalent to √-1, and the Greek letter Σ (uppercase sigma) to express summations. He gave the current definition of the constant e, the base of the natural logarithm, still known as Euler's number.Euler also revolutionized the field of physics by reformulating Newton's classic laws of physics into new laws that could explain the motion of rigid bodies more easily, and made significant contributions to the study of elastic deformations of solid objects.

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  1. S

    Euler angles and symmetric top

    Homework Statement Check out problem 5.7 part a I want to express the exterior gravitational potential in terms of the Euler angles so that I might eventually use (dV/dB)B=B0 = 0 - the condition for equilibrium. I am therefore expecting the Lagrangian to be cyclic in terms of the other two...
  2. S

    Fortran Euler Method in Fortran - HELP

    Euler Method in Fortran - HELP! Using Euler method I want to calculate the equation below, deltat=0.3, 10 times. Problem: When n=1, Yn1(1) is calculated right. But, in the following steps it should assume that Yn(2)=Yn1(1) and the program is assuming Yn(2)=0, as well as Yn(3)=0,...and so on...
  3. M

    Is There a Way to Regularize Euler Products on Primes?

    although is not valid in general (since an Euler product usually converges only when Re(s) >1) \frac{ d \zeta(1/2)}{\zeta (1/2)}= -\sum_{p} log(p)(1-p^{1/2}
  4. Y

    How do I solve for w and p in an incompressible flow using Euler's equation?

    I'm stumped on a HW question that I just can't seem to proceed on. Homework Statement An incompressible ( rho = constant ) flow in 2 dimensions [x = (x,z)], with F = (0,-g), satisfies Euler's equation. For this flow, the velocity is u = (u0,w(x)), where u0 is a constant, with w = 0 on x...
  5. N

    Solving Euler Theorem Doubts with Partial Derivatives

    firstly, all d's i use will mean partial derivative 'do' because i don't have the font installed. sorry :( please help me with these.. u got to use euler theorem 1. if z= xf(y/x) + g(y/x), show that x2(d2z/dx2) + 2xy(d2z/dxdy) + y2(d2z/dy2) =0 2. if z= (xy)/(x-y), PT (d2z/dx2) + 2(d2z/dxdy)...
  6. B

    MATLAB Euler Method in MATLAB: Solving a Double Pendulum

    Hi all Im working on a systems dynamics problem which involves modelling a double pendulum, a chaotic system I have a non linear system of 4 1st order differential equations which I need to solve using numerical methods in MATLAB The methods are ode45, ode23s and euler method Ive...
  7. L

    Euler Angles between two versors

    Dear All, Suppose you have two versors x and y. You would like to calculate the 3 Euler angles between them. How would you do that? You know that x=Ry where R is the rotation matrix (see e..g Goldstein). Now the point is that I know both x and y and want to get phi, theta and psi (no...
  8. P

    Solving Euler Equation for F(K) and U(C) with Initial and Terminal Conditions

    The functions K maps to f(K) and C maps to U(C) are given as: F(K) = aK and U(C) = -C^2 +b here a and b are positive constants. the initial and teriminal conditions with the triminal time T = 1/a are K(0) = K_0 and K_T = K(1/a) here K_0 and K_T are positive constants. Write Euler...
  9. M

    Givens rotations versus Euler angles

    Trying to implement QR decomposition using Givens rotations, I calculate G1 to zero n32 of original matrix A, then G2 to zero n31 of G1 * A, then G2 to zero n21 of G2 * G1 * A. Residual matrix, R = G3 * G2 * G1 * A comes upper triangular as expected, so I believe my code is correct. Looking...
  10. T

    Is the Advanced Euler Method Accurate for Solving Differential Equations?

    Homework Statement integrate dy/dx = (1+e^y)(1+x) from x=0 using the improved euler method taking step sizes of 0.125. compare the numerical solution of y(0.5) with the exact value The Attempt at a Solution Firstly to find the exact value I use separation of variables and the initial...
  11. J

    Euler and lagrange frame of reference

    can anybody please clearly explain me the difference between these two frames of reference with few examples. my exames are closing up. please help me.
  12. K

    Validity Using Euler Circles and Truth Tables

    I'm so confused on how to tackle this problem: 1. Truth tables are related to Euler circles. Arguments in the form of Euler circles can be translated into statements using the basic connectives and the negation as follows: Let p be “The object belongs to set A. “Let q be...
  13. M

    Runge-Kutta vs Euler: Solving Two-Dimensional Differential Equation

    Hi, I don't know if this is the right forum to adress, but I will try nevertheless Im solving a simple two-dimensional differential equation: dx/dt = (-y,x) which will give a circle when integrating over time. Now, the problem is that the simple euler scheme seems to be a lot more...
  14. T

    How did Euler know this?

    In the general DE form, y"+y=0 and where the initial conditions are y(0)=2 and y'(0)=0, Euler realized that y(x)=e^ix+e^ix. How did he know that it's a cosine graph when there's no indication in the equation that any solution is possible? y= c1+c2=2 but y" gives 0.
  15. X

    Does d dividing n imply phi(d) divides phi(n)?

    Prove that if d divides n then phi(d) divides phi(n). Thanks
  16. P

    Using Euler Lagrange in General Relativity

    I'm trying to follow a professor's notes for finding Christoffel symbols for a two-sphere. He gives the following two equations: The Lagrangian for a two sphere:L = \left( \frac{d\theta}{ds} \right)^2 + sin^2\theta \left( \frac{d\phi}{ds} \right)^2 The Euler Lagrange equation: \frac{d}{ds}...
  17. J

    Solving non-linear equations simultaneously with DE's using Euler integration

    I have 15 equations with 15 unknowns describing a dynamic process. I would like to know how I can conbine solving non-linear equations together with ordinary differential equations (1st and 2nd order) simultaneously, without using ode45, but Euler's method.
  18. B

    Drawing Direction Fields for Non-Autonomous Differential Equations

    Direction field Do anybody have a hint for drawing direction field of a non-autonomous differential equation? I mean do I have to calculate as many slopes of points as possible, then draw it? Also, Can I conclude that if we use Euler's Method to estimate a CONCAVE UP/ CONCAVE DOWN...
  19. C

    Euler expansion of double exponential?

    Simple question, I have used the euler expansion to estimate a variable that grows as a single exponential. adapt = Amax * exp(-tau*X); In excerpted form: for (i=1;i<npts; i++) { adapt[i] = adapt[i-1] + (Amax -adapt[i-1]) * dt / tau; } where dt is the step size and tau is...
  20. J

    Conditions of the Euler Equation

    Hi everyone :D This is my problem: Find conditions on \alpha and \beta in the Euler equation x^{2}y'' + \alphaxy' + \betay = 0 such that: a) All solutions approach zero as x \rightarrow 0 b) All solutions are bounded as x \rightarrow 0 c) All solutions approach zero as x \rightarrow\infty I...
  21. M

    Can someone explain Euler angles?

    Can someone explain "Euler" angles? From what I read, "Euler" rotations are composed out of matrices like * * 0 1 0 0 * * 0 * * 0 0 * * * * 0 0 0 1 0 * * 0 0 1 which is pretty distinctive in that they rotate around same axis twice, and makes sense for devices like this...
  22. E

    What is the Proof of the Euler Lagrange Equation?

    [SOLVED] Euler Lagrange Equation Hi there , I am missing a crucial point on the proof of Euler Lagrange equation , here is my question : \frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{df}{dy^{'}}\right)=0 (Euler-Lagrange equation) If the function "f" doesn't depend on x explicitly...
  23. C

    Betti numbers and euler characterstic?

    Let's say you have some type of simplicial complex that is made only of 2 simplices. What happens if all those 2 simplices are adjacent to a single edge (creating a type of book shape), so that this complex can only be embedded in dimensions 3+? Would this complex have the same 2nd betti number...
  24. H

    Translation of euler angles into rotation around arbitrary axis

    i have an orientation of a 3d object in space given by theeta, si and phi i.e. angles which the objects makes with respect to three axis. Now i want to translate the problem such that i get an arbitrary axis rotation about which to some calculated degrees would produce same orientation...
  25. E

    Solving the Euler Cauchy Equation: Finding the General Solution

    Homework Statement Find the general solution of x^2y" - 2y = 0 Homework Equations The Attempt at a Solution Can anyone tell me how to find the general solution of the Euler Cauchy equation. How do we make it into one?? Thanks.
  26. J

    Euler rotation of objects with velocity vectors

    I have a number of objects (points) in a 3D space. I need to rotate this space using euler angles (or equivilent) and place it in another coordinate system. (ie i start with objects placed within the confines of a cylinder aligned with the z axis, and after rotation have a cylinder of objects at...
  27. R

    Explore Project Euler: A Fun Challenge for Math & Programming Lovers

    Hi people! Well i don't know if all here know about the Project Euler (http://www.projecteuler.net/) but i think it is very useful because for all of us here know how to program numerical algorithms fine is very important. "Project Euler is a series of challenging mathematical/computer...
  28. T

    Question on a form of the Euler constant

    I was wondering how I would go about proving this equation: \int_{1}^{\infty}\frac{u-[u]}{u^2}du=1-\gamma where \gamma is the Euler Constant, and [u] is the floor function
  29. S

    Usefulness of Euler line in triangles?

    Is the Euler line in triangles USEFUL for anything in real life? This is the line which contains the concurrency points for the intersections of the triangle perpendicular bisectors, the medians, the altitudes, but not the angle bisectors. Interesting stuff, but are these points which occur on...
  30. MathematicalPhysicist

    Calculate an integral using euler substituition.

    i was asked to calculate the integral: \int\frac{dx}{x+\sqrt{x^2-x+1}} by using euler substituition (i.e, finding a line which intersects sqrt(x^2-x+1) through one point and then the equation of the line will be y-y0=t(x-x0) where (x0,y0) is one point of intersection, and then substituing x for...
  31. O

    Why is the Euler Sum Infinite for Zeta=1?

    This may be a stupid question, but I do not understand why the Euler sum is infinite for zeta=1. Why is "1+1/2+1/3+1/4+... " infinite, but zeta=2 (1+1/4+1/9+...) not?
  32. K

    Graphing Euler Relationship as exponential spiral

    Hi everyone, I am trying to get an intuitive grasp of the Euler Relationship e^i(theta)=cos(theta)+i sin(theta) and also understand how to graph the exponential spiral, as demonstrated on this web page: http://www-math.mit.edu/daimp/ComplexExponential.html" Ok, first the neuron...
  33. D

    Proving Euler Equations and Solving Complex Numbers in Signal and Systems Course

    Hi i am reading about signal and systems course . What i want to prove is not a problem that i have to solve is something that the books take for granted and i want to prove it so i ll be able at exams to reprove so i won't have to remember it, (if u don't believe me i can give u the course's...
  34. quasar987

    The Klein bottle using the Euler caracteristic + orientability caracterisation

    I went off on my own to study the Euler caracteristic + orentability caracterisation of closed surface and I must have gotten lost somewhere, because I do not find that the Klein bottle is homeomorphic to \mathbb{R}P^2\#\mathbb{R}P^2 as I should. I started with the result that for two surfaces...
  35. B

    Differential equations - 2nd order euler eq'n

    (a) Let alpha (a) and beta (b) be given constant. show that t^r is a solution of the Euler equation t^2 d^2y/dt^2 + at dy/dt + by = 0 , t>0 if r^2 + (a-1)r + b = 0 (b) suppose that (a-t)^2 = 4b. Show that (ln t)t^(1-a)/2 is a second solution of Euler's equation. please help, i have no idea...
  36. Bob3141592

    Dr Euler and characteristic equations

    In "Dr. Euler's Fabulous Formula" by Paul Nahin, early in chapter 1 is discussed characteristic polynomials of a square matrix and the Cayley-Hamilton theorem, that any square matrix A satisfies its own characteristic equation. On page 21 it states p(lambda) = lambda^2 + a1*lambda + a2 = 0 and...
  37. D

    Results same w/ Euler, Improved Euler, & Runge-Kutta

    I've been trying to do some simulations using the Euler, improved Euler, and Runge-Kutta methods. My results for the improved Euler and Runge-Kutta are very close to my results with the plain old Euler method. If there is any improvement, it is negligible. This lack of improvement surprises...
  38. cepheid

    Solving Euler Integrals Homework: Show B(a,b)

    Homework Statement For fun: show that B(a,b) = \int_0^1{x^{a-1}(1-x)^{b-1}\,dx} = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} where a > 0 , b > 0 . Hint: start from the product \Gamma(a)\Gamma(b) and switch to polar coordinates. The radial integral is proportional to \Gamma(a+b).Homework...
  39. G

    Euler was pronounced with a long u sound

    "Euler" was pronounced with a long "u" sound Ok, Up until last week I thought "Euler" was pronounced with a long "u" sound (like Euclid). Since most of the famous names of science/math are names I have read, not heard, I was wondering if someone could give me the correct pronunciation of...
  40. H

    Solve Euler's Equation x^7 ≡ 21 mod 66

    Hi How do I solve the equation x^7 \equiv 21 modulo 66 66 = 2.3.11 so try solving it mod 3 and mod 11 (mod 2 doesn't gives any new information). This tells us x is divisible mod 3, and x^7 = -1 mod 11. One solution to this is x = -1 mod 7. The smallest solution of these two congruences...
  41. M

    Euler: Please verify my result

    Hi I got two tasks which I have some trouble with. 1) A guy has 1770 dollars to shop food for. One bread costs 31 dollars and a jar of jam costs 21 dollars. How many loafs of bread and jar's of jam can the guy buy? I'm suppose to calculate it using Euler Algebra 31x + 21y...
  42. H_man

    MATLAB Matlab and the Euler Equations

    Hi, I was wondering if Matlab was the sort of program I'd want to solve the Euler Equations (fluid dynamics). And if it is, I am sure this must be a very standard problem.. does anybody know of any tutorials for this sort of problem as I have never used matlab? :tongue2:
  43. B

    Proving the Divisibility of the Euler Function: A Number Theory Problem

    The Euler Function \varphi (n) is defined as the number of natural numbers less than n that are relatively prime to n. That is, \varphi (n) = | \{ a \in \mathbb{N} | a < n and gcd (a, n) = 1 \} | I have been asked to show that if a number d divides a number n, then \varphi (d) divides...
  44. K

    Vector rotation (possibly Euler angles)

    I'm looking for a method to rotate a 3D vector, and place it at an arbitary 3D point (x,y,z) without changing the vectors magnitude. I have briefly investigated eulers angles (mainly through wikipedia links etc), but don't fully understand the process yet. As an example, given the vector ...
  45. M

    The Effects of the Euler Force

    Can anyone describe what the effects of the Euler force are? If you're not familar with it, it is a fictitious force that arises from a rotating object undergoing a change in rotation speed. Can someone tell me what the effects of this fictitious force are?
  46. E

    Euler product and Goldbach conjecture

    In an anlaogy with the Euler product of the Riemann function we make: \prod_{p}(1+e^{-sp})=f(s) of course we have that: f(p1+p2+p3)=f(p1)f(p2)f(p3) f(x)=exp(-ax) if Goldbach Conjecture is true then p1+p2= even and p5+p6+p8=Odd for integer n>5? then this product should be equal to...
  47. A

    Earth's Euler angles - nutation?

    Hi, I would like to see the resemblance between planet Earth and a spherical top. I draw here the x-convention Euler angles as I know them: 24 hours to complete one complete one revolution around it's own z' axes. And the 23.5° Earth axes revolves around the "sun's z axes" - the line...
  48. V

    Proving Euler Line Parallel to Side BC in Triangle ABC

    Can anyone help me solve the following question. ABC is any triangle. XAB, YBC and ZAC are equilateral triangles formed on this triangle. Prove that AY, BZ and CX are concurrent. In a triangle ABC, the Euler line is parallel to side BC. Prove that tanB*tanC = 3. I just need a hint. I...
  49. M

    Solving Classic Euler Question on Buying Rice and Corn with 1770 Dollars

    Hi I'm told that the the following can be presented as a classic euler question: A farmer goes to marked with 1770 dollars to buy rice and corn. One bag of corn costs 10 dollars and one of rice cost 20 dollar. How much corn and rice can the farmer buy? This can be presented in the following...
  50. K

    Generating function for terms of Euler triangle?

    I'm sure this is relatively easy, but after an hour or so googling, I can't seem to find the formula for generating terms of the http://steiner.math.nthu.edu.tw/chuan/123/test/euler.htm Is this known by some other name? Maybe that's why I can't find it? Thanks
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