Euler Definition and 414 Threads

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. Anonymous Vegetable

    B What is the Error in Calculating (e^(iπ))^i?

    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...
  2. 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...
  3. 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...
  4. 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...
  5. L

    I Graph Theory: (proof)conditions for Euler Circuit in Digraph

    I have read in many places that one necessary condition for the existence of a Euler circuit in a directed graph is as follows. Theorem: "A directed graph has an eulerian circuit if and only if it is connected and each vertex has the same in-degree as out-degree." However, I don't understand...
  6. R

    I Body rates from Euler angles....

    Referring to slides 3-4 (page 2) of this link: https://www.princeton.edu/~stengel/MAE331Lecture9.pdf The author states the relationship between body rates [p q r] and Euler angle rates [φ_dot θ_dot ψ_dot]. I want to verify this but have been failing miserably... My reasoning: 1) p, q, and r...
  7. NickTesla

    Capacitor Discharge Time Formula Euler

    I would like to understand how the Euler constant elevated to fractional exponent gave this result? 958mV someone please!?
  8. H

    I Explorer-1 tipped over, but Euler equation doesn't agree

    \dot{\boldsymbol{H}} = \dot{\omega} + \boldsymbol{\omega}\times\boldsymbol H Suppose body's got an impulse and 's started to rotate about its principle axis, say z, No more external moment from this time. So, Euler equations become 0 = I_{xx}\dot{\omega}_x − (I_{yy} − I_{zz}) \omega_y \omega_z 0...
  9. Mr. Rho

    Mathematica Rotation of 3D Plot using Euler angles

    So, I'm trying to plot a 3D "dipole" (an arrow with a small torus around it basically) in mathematica, and want to rotate it according to Euler angles... I use this code for the rotation matrix: rot[a, b, g] := RotationMatrix[g, {1, 0, 0}].RotationMatrix[b, {0, 1, 0}].RotationMatrix[a, {0, 0...
  10. P

    Euler Formula: Understanding (4.25) to (4.26)

    can someone explain how you go from (4.25) to (4.26) using Euler's formulas in the attachment?
  11. bananabandana

    Euler Lagrange Derivation (Taylor Series)

    Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way: $$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...
  12. M

    Euler angles in latitude longitude space

    In most physics introductions Euler angles(pitch, roll, yaw) are defined with respect to Cartesian coordinate system. If I chose not to use a Cartersian coordinate system but instead use a latitude, longitude and a proprietary vertical coordinate(and no back transformations to Cartersian...
  13. Hijaz Aslam

    Euler Representation of complex numbers

    I am bit confused with the Eueler representation of Complex Numbers. For instance, we say that e^{i\pi}=cos(\pi)+isin(\pi)=-1+i0=-1. The derivation of e^{i\theta}=cos(\theta)+isin(\theta) is carried out using the Taylor series. I quite understand how ##e^{i\pi}## turns out to be ##-1## using...
  14. V

    What is the "Book proof" of Euler's formula?

    The eccentric mathematician Paul Erdos believed in a deity known as the SF (supreme fascist). He believed the SF teased him by hiding his glasses, hiding his Hungarian passport and keeping mathematical truths from him. He also believed that the SF has a book that consists of all the most...
  15. F

    Proving Euler Lagrange equations

    Hi everybody; I am looking for the deduction of the euler lagrange equations (d/dt)(∂L/∂v)-(∂L/∂x) from the invariance of the action δ∫Ldt=0. Can someone please tell me where can I find It? Thanks for reading.
  16. Geofleur

    Relativistic Euler Equation in Spherical Coordinates

    I just wanted to check that I am thinking about the coordinate transition correctly. The relativistic generalization of Euler's equation is (from Landau & Lifshitz vol. 6) ## hu^\nu \frac{\partial u_\mu}{\partial x^\nu} - \frac{\partial P}{\partial x^\mu} + u_\mu u^\nu \frac{\partial...
  17. S

    Query on the Euler Theorem for Rigid Body Rotation

    Hi, I am having some problems conceptualizing the Euler's Theorem. Any help will be greatly appreciated. In Goldstein's book the Euler's theorem is stated as 'Any displacement of a rigid body, whose one point remains fixed throughout, is a rotation about some axis', then he has proven that the...
  18. Y

    Solving an RC circuit using explicit euler

    Homework Statement Hi there. I have a simple RC circuit with a battery voltage of 10 V, R = 1 Ω, C = 1 F and a switch. I want to use the Explicit Euler (forward divided difference) to solve the equation and check for stability, rather than using a ODE. I am finding the equation for when the...
  19. 9

    Coordinate transformation - NED and ECEF frames

    Hi, I have a reference device that outputs euler angles, which are angles that relate the sensor body frame to the north east down frame. These angles are called pitch roll and yaw. The sensor is an accelerometer. I know how to get the rotation matrix that will put accelerations from the...
  20. Vinay080

    What is the Euler's stand on infinitesimals?

    Euler was the master in analysisng anything. This can be seen in his words in the preface of his book "Mmathematica" (translated by Ian Bruce), where he speaks on the text of Hermann "Phoronomiam": Euler has given many insightful words on analysisng things in his preface of many other books...
  21. B

    Euler, Tait, Gyroscope: Rotations that cover it all

    I understand that a gyroscope undergoes precession, nutation, spin. And that the order of the rotations are such that the precession and spin share a common "local axis." I also understand there are, for totally different purposes, Euler angles to model rotations. In this case, the order of the...
  22. B

    Optimizing Euler Method for Differential Equations with Large Coefficients

    Homework Statement Hello, I have a question about using Eulers Method to approximate a solution to a differential equation. The problem lists forces that would be applied on an object and influences its velocity and therefore its position. I believe I am doing the Euler method correct to...
  23. R

    Simple Euler (Tait-Bryan) rotation question

    Suppose an aircraft attitude is given by Heading=040 degrees Pitch=15 degrees Up Bank=20 degrees left bank And then the aircraft “yaws”, or rotates 30 degrees left about its yaw axis. What are the new Heading, Pitch and Bank angles? What do the rotation matrices look like that describe...
  24. M

    How to convert Euler Equations to Lagrangian Form?

    I am not entirely sure how to convert the conservation of mass and momentum equations into the Lagrangian form using the mass coordinate h. The one dimensional Euler equations given by, \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0...
  25. M

    Exploring the Euler Formula: Is e(-iπ) + 1 = 0?

    For Euler formula : e(iπ) + 1 =0 is it the same if we say e(-iπ)+1 = 0 or not ? (minus sign is included in the exponent)
  26. M

    Calculate the angular frequency w using the Euler equations

    Homework Statement Consider the Earth as a rigid body with moment of inertia I1, I2 and I3. The Earth is symmetric around the z-axis (I1 = I2). Calculate the angular frequency w using the euler equations Homework EquationsThe Attempt at a Solution
  27. SSGD

    What is this differential equation? I'm going crazy

    I have been working on a math problem and I keep getting the some type of PDEs. x*dU/dx+y*dU/dy = 0 x*dU/dx+y*dU/dy+z*dU/dz = 0 ... x1*dU/dx1+x2*dU/dx2+x3*dU/dx3 + ... + xn*dU/dxn= 0 dU/dxi is the partial derivative with respect to the ith variable. Does anyone know about this type of PDE...
  28. RJLiberator

    Show that Odd Euler Numbers are 0

    Homework Statement Complex Analysis Problem: The euler Numbers E_n, n=0, 1, 2,..., are defined by 1/cosh(z) = the sum from n=0 to n=infinity of E_n/n! z^n (|z|<pi/2). show that E_n=0 for n odd. Calculus E_0, E_2, E_4, E_6 Homework Equations Not entirely sure what to put here for this one...
  29. N

    Python Project euler 1 understanding the python code

    If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000. This is the code for python I found (didn't create) which I believe is correct: max = 1000 result = 0 for...
  30. hideelo

    Deriving Commutation of Variation & Derivative Operators in EL Equation

    I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality δ(dq/dt) = d(δq)/dt Where q is some coordinate, and δf is the first variation in...
  31. G

    What is the result of using Euler's equation for Fourier transform integrals?

    when I am using Euler equation for Fourier transform integrals of type \int_{-\infty}^{\infty} dx f(x) exp[ikx] I am getting following integrals: \int_{-\infty}^{\infty} dx f(x) cos(kx) (for the real part) and i* \int_{-\infty}^{\infty} dx f(x) sin(kx) (for its imaginary part) I am...
  32. Saitama

    MHB How to Efficiently Compute Euler's Totient Function for Large Ranges?

    Following is the problem I am trying to solve: SPOJ.com - Problem ETFS Here is my code: #include <iostream> #include <cstdio> #include <vector> using namespace std; typedef long long ll; ll solve(ll n) { ll i, ans=n; for(i=2; i*i<=n; ++i) { if(n%i==0) ans-=ans/i; while(n%i==0)...
  33. almarpa

    Euler equations in rigid body: Taylor VS Kleppner - Kolenkow

    Hello all. After reading both chapters on rigid body motion both in Kleppner - Kolenkow and Taylor books, I still do not undertand the physical meaning of Euler equations. Let me explain: In Kleppner - Kolenkow, they claim (page 321 - 322) that in Euler equations, Γ1, Γ2 and Γ3 are the...
  34. C

    3D Coordinate transformation and Euler Angles

    Hello, I'm running a galaxy formation simulation. The output specifies the coordinates in (x, y, z) of all the particles in a galaxy, which usually fall in a disk. The orientation of the disk depends on the initial conditions, but it is generally not aligned with any of the coordinate axes...
  35. evinda

    MHB Bound of Euler method- nonuniform partition

    Hello! (Wave) Consider a nonuniform partition $a=t_0< t_1< \dots < t_{\nu}=b$ and assume that if $h_n=t^{n+1}-t^n, 0 \leq n \leq N-1 $ is the changeable step, then $\min_{n} h_n > \lambda \max_{n} h_n, \lambda>0$ independent of $n$. Show a bound of the error of Euler method analogous to...
  36. Q

    Nonlinear Mass Spring Damper with Euler Bernoulli Beam

    I'm trying to find a solution to a system in which a clamped free Euler-Bernoulli Beam system rests on top of a mass-spring-damper system. The MSD system has nonlinearities in both the spring and the damper and is of the form: I have extended the nonlinear restoring force to its 3rd term and...
  37. T

    Exploring the Reliability of Euler Method for Orbital Gravity Equations

    Hello everyone, I am curious as to if it is possible to use the Euler Method to solve the gravity differential equations? Would the approximations quickly diverge to inaccurate solutions, or would it stay relatively reliable? Thanks
  38. BiGyElLoWhAt

    Excellent video series raises good question:

    www.youtube.com/watch?v=oW4jM0smS_E That's the video I'm referencing in particular, but 1 and 3 are necessary prereqs if you're new to the matter (as I am). He goes through and derives the product rule and power rule for polynomials using algebra. My question is this: why don't we teach...
  39. T

    Direction of Forces on an Euler Spiral Path | Intuitive Explanation

    Trying to figure out direction of forces of an object traveling on an Euler spiral path. As an example if you had an astronaut with a jetpack and he wanted to change his direction 90 degrees he could aim his thrusters outward from the center of a circle and he would turn at a constant rate with...
  40. B

    Euler Lagrange equation of motion

    Homework Statement Find the equations of motion for both r and \theta of Homework Equations My problem is taking the derivative wrt time of and \dfrac{\partial\mathcal{L}}{\partial\dot{r}}=m \dot{r} \left( 1 + \left( \dfrac{\partial H}{\partial r}\right)^2 \right) The Attempt at a...
  41. AdityaDev

    Prove that f is a constant function

    Homework Statement Suppose that f:R->R satisfies the inequality ##|\sum\limits_{k=1}^n3^k[f(x+ky)-f(x-ky)]|<=1## for every positive integer k, for all real x, y. Prove that f is a constant function. Homework Equations None The Attempt at a Solution I tried taking f(x)=sinx and then using...
  42. evinda

    MHB Proof of Euler Tour in Graphs: In-degree($v$)=Out-Degree($v$)

    Hello! (Wave) I am looking at the proof that $G$ has an Euler tour iff in-degree($v$)=out-degree($v$), that I found at this site: https://www.cs.duke.edu/courses/fall09/cps230/hws/hw3/headsol.pdf (Problem 2)A simple cycle is a path in a graph that starts and ends at the same vertex without...
  43. spovolny

    Free Beam Bending: Find Complete Answer & More

    Consider a beam with an upwards concentrated force applied to its center. This is equilibrated by a distributed downwards force. There are no displacement boundary conditions. I've tried approaching this with simple beam theory, but I can't get a complete answer (shear, moment, slope...
  44. T

    Understanding the Euler Lagrange Equation and Its Boundary Condition

    I am trying to derive it but I am stuck at the boundary condition. What is this boundary comdition thing such that the value must be zero?
  45. G

    Euler and the youth of mathematicians

    Euler went blind in the year 1771 but his productivity increased after he went blind according to this lecture: Does he defy the common wisdom that math is a young man's game?
  46. B

    Euler Angle from Body Frame to Inertial Frame

    Hi, This is not really a homework problem, but a project I'm working on. So, I am trying to build a Simulink model for my quadcopter. I derived the equations of motion using the Newtown-Euler method in the body frame to get transnational and angular acceleration. For the transnational part, I...
  47. evinda

    MHB Can the Euler method accurately approximate solutions for stiff systems?

    Hello! (Wave) We consider the initial value problem $$\left\{\begin{matrix} y'=\lambda y, & t \in [0,\infty), \lambda \in \mathbb{C}, Re(\lambda)<0 \\ y(0)=1 & \end{matrix}\right.$$ Since $y^n=(1+h \lambda)^n, n \in \mathbb{N}_0$ is the sequence of approximations that the Euler method...
  48. Q

    Direction cosine matrix of rolling disk on circular ring

    Hey all, I'm stuck on this problem and not sure how to proceed/if I'm in the right direction. Problem: One reference frame N sits at the origin (inertial frame) while another frame, B, describes a disk rolling on a circular ring about the other frame. Picture below (A) find the direction...
  49. J

    Bucklig: deflection at Euler load

    Hello I was trying to calculate the horizontal deflection of the free end of vertical clamped beam. The beam would be loaded at the free end with a horizontal force H and a vertical force P. My idea was to calculate an initial deflection due to the force H. Then calculate the additional...
  50. J

    Can Euler-Lagrange Equations Explain Mirages?

    Homework Statement On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...
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