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. Y

    Number Theory - How to Prove n^7 is Congruent to n Mod 63

    number theorm -- Euler theorem Homework Statement let be an integer that not divisible by 3. Prove that n^7\equivn mod 63 Homework Equations none The Attempt at a Solution it is suffice to prove that n^7\equivn mod 7,n^7\equivn mod 9, i get n^7\equivn mod 7 by Euler theorem ...
  2. V

    Solving for Euler angles and 3-D coordinate Rotations.

    Hi, (attachment with visuals is included) I have a 3-D vector dataset that is measured in a reference frame (measurement reference frame) that is oriented relative to a horizontal coordinate system. In this dataset I have x-y- and z-component data for the vectors relative to a coordinate...
  3. M

    Need to understand: derivating euler rotation from fixed axis rotation

    Homework Statement Homework Equations given in question The Attempt at a Solution I already know the solution but I don't understand what it means the process is simply fixed(c,b,a) to euler(a,b,c) Rz(a)=Rz(a) Ry(b)=Rz(a)Ry(b)Rz'(a) Rz(c)=Rz(a)Ry(b)Rz(c)Ry'(b)Rz'(a) write it in fixed...
  4. J

    Euler angles. Quantum Mechanics Question

    Homework Statement Let U = e^{iG_{3}\alpha}e^{iG_{2}\beta}e^{iG_{3}\gamma} where ( \alpha, \beta, \gamma ) are the Eulerian angles. In order that U represent a rotation ( \alpha, \beta, \gamma ) , what are the commutation rules satisfied by the G_{k} ?? Relate G to the angular...
  5. K

    How to calculate the Euler class of a sphere bundle?

    I have read the section about sphere bundle in Differential Forms in Algebraic Topology,but I still don't understand the Euler class very clear.I don't know how to calculate it for a sphere bundle,for example the sphere bundle of S^2. And I can't work out the exercise at the end of the...
  6. H

    MATLAB Matlab programming using shooting method, Euler and Runge Ketta methods

    I need help to solve this coursework: MATLAB PROGRAMMING COURSEWORK OBJECTIVES:  Learn to solve engineering problems using MATLAB  Write Euler and Runge-Kutta initial-value ODE solvers  Write a Shooting Method boundary-value ODE solver  Investigate the properties of the solvers ...
  7. M

    Finding Functional for Euler Lagrange ODE

    Hello there, I am interested in the following matter. Given an ODE, can one always find a functional F such that the ODE is its Euler Lagrange equation? I am thinking at the following concrete case. I have the ODE y' = a y I would like a functional given by the intergral over a...
  8. K

    The Euler class of the unit tangent bundle to S^2

    This is an example from Bott and Tu 's book DFAT(page 125).The example is in the image.I don't understand why can we get the local degree of the section s by constructing an vector field by parallel translation and calculate the rotating number of it.And why the local degree is 2? Could...
  9. S

    Euler Lagrange derivation in book

    Hello Can any1 recommend a book that will show the derivation of the Euler-Lagrange equation. (I am learning in the context of cosmology ie. to extremise the interval). Ideally the derivation would be as simple/fundamental as possible - my maths is not up to scratch!
  10. B

    Euler Lagrange Equation trough variation

    Homework Statement "Vary the following actions and write down the Euler-Lagrange equations of motion." Homework Equations S =\int dt q The Attempt at a Solution Someone said there is a weird trick required to solve this but he couldn't remember. If you just vary normally you get \delta...
  11. L

    Initial value problem Euler equation

    Question: Find y as a function of x: x^2 y'' + 8 x y' - 18 y = x^8 y(1)=3, y'(1)=2 Attempted solution: I found the general equation to be Ax^(-9)+Bx^2+Cx^8. However when I try to solve the initial value problem for this equation I have 3 unknowns.
  12. B

    How did Euler come up with his famous formula?

    e^{ix}=cos(x)+isin(x) Simple enough? Well I am biting my head off because I don't know how he did it. Why is this bothering me? ex is not a periodic function. How in the world magic happens when you put an i into exponent, and it gets periodic. I want to know this so badly.Why is this...
  13. F

    What does the expansion step for 1D Euler Equations for unsteady gas flow mean?

    Expansion 1D Euler Eq.?? Trying to figure out an expansion step for 1D Euler Equations for unsteady gas flow. Continuity: \frac{\partial(\rho F)}{\partial t}+\frac{\partial (\rho uF)}{\partial x}=0 After Expansion: \frac{\partial(\rho)}{\partial t}+\frac{\partial (\rho u)}{\partial...
  14. K

    Calculate Euler Buckling Load: Formula & Tutorial

    hi, i am a beginner of this forum. i am not really sure how to use this forum actually. i would like to ask, how to find euler buckling load? what is the formula i need to use?
  15. L

    Euler Lagrange method, just not getting it at all

    Homework Statement In Classical mechanics 2 i have an assignment based on the Euler Lagrange method and i cannot seem to grasp the concept, even with all the internet resources i can find as well as my two textbooks which have a chapter on it. (Boass (Mathematical methods in teh physical...
  16. J

    Eliminating the Euler Angle singularity without quaternions?

    Hi all, I've formulated using Lagrangian formalism the equations of motion for a spinning top. I know about the gimbal lock/singularity that occurs at theta=0 and I was wondering if there was any other way to do it without dwelving into quaternions. Yogi published a paper "A Motion of Top...
  17. Y

    Integral representation of Euler - Mascheroni Constant

    The definition of the Euler - Mascheroni constant, \gamma, is given as \gamma = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k} - \ln(n) or equivalently in integral form as \gamma = \int_{1}^{\infty}\frac{1}{\left\lfloor x\right\rfloor} - \frac{1}{x}\ dx I saw a seeming related integral...
  18. K

    Who are the current greats that will go down as Gods equal to Gauss, Euler, etc.)?

    Who are the current greats that will go down as "Gods equal to Gauss, Euler, etc.)? Each generation of mathematics (dunno how long a generation is) have great mathematicians. Last generation was Hilbert, Poincare, cantor, and prolly 1-2 more. Alot of the works that these mathematicians have...
  19. D

    Showing that the euler lagrange equations are coordinate independent

    so i know for example that d/dt (∂L/∂x*i) = ∂L/∂xi for cartesian coordinates, where xi is the ith coordinate in Rn and x*i is the derivative of the ith coordinate xi with respect to time. L represents the lagrangian. so using an arbitrary change of coordinates, qi = qi(x1, x2, ..., xn) i...
  20. N

    Euler substitution (Good sources)

    Anyone know any good sources to read about Euler substitutions from? Both online and books would be suffice. Also videos, if there are any. All help would be greatly appreciated =)
  21. F

    Should I live a life of solitude like Newton and Euler?

    Yeah is there any correlation between being single for the rest of your life and very successful in Math/physics?
  22. R

    Most Efficient way to solve for Euler Angles

    Hi guys, could you please help me out? Essentially, a laser is pointing at a certain point in 3 dimensional space. There is a fixed target that the laser is specifically supposed to point to. My job is to find the most efficient way to solve for the euler angles(yaw pitch and roll) in order...
  23. fluidistic

    Fortran Fortran 90. Euler and Runge-Kutta's method

    I've successfully (well I think so) made 2 different programs that can numerically solve an ODE using Euler and Runge-Kutta's methods. Here they are: program test implicit none real(8)::a,b,h,y_0,t write(*,*)"Enter the interval a,b, the value of the step-size h and the value of y_0"...
  24. W

    Converting Complex Impedance to Euler Form: Is it Applicable to Just Cos?

    1. I have a complex Ohm question in which u(t) is given as Umax*\sqrt{2}cos(\varpi+\varphi), i know how to convert from trigo to euler form if i have both sin and cos but this doesn't. Is it possible to convert just a cos to Euler form ? 3. Since it is a complex impedance i tried to...
  25. R

    Euler lagrangian equation associated with the variation of a given functional

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  26. U

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    hello, I have read in a number of papers that if we have a cantilever beam and are only interested in the movement of the tip when the base is being excited at the frequency of the first eigen mode , then the whole beam can be replaced by a spring mass system. Can anyone tell me of a...
  27. K

    What is the equation for my graph and how can I use Eulers formula to help me?

    Homework Statement Since the exercise has a graph I uploded it here :http://imageshack.us/photo/my-images/833/img9845wz.jpg/ fmax(t)=1 T=2Pi h=1,3,5,7,... Also I was told that I could use Eulers formula here. Homework Equations maybe someone could give me some tips how to make the...
  28. S

    Particular solution of nonhomog. euler equation Messy absolute values in integral

    Homework Statement Solve the IVP (x^2)y'' + 4xy' - 40y = x^6 for y(1) = 10, y'(1) = 1Homework Equations not so much "equations" but here I try to use variation of parameters to get the particular solution.The Attempt at a Solution FOR THE HOMOGENEOUS SOLUTION: using the substitution y = x^r...
  29. S

    Convergence of implicit Euler method

    Homework Statement The implicit Euler method is yn = yn-1 + hf(xn,yn). Find the local truncation error and hence show that the method is convergent. Homework Equations The Attempt at a Solution I found the error to be ln = (-h2/2)y''(xn-1) + O(h3). For convergence I am up to...
  30. M

    Calculating Euler Angles from Two Frames of Reference

    Fairly straight forward question. If you have a set of three vectors specifying a frame of reference and a second set of 3 vectors stating another frame of reference. How do you get the Euler angles associated with that rotation? More generally I am considering the relative orientation of one...
  31. A

    C/C++ Solve ODEs Backwards in Time with C++ Euler Method

    Hi Everybody I am beginner in c++ and I need your help please. I implemented euler method for solving simple ODEs (y' = x -y, y(0)=1)and it is forward in time(from t=0 to t=1) and it worked well, my question is : I want to run this code backward in time(t=1 to t=0) what i have to change in my...
  32. T

    Solving Diff. Eq. with Euler Method: x=0, y=5

    Homework Statement Using the Euler method solve the following differential equation. At x = 0, y = 5. y' + x/y = 0 Calculate the Numerical solution using step sizes of .5; .1; and .01 The Attempt at a Solution From my textbook I have coded Euler's method function [t,y]...
  33. M

    Euler Lagrange equation - weak solutions?

    Hello there, I was wondering if anybody could indicate me a reference with regards to the following problem. In general, the Euler - Lagrange equation can be used to find a necessary condition for a smooth function to be a minimizer. Can the Euler - Lagrange approach be enriched to cover...
  34. U

    Damping term in Euler Bernoulli equation

    hello, I have made an FEM simulation of a cantilever beam in Matlab. I have included the damping using damping matrix C= alpha x M + beta x K. Problem is that I want to compare my result with this paper http://flyingv.ucsd.edu/rvazquez/Journal/nano.pdf (See eq.1) where the...
  35. T

    Calculating Euler Buckling of a Steel Rod: End Fixities & Failure Loads

    a steel rod, 40mm in diameter and 1.00m long, is pinned at each end i) calculate the euler buckling for the rod ii) identify three other possible end fixity conditions for the rod and demonstrate how euler buckling load would be affected in each case iii) explain the relation between the...
  36. M

    How can I find the y(x) that minimizes the functional J?

    Hello there, I am dealing with the functional (http://en.wikipedia.org/wiki/First_variation) J = integral of (y . dy/dx) dx When trying to compute the Euler Lagrange eqaution I notice this reduces to a tautology, i.e. dy/dx - dy/dx = 0 How could I proceed for finding the y(x) that...
  37. Telemachus

    Modified Euler equations doubt

    Homework Statement Hi there. I'm not sure if this question corresponds to this subforum, but I think you must be more familiarized with it. The thing is I don't know how to get from: M_x=(I_0-I)\dot\Psi^2\sin\theta\cos\theta+I_0\dot\Phi\dot\Psi\sin\theta to...
  38. M

    Proving the Euler Phi Equation Divisible by 2: A Step-by-Step Guide

    Prove that the Euler phi equation is always divisible by 2. If n > 2.? I don't understand how this proof works: I think I need to show that the inverse of a and a are both generated by the same group. Therefore, there are at least 2 elements that generate all other elements in the group...
  39. P

    Error in Numerical Solution of ODE by Euler Method - Patrick

    Hi, I recently need to do some numerical simulation by Euler method to solve a PDE. However, I noticed that there are some errors which are obtained with bigger numerical steps, when applying Euler scheme. Since my major is not mathematics, I do not know what this phenomenon is called. I...
  40. B

    Lagrangian mechanics - Euler Lagrange Equation

    Euler Lagrange Equation : if y(x) is a curve which minimizes/maximizes the functional : F\left[y(x)\right] = \int^{a}_{b} f(x,y(x),y'(x))dx then, the following Euler Lagrange Differential Equation is true. \frac{\partial}{\partial x} - \frac{d}{dx}(\frac{\partial f}{\partial y'})=0...
  41. WannabeNewton

    Euler-Lagrange Equation for a Stationary Action

    Homework Statement If L(y, y', x) = y^{2} + y'^{2} then find the appropriate Euler Lagrange Equation. I have absolutely no idea how to solve this. I used the differential form of the Euler Lagrange equations for a stationary action but the answer i got was nothing like the answer in the book...
  42. L

    Solving 2nd Order ODEs with Euler & Runge-Kutta 4

    For weeks I've been coding a program to simulate waves on a string. Currently I am working on some numerical methods. Namely Euler and Runge-Kutta. I have Euler working fine as far as I can see for a variety of input waveforms (of course after it runs for a while it goes a bit beserk due to...
  43. Y

    How to derive Euler DE for n=0

    x^2 y'' + x y' + n^2 y = 0 \; Is Euler equation and solution is y=x^m I understand the three cases with different solution. But my question is if n=0. If I use y=x^m \;\Rightarrow\; m(m-1)+m=0 \;\Rightarrow m^2 = 0 \;\Rightarrow m=0 That would not work. I know the answer is y=C_1...
  44. D

    Third order Euler Differential Equation

    Homework Statement x^3y'''-3x^2y''+7xy'-8y=x+e^2x Edit : Got it thanks
  45. F

    Solving Differential Equations: the Euler D.E. and Linear D.E.s

    Hello guys, I am new here to the forum and I was wondering if you could help me with some trouble I am having with differential equations. For your information: I am a second year applied physics student from the Netherlands and I loveee everything about physics and mathematics! The problem...
  46. C

    What Are Embedded Axis Frames in Euler Equations?

    my book says that it is actually difficult to get the true motion of a body by using these equations because it says that euler equations are written in embedded axis frame ... what is an embedded axis frame?where is it different from normal frames that i used in before?after solving euler...
  47. T

    Advanced Engineering Mathematics: Euler Method

    Do 10 steps. Solve the problem exactly. Compute the error (Show all details). The problems says do 10 steps, but 3-4 steps will suffice! Problem: y(prime) = (y-x)^2 y(0) = 0 h = 0.1 I don't understand how to get the exact solution and what to do from there! I know that, f(x,y) =...
  48. P

    How can the solution to a non-exact differential equation be derived?

    Suppose we have the differential equation: \frac{dx}{\sqrt{1-x^2}} + \frac{dy}{\sqrt{1-y^2}} = 0 It can be rewritten as: \sqrt{1-y^2}dx + \sqrt{1-x^2}dy = 0 One solution of this equation (besides arcsin(x) + arcsin(y) = C), is given by: x\sqrt{1-y^2} + y\sqrt{1-x^2} = C...
  49. K

    What is the uniqueness of Euler angles?

    Hi.. The wikipedia article on euler angles claims that the Euler angles in zxz convention are unique if we constrain the range they are allowed to take (except in the case of the gimbal lock). This seems reasonable. But can someone give me a reference... a book or a paper where this is...
  50. T

    Odd result from an eigenvalue problem in the Euler equations

    Given the Euler equations in two dimensions in a moving reference frame: \frac{\partial U}{\partial t} + \frac{\partial F\left(U\right)}{\partial x} = 0 U = \left(\rho , \rho u , \rho v , \rho e \right) F\left(U\right) = \left(\left(1-h\right)\rho u , \left(1-h\right)\rho u^2...
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