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

    Does Euler Force Affect Flight Experiments on Earth?

    Hello! I am currently starting my thesis, and I would be pleased if you could help me. As it is an ficticious force for non inertial frame, the Euler force affects every movement measured from the surface of the Earth. But, it is related to the angular acceleration changes of our planet, so...
  2. P

    Euler-Lagrange Equation of $$ds^2=-f(x)dt^2+g(x)dx^2+2l(x)dxdt$$

    If metric is $$ds^2 = -f(x)dt^2 + g(x)dx^2 + 2l(x)dxdt $$ Then we have this Lagrangian: $$L= \frac{1}{2}(-f(x)\dot{t}^2 + g(x)\dot{x}^2 + 2l(x)\dot{x}\dot{t}).$$ The Euler-Lagrange equation for $$t$$ is: since $$t$$ is not there in the Lagrangian then $$\partial L/ \partial t=0$$ This implies...
  3. SalfordPhysics

    FORTRN90: Euler Midpoint Method for SHO

    Homework Statement Write a program to simulate motion of simple harmonic oscillator. Initial conditions: Let ω = 1, x(t=0) = 1, v(t=0) = 0. Integrate over 30 seconds in intervals of 0.05s. Homework Equations δ2x / δt2 = -ω2x As set of 2 coupled ODE's; x' = v, v' = -w2x The...
  4. P

    Euler Equations & Rigid Lamina Moment of Inertia

    "A rigid lamina (i.e. a two dimensional object) has principal moments of inertia about the centre of mass given by ##I_1=u^2-1##, ##I_2=u^2+1##, ##I_3=2u^2## Choose the initial angular velocity to be ##ω = µN \hat{e_1} + N \hat{e_2}##. Define tan α = ω2/ω1, which is the angle the component of ω...
  5. F

    MHB Solving Cauchy-Euler Equation with Constant Coefficients

    I'm trying to solve a(x-x_0)y''+b(x-x_0)y'+cy-c=0 So I let y=(x-x_0)^m then y'=m(x-x_0)^{m-1} and y''=m(m-1)(x-x_0)^{m-2} plugging in gives a(x-x_0)m(m-1)(x-x_0)^{m-2}+b(x-x_0)m(x-x_0)^{m-1}+c((x-x_0)^m-1)=0 now I want to find the values of m that make the equation 0, but factoring seems to be...
  6. B

    Integration using Euler Substitution

    Does anyone know of a derivation or justification of Euler's substitution formulas for evaluating irrational expressions? In other words, to evaluate integrals of the form: \int R(x,\sqrt{ax^2+bx+c}) You can use Euler's substitutions: 1. \sqrt{ax^2+bx+c} = t \pm \sqrt{a}x, a>0 2...
  7. W

    Formal Computations of Euler Classes

    Hi All, The Euler class of the tangent bundle of a compact, oriented manifold agrees with the evaluation of the top homology class on the fundamental class (which is represented by the manifold itself), and maybe also figure out how to do actual computations using Poincare duality (to figure out...
  8. D

    Euler Lagrange equations with viscous dissipation

    Homework Statement The system can pivot at point O and I am taking small angle approximations. I am trying to determine the Lagrangian, ##\mathcal{L} = T - U## for the following system: Homework Equations E-L equation with dissipation: ##\frac{\partial\mathcal{L}}{\partial q_i} -...
  9. P

    Frobenius method and Euler equations

    Hi, I'm having trouble with this one. Homework Statement Find a particular solution of the second-order homogeneous lineal differential equation x^2y'' + xy' - y = 0 taking in account that x = 0 is a regular singular point and performing a power series expansion. Homework...
  10. Ascendant78

    Use the Euler identity to prove sin^2x+cos^2x=1

    Homework Statement Just like my title says, we are to prove the trig identity sin^2x+cos^2x=1 using the Euler identity. Homework Equations Euler - e^(ix) = cosx + isinx trig identity - sin^2x + cos^2x = 1 The Attempt at a Solution I tried solving the Euler for sinx and cosx...
  11. A

    Euler Equations, Sod shock tube & conservation

    Is momentum conserved? I am considering the Euler equations in conservative form and solving the Sod shock tube problem I have written a Godunov finite volume type solver. It solves for density ρ, momentum ρu, and total energy E; therefore, I would expect all of these quantities to be...
  12. F

    Question on Elements of Algebra by Euler

    I like this book very much. Euler is a brilliant mathematician no doubt. He explains everything very well without holding back significant information with his exposition. However, I bump into an unfamiliar topic. I do believe it has something to do with advanced mathematics. I googled it, and I...
  13. N

    Project Euler 12: Highly divisible triangular number

    After running into the dilemma of having nothing to do a little while ago, I decided to try working on a Project Euler problem with a mathematical approach. Not being a mathematician, I soon found myself in a rut. Now, what I have so far is this: Let T(n) be the nth Triangle Number, n > 0...
  14. P

    Help with implicit euler method

    Hey I don't understand this backward euler solution, in particular why the f(y,t+h) is equal to y(t+2)
  15. T

    Diagonalization of 8x8 matrix with Euler angles

    I am trying to diagonalize the following matrix: M = \left( \begin{array}{cccc} 0 & 0 & 0 & a \\ 0 & 0 & -a & 0 \\ 0 & -a & 0 & -A \\ a & 0 & -A & 0 \end{array} \right) a and A are themselves 2x2 symmetric matrices: a = \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{12} & a_{22}...
  16. F

    2nd Order Differential Equation with Improved Euler Method (Heun's)

    Homework Statement I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct...
  17. skate_nerd

    MHB Euler Lagrange equation of motion

    I have a system with one generalized coordinate, x. In the potential energy part of the lagrangian, I have some constants multiplied by the absolute value of x. That is the only x dependence the lagrangian has, so when I take the partial derivative of the lagrangian with respect to x (to get the...
  18. W

    Implicit Euler scheme and stability

    Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation y'=y(1-y) and investigate their stability? => implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1})...
  19. R

    MHB Implicit Euler Scheme and stability

    Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their stability? => implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation}...
  20. Z

    Integration of partials, specifically Euler to Bernoulli Equation

    Hi! I am having trouble following the derivation from Euler's Equation to Bernoulli's Equation. The trouble lies in the math, not the physics part. Especially the step when partial derivatives are being integrated. I have attached the relevant part as a screenshot. How does the...
  21. R

    MHB Harmonic oscillator and symplectic Euler method

    Given the equations for the harmonic oscillator $\frac{dy}{dz}=z, \frac{dz}{dt}= -y$if the system is approximated by the symplectic Euler method, then it gives$z_{n+1}= z_{n}-hy_{n}, \\ y_{n+1}= y_{n}+hz_{n+1}$which shows that the circle $y^2_{n} + z^2_{n} = 1$ is mapped into an ellipse...
  22. D

    What has Euler got on his head here?

    This is a very well-known portrait of Leonhard Euler, which I'm sure most people have seen before, but does anyone know what the heck he's wearing on his head? http://upload.wikimedia.org/wikipedia/commons/d/d7/Leonhard_Euler.jpg It looks like someone threw a dirty old rag on him - does anyone...
  23. N

    Euler equations for ideal fluids, approximations

    Homework Statement The Euler equations for ideal compressible flow are given by \partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p \\ \partial_t \rho + \nabla \cdot(\rho v) = 0 In my book these are written in terms of the small-value expansions \rho = \rho_0 + \delta \rho, p = p_0 +...
  24. S

    Forward Euler Method for ODE system

    Homework Statement Solve the following system for 0<t<5 u^\prime = u-e^{-2t} v, u(0) = 1 v^\prime = u+3v, v(0) = -2 using Forward Euler method and implement the numerical scheme into a MATLAB code. Homework Equations Forward Euler : \vec x^(\prime)_{n+1} = \vec F(t,\vec x)...
  25. J

    Project Euler programming challenges: Am I missing something?

    I recently was looking for programming challenges and it was suggested to me that I check out Project Euler. http://projecteuler.net/problems These all seem straight-forward to me. Or there's something I'm not understanding about them. For instance, problem 1: Is there supposed to be...
  26. S

    Finding general solution to Euler equation via variation of parameters

    Homework Statement The problem is attached as TheProblemAndSolution.png, and everything is typewritten, so it should be easily legible (but you will likely need to zoom into read the text since the image's height is significantly larger than its width). Homework Equations Differential...
  27. polygamma

    MHB The Euler Maclaurin summation formula and the Riemann zeta function

    The Euler-Maclaurin summation formula and the Riemann zeta function The Euler-Maclaurin summation formula states that if $f(x)$ has $(2p+1)$ continuous derivatives on the interval $[m,n]$ (where $m$ and $n$ are natural numbers), then $$ \sum_{k=m}^{n-1} f(k) = \int_{m}^{n} f(x) \ dx -...
  28. H

    Variational calculus Euler lagrange Equation

    I am trying to understand an example from my textbook "applied finite element analysis" and in the variational calculus, Euler lagrange equation example I can't seem to understand the following derivation in one of its examples ∫((dT/dx)(d(δT)/dx))dx= ∫((dT/dx)δ(dT/dx))dx= ∫((1/2)δ(dT/dx)^2)dx...
  29. K

    Fortran How to fix the Euler method error?

    Hi everybody, I am programming a new code for a problem. The problem is numerically solving the Simple Harmonic Motion using the Euler method. This approach is just an approximate solution and not a exact solution, however when I run the code successfully and plot my data, it comes up as an...
  30. matqkks

    Euler Totient or Phi Function

    What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
  31. matqkks

    MHB Exploring the Practical Applications of the Euler Totient Function

    What is most motivating and tangible way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
  32. H

    Euler Solution to the Basel Problem

    I found what looks like an interesting document on a solution to the Basel Problem by Euler. http://eulerarchive.maa.org/pages/E063.html What stinks is that the document is in German and French(I think) so I can't read it apart from the math. I can't seem to find a translated copy of this...
  33. K

    Fortran Fortran program for oscillator using Euler method

    I am trying to run a program with fortran. The program is about solving the Oscillator using Euler Method. I am trying to run this code and applying array arguments (as I want to extend it to 3 dimensions afterwards). When I try to compile, it comes up with an error "Unclassifiable statement at...
  34. alyafey22

    MHB What is the Interesting Euler Sum Proven by this Equation?

    Prove the following Euler sum \sum_{k\geq 1}\left(1+\frac{1}{3}+\cdots +\frac{1}{2k-1} \right) \frac{x^{2k}}{k}=\frac{1}{4}\ln^2\left( \frac{1+x}{1-x}\right)
  35. S

    Euler Method with h=0.05 for y'=3+t-y, y(0)=1 at t=0.1, 0.2, 0.3, and 0.4

    Homework Statement Use the Euler method with h=0.05 to find approximate values of y'=3+t-y, y(0)=1 at t=0.1, 0.2, 0.3, and 0.4. Homework Equations I don't even know what formula and how to use it. The Attempt at a Solution The answers are 1.1975, 1.38549, 1.56491, 1.73658.
  36. D

    MHB Coupled ODEs from Euler Lagrange eq

    Given \(F = A(x)u_1^{'2} + B(x)u'_1u'_2 + C(x)u_2^{'2}\). \[ \frac{\partial F}{\partial u_i} - \frac{d}{dx}\left[\frac{\partial F}{\partial u_i'}\right] = 0 \] From the E-L equations, I found \begin{align*} \frac{d}{dx}\left[2Au_1' + Bu_2'\right] &= 0\\ \frac{d}{dx}\left[2Cu_2' + Bu_1'\right] &=...
  37. D

    MHB What is the integral of f(x) and q(x) in the Euler Lagrange equations?

    Given this \(F = p(x)y^{'2}-q(x)y^2+2f(x)y\). What would be the integral of \(f(x)\) and \(q(x)\)? \begin{align*} f(x) - q(x)y - \frac{d}{dx}\left[p(x)y'\right] &= 0\\ \frac{d}{dx}\left[p(x)y'\right] &= f(x) - q(x)y\\ y'p(x) &= \int f(x)dx - y\int q(x)dx \end{align*}
  38. S

    Euler Angle transformation, help

    I'm doing a research project currently and basically what I have is a camera measuring a probe. I have designed the camera to give the orientation of the probe using euler angles in the camera's frame of reference. This was working for most of my data, but now I need a 3-D visualization of what...
  39. R

    Mapping Euler spiral (clothoid) parameters

    Hi... first post here! Sorry if not in the right place. I am trying to decode the parameters for an xml file format and I would appreciate help in interpreting some parameters. I know the thing specified is a "transition curve" or clothoid curve, as the transition between a straight path and...
  40. P

    Midpoint Euler method, second order system

    Hello I need to plot this simple system: x'' = -x using midpoint Euler. u1 = -x , u2 = -x' u1' = u2 u2' = -x u1(n+1) = u1(n) + h*? u2(n+1) = u2(n) + h*f((1/2)*(u1(n) + u1(n+1)) We don't know u1(n+1). I tried approximating it with u1(n+1) = u1(n) + h*u2(n) u2(1+i) =...
  41. R

    Euler Angles/ Rigid Body application

    I apologize upfront, as I have no experience with math to begin with. However, I have a real life problem I am trying to figure out here at work and I would appreciate some help. Here is my situation: Homework Statement In a 3 dimensional plane, I have an artillery piece pointing at...
  42. A

    MHB How Does the Euler Circuit Algorithm Solve Mazes?

    I didn't understand the algorithm explained in my textbook. ("Introduction to combinatorics" P.165) I would an alternative explanation with an example. Here is the algorithm: (i) from a new vertex, any edge may be taken; (ii) from an old vertex, if the edge just traversed was nes, then turn...
  43. E

    Euler - Lagrange Equation(changing variable)

    Create the Euler-Lagrange equation for the following questions (if it's necessary change the variables). Homework Statement $$\tag{1}\int _{y_{1}}^{y2}\dfrac {x^{'}{2}} {\sqrt {x^{'}{2}+x^{2}}}dy$$ $$\tag{2}\int _{x_{1}}^{x_{2}}y^{3/2}ds $$ $$\tag{3} \int \dfrac {y.y'} {1+yy{'}}dx...
  44. M

    Euler Lagrange Equations for 1 particle in 3-dimensions

    Homework Statement Do the Euler-Lagrange equations set to zero for each of the 3 orthogonal coordinates or do you sum them all equal to zero. Do the coordinates have to be orthogonal in order to write separate E-L equations? Or is there no such thing as non-orthogonal coordinates to analyze a...
  45. M

    MHB Backward Euler Method for 2x2 systems

    Hi! I want to write a code in Matlab for the Backward Euler Method for 2x2 systems, using the fixed point iteration to find the yn+1. y1n+1=y1n+h*f(tn+1,y1n+1,y2n+1) (1) y2n+1=y2n+h*g(tn+1,y1n+1,y2n+1) (2) Could you tell how I use the fixed point iteration?? At (1) the fixed point iteration...
  46. J

    Modeling Rigid Body - Unsure about Euler angles and angular velocity

    I'm modeling a single 3D rigid body in preparation for some more complicated modeling in order to gain a better understanding of Euler angles, the angular velocity vector and the rotating coordinate system. The body is rotated in inertial frame by an intrinsic ZXZ rotation, with respective...
  47. E

    Euler characteristic as a total derivative

    We all know that the Euler characteristic is a topological invariant. But let's suppose that we don't know this or anything else about algebraic topology for that matter. We are given only the Gauss-Bonnet theorem, which expresses the Euler characteristic in geometrical terms. In his string...
  48. G

    Generality of the Euler Lagrange equations

    Hi I wanted to know for which cases the Euler Lagrange equations are applicable? 1.) Imagine that we have a kinetic Energie T(q,q') and a potential that also depends on velocity V(q,q'). As far as i know the Euler Lagrange equations for a particle still hold in this case, is that true...
  49. H

    Limit of the Euler totient function

    My question is relatively breif: is it true that \displaystyle \lim_{n \rightarrow \infty}(\varphi(n))=\lim_{n \rightarrow \infty}(n) \cdot \prod_{i=1}^{\infty}(1-\frac{1}{p_i}) Where p is prime? Pehaps \varphi(n) is too discontinuous to take the limit of, but it would seem that as it increases...
  50. T

    Finding Euler angles from rotation about arbitrary axis

    Homework Statement An object is rotated 45 degrees about an axis whose + direction is that of (i-k). Find zxz Euler angles (that is, Euler angles as introduced by Goldstein) for a set of three active rotations that gives the same net motion of the object. Homework Equations...
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