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. Abhishek11235

    Euler Equations for Dynamics of rigid body

    I have been studying the dynamics of free top from Morin's book. In his book when describing the dynamics,he writes down the equation of motion as shown in screenshot. However,I am not able to understand which term refers to which coordinate system. For eg: Here ##\omega## refers to angular...
  2. S

    How to write the complex exponential in terms of sine/cosine?

    I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know. 1. Homework Statement Using Euler's formula : ejx = cos(x) + jsin(x)...
  3. QuasarBoy543298

    I Understanding 3D Coordinate Rotations with Euler Angles

    I'm trying to wrap my head around the concept. we use 3 rotations to transfer our regular cartesian coordinates (3 x,y,z unit vectors) to other 3 unit vectors. each rotation is associated with an angle. so far I'm good. but now I saw in Landau's and Lifshitz's "mechanics" book this thing...
  4. M

    Euler Line theoram - part of the proof is not clear to me

    This is not homework. I am reading a book: "The art of infinite: The Pleasure of Mathematics" and pages 119-120 give a proof of the Euler Line theoram: the circumcenter, centroild and orthocenter of a triangle are always colinear (see the attached files). 1. Homework Statement Page 119 shows...
  5. T

    I Exploring the Connection between Trigonometric and Exponential Functions

    Hi all: I really do not know what to ask here, so please be patient as I get a little too "spiritual" (for want of a better word). (This could be a stupid question...) I get this: eiθ=cosθ+isinθ And it is beautiful. I am struck by the fact that the trig functions manifest harmonic...
  6. WMDhamnekar

    MHB Euler equations having double roots as a solution

    If the Euler equations have double roots as it's solution, second solution will be $y_2(x)=x^r\ln{x}$. what is its proof? or how it can be derived?
  7. T

    A How do I KNOW that Euler angles are sufficient?

    Hello Before I "phrase" my question (and that may be my problem), may I first state what I do know. I understand that a Rotation matrix (a member of SO(3)) has nine elements. I also understand that orthogonality imposes constraints, leaving only three free parameters (a sub-manifold) I also...
  8. Edge5

    I Solution of Quantum differential equation

    (I think I couldn't add the image) you can see my answer in link https://pasteboard.co/HPKZ6KD.jpg (Please first see my answer in the link) But in answer it is φ= Asin(kx) + Bcos(kx) I know that euler formula is eix = cosx +isinx But I can't get this answer can you help me?
  9. R

    A Integration with Euler angle of rotation matrixes

    Hello, I was struggling with solving a specific integral. I know that I can rewrite the exponential matrices and the range of the three Euler angles. However, I am not sure I should I write in terms those three Euler angles.
  10. R

    Equation of an oscillating system without any starting values

    Homework Statement A mass m1 is located on a platform with mass M. The platfrom is located on springs with total constant k such that it can swing vertically in direction x. a) Write down the equations of motion assuming mass m1 will always be connected to the platform. Write it as x(t) b)...
  11. Arman777

    Python How Does One Solve the Minimal X for Pell's Equation with D Up to 1000?

    Consider quadratic Diophantine equations of the form: x2 – Dy2 = 1 For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 1. It can be assumed that there are no solutions in positive integers when D is square. By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we...
  12. Arman777

    Python Optimizing Prime Pair Combinations for Project Euler Problem 60

    https://projecteuler.net/problem=60 I wrote a piece of code to solve the problem but it still takes too much time to solve it Any ideas how can I improve my code, like adjustments that I can make ? import itertools import time start = time.perf_counter() def prime(N): if N == 0 or N == 1...
  13. S

    A Is the Pressure in Paradoxical Euler Flow Fictitious?

    Consider the following flow: x = (1+ct)x0. Let the density rho(t) = rho0/(1+ct) so that the flow conserves mass. Physically, this is just a bunch of fluid elements on the positive x0-axis each given initial velocities that are proportional to their initial positions. Each fluid element should...
  14. Arman777

    Project Euler Question 58 -- ratio of primes along both diagonals of a spiral....

    Homework Statement https://projecteuler.net/problem=58 Homework EquationsThe Attempt at a Solution def prime(N): if N == 1: return False y = int(N**0.5) for i in range(2,y+1): if N%i == 0: return False return True def finder(N): L = len(N)...
  15. Arman777

    Finding Prime Families by Digit Replacement

    Homework Statement By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime. By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes...
  16. A

    A Euler Angles Transform: Rotating a Body in 3D Space

    Only recently started to understand Euler angles and rotation matrices, and I am reasonably comfortable with the concepts already posted here. I am pretty sure I am missing something obvious, but I cannot figure out the way to solve this problem: A body in 3D space with a orientation defined by...
  17. wirefree

    I Euler, De Moivre and a printing error

    to all members of the forum. In the attached image is equation numbered 3.23 which, by the application of Euler’s Identity - called De Moivre Theorem one line below - leads to equation 3.24. Above is an a textbook frought with errors - printing ones. I would be highly appreciative of a...
  18. H

    I Intuitive understanding of Euler's identity?

    I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and...
  19. Delta31415

    I Question about the Divisor Function/Sums and Project Euler

    So I am kind of lost... I don't really know how to ask this. Project Euler is a website that hosts multiple programming contests and I am interested in this problem https://projecteuler.net/problem=608 but my question isn't truly about this problem but a more solution. I know that the Divisor...
  20. petterson

    A Maximization problem using Euler Lagrange

    Hi, I'm trying to solve the following problem ##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##. I have only little experience with calculus of variations - the problem resembles something like ## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt## but I don't know about the...
  21. S

    I Euler Lagrange formula with higher derivatives

    I was trying to Extrapolate Eulers formula , after deriing the basic form I wanted to prove: ∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0 Here is my attempt but I get different answers: J(y) = ∫abF(x,yx,y,yxx)dx δ(ε) = J(y+εη(x)) y = yt+εη(x) ∂y/∂ε = η(x) ∂yx/∂ε = η⋅(x)...
  22. JTC

    I Using Complex Numbers to find the solutions (simple Q.)

    Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function. We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time. Then, once done, if the forcing function was cosine, we choose the real part as the...
  23. F

    I Deduce Geodesics equation from Euler equations

    I am using from the following Euler equations : $$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$ with function ##f## is equal to : $$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$ and we have...
  24. riveay

    Euler angles in torque free precession of a symmetric top

    Is calculating the Euler angles analitically possible? I am trying to obtain the angles to transform the body-fixed reference frame to the inertial reference frame. I can get them without problems with numerical methods. But I would to validate them analitically, if possible. I followed the...
  25. M

    Solving the Implicit Euler ODE with Boundary Conditions

    Homework Statement Write an implicit Euler code to solve the system ##c'(x) = \epsilon c''(x)-kc(x)## subject to ##1-c(0)+\epsilon c'(0) = 0## and ##c'(1)=0##. Homework Equations Nothing out of the ordinary comes to mind. The Attempt at a Solution In the following code, there is central...
  26. MathematicalPhysicist

    Equilibrium Statistics -- Euler summation formula

    Homework Statement In the calculation in high temperatures of ##Z_{rot} = (\sum_{j=0}^\infty (2j+1)\exp{j(j+1)\theta_{rot}/T})^N##; they use Euler summation formula: $$\sum_{n=0}^\infty f(n) = \int_0^\infty f(x)dx+\frac{1}{2}f(0)-\frac{1}{12}f'(0)+\frac{1}{720}f^{(3)}(0)+\ldots$$ for ##f(x) =...
  27. PsychonautQQ

    I Euler Characteristic of the Projective plane and sphere?

    The Euler Characterist of the projective plane and sphere is given by V - E + F. V is vertices, E is edges, F is faces. A presentation of the projective plane is {a | aa} and a presentation of the sphere is {b | bb^1} Yet the Euler characteristic is 2 for the sphere and 2-n for the connected...
  28. SeattleDrew

    DE application with possible Euler steps

    Homework Statement A mouse starts at the origin and runs up the y-axis with a speed a. At the same time, a cat running with speed b, starts at the point (c,0) and pursues the mouse. i. What is the path of the cat? ii. Assume a<b and solve for y(x). How far does the mouse run before being...
  29. PrathameshR

    I Derivation of Euler Lagrange's equations from D'alemberts principle

    In the derivation given in Goldstein's book it is given I can't understand from where it comes. It's not at all trivial for me but it's presented as if it's trivial.
  30. C

    Why does setting the condition to x*x < n not work in finding prime factors?

    I've just started with project Euler. The problem I just finished is phrased as follows: "The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ?" The method i used was trial division. Here is my code in C: #include<stdio.h> int main() {...
  31. Alexanddros81

    Determine stopping distance of a train - modified Euler method

    Homework Statement 12.81[/B] A train traveling at 20m/s is brought to an emergency stop. During braking, the acceleration is a=-(7/4)+(t/16) m/s^2, where t is the time in seconds measured from when the brakes were applied. (a) Integrate the acceleration from t=0 to t=16s using Euler's method...
  32. C

    A Derivation of Euler Lagrange, variations

    What is wrong with the simple localised geometric derivation of the Euler Lagrange equation. As opposed to the standard derivation that Lagrange provided. Sorry I haven't mastered writing mathematically using latex. I will have a look at this over the next few days. More clarification. I...
  33. W

    Complex Numbers: Euler's formula problem

    Homework Statement Homework EquationsThe Attempt at a Solution I attempted to use the formula zj = xj + iyj to substitute both z's. Further simplification gave me (x1 + x2)cosθ + (y2 - y1)sinθ or, Re(z2 + z1)cosθ + Im(z2 - z1)sinθ. Is this a valid answer? Or are there any other identities...
  34. binbagsss

    Zeros of Riemann zeta function, functional equation and Euler product

    Homework Statement Question Use the functional equation to show that for : a) ##k \in Z^+ ## that ## \zeta (-2k)=0## b) Use the functional equation and the euler product to show that these are the only zeros of ##\zeta(s) ## for ##Re(s)<0## . And conclude that the other zeros are all located...
  35. B

    A Integral representation of Euler constan

    I am working on the integral representation of the Euler-Mascheroni constant and I can't seem to understand why the first of the two integrals is (1-exp(-u))lnu instead of just exp(-u)lnu. It is integrated over the interval from 1 to 0, as opposed to the second integral exp(-u)lnu which is...
  36. Elroy

    Linear Algebra Problem: Solving for Euler between two ordered bases

    Homework Statement Linear Algebra Problem: Solving for Euler between two ordered bases I've got a problem I need to solve, but I can't find a clean solution. Let me see if I can outline the problem somewhat clearly. Okay, all of this will be in 3D space. In this space, we can define some...
  37. M

    Solving an Euler differential equation

    Homework Statement Solve the differential equation ##(2x+1)^2y'' + (4x+2)y' - 4y = x^2## Can someone verify whether my solution is correct? Homework EquationsThe Attempt at a Solution We perform the substitution ##t = \ln|2x+1|##. Then, ##e^t = |2x+1|## and ##x = \pm(e^t -1)/2## Without...
  38. Mario

    Circles and Euler spiral (repost from general math)

    Hi, i have this problem..., giving two circle (example radius 1 = 500 units, radius 2 = 200 units, distance between centers = 275.73 units) find Euler Spiral (aka Cornu spiral, aka Clothoid https://en.wikipedia.org/wiki/Euler_spiral) tangent giving circle (unknown tangent points). For this...
  39. T

    Rigid body orientation using Euler angles confusion

    Hello, Homework Statement I'm given the following exercise: "A rod with neglected thickness exists. What is the relation between the α,β angles to Euler angles of orientation? α is defined as the angle between the rod and its projection on the XY plane. β is defined as the angle between the...
  40. F

    A About the nodal line in defining Euler angles

    I hope someone can explain this to me. In the definition of Euler angles, the body-fixed-azimuthal angel γ is measured from the nodal line that defines the intersection of the body-fixed-XY plane and the space fixed-xy plane to the body-fixed-Y axis. This is the green line in this image from...
  41. Rectifier

    Finding anitderivative using complex numbers and Euler

    I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...
  42. F

    Euler Lagrange equation issue with answers final form

    Homework Statement For the following integral, find F and its partial derivatives and plug them into the Euler Lagrange equation $$F(y,x,x')=y\sqrt{1+x'^2}\\$$ Homework Equations Euler Lagrange equation : $$\frac{dF}{dx}-\frac{d}{dy}\frac{dF}{dx'}=0$$ The Attempt at a Solution...
  43. Cocoleia

    Second order non homogeneous ODE, IVP

    Homework Statement I need to solve: x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9 Homework EquationsThe Attempt at a Solution I know that the answer is: y=x^2+2x^3+x^3lnx Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...
  44. A

    A Euler beam dynamic equation under point load

    Hi So the problem I have is I want to get the equation of motion of a vibrating beam under a nonlinear "Point" force. The equation would be like this for a distributed load (Which is not the case) But I want the load to be at a point at x=L So I have to options. Add an impulse dirac function...
  45. O

    A How do you KNOW the Euler (Tait) angles cover orienations

    Well, that question just about states my issue. We have a body and we rotate about, say, the 3-axis of its body frame. Then, we must do the next rotation about the 1 or 2 axis. Let me say we choose the 1-axis Then we have a choice: continue on to the 2 axis or repeat the 3 axis. One set is...
  46. MidgetDwarf

    Algebra Best Version of Euler: Elements of Algebra

    So I want to read Euler's : Elements of Algebra. However, I see many editions on Amazon. What is the best version of this book?
  47. Anonymous Vegetable

    B Implications of e^i*pi = -1

    Before I start, there are only really two pieces of information this concerns and that is the idea that 1x = 1 and that ei*π = -1 So it would follow that (ei*π)i = -1i And so that would mean that i2i = e-π which doesn't seem to be right at all. Where is the issue here as there must be one but I...
  48. F

    I How does the angle γ change under inversion in Euler angles?

    The well known Euler angles (αβγ) are defined as in the image It is easy to see that under inversion α → π+α β → π-β but I cannot figure out how γ transforms under inversion. actually I am stuck at the question whether I should measure it from the same intersection line ON (thence γ →π+γ) or...
  49. Arij

    Euler Equation -- diff. equations hw

    Homework Statement [/B] Consider the differential equation (t-2)2y"-2(t-2)y'+2y=0, t>2 Find r+,r-roots of the indicial polynomial of the equation above. Homework Equations [/B] The Attempt at a Solution y'-[2/(t-2)]y'+[2/(t-2)^2]y=0 but then I am confused on how to set p(r), what do I...
  50. mooncrater

    Euler circuit in a directed multigraph

    Homework Statement So the question is: Show that a directed multigraph having no isolated vertices has an Euler circuit if and only if the graph is weakly connected and the in degree and out degree of each vertex are equal.Homework Equations Euler circuit: A circuit that has all edges of the...
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