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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. P

    MHB Can the Euler Totient Function Ever Divide n/5 for Any Positive Integer n?

    prove that there is no positve integer n such that g(n) dividies n/5, where g is the euler totient function.
  6. M

    Analytical solution of the Euler equations in 1D?

    Hi. Since these equations are approaching three hundred years old I'm pretty sure someone must have solved them somewhere before. However I have not been able to find any text-books or papers that actually show me how to solve these equations. So I'm wondering if anyone here know where I can...
  7. H

    Euler Lagrange Equation Question

    Homework Statement Consider the function f(y,y',x) = 2yy' + 3x2y where y(x) = 3x4 - 2x +1. Compute ∂f/∂x and df/dx. Write both solutions of the variable x only. Homework Equations Euler Equation: ∂f/∂y - d/dx * ∂f/∂y' = 0 The Attempt at a Solution Would I first just find...
  8. O

    Backward euler method for heat equation with neumann b.c.

    I am trying to solve the following pde numerically using backward f.d. for time and central di fference approximation for x, in MATLAB but i can't get correct results. \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}},\qquad u(x,0)=f(x),\qquad u_{x}(0,t)=0,\qquad...
  9. G

    Euler sum of positive integers = -1/12

    My question arises in the context of bosonic string theory … calculating the number of dimensions, consistent with Lorentz invariance, one finds a factor that is an infinite sum of mode numbers, i.e. positive integers … but it really goes back to Euler, and his argument that the sum of all...
  10. J

    How can the Improved Euler Method be used in programming assignments?

    The "improved" Euler method Homework Statement Using it on a programming assignment. The description in our course notes is a little confusing, so I need to know whether I'm doing it correctly. Homework Equations Go to p. 22 of this, if you're so inclined...
  11. G

    How Do You Linearize the Euler Equations for Fluid Motion?

    1. Homework Statement Linearize the euler equations of fluid motion, write as a single partial differential equation for example the pressure pertubation Homework Equations The euler equations of fluid dynamics The Attempt at a Solution Not sure how I would be able to do this.
  12. C

    Comp Sci FORTRAN: second-order ODE with Euler Method

    Homework Statement Dear all, please help. I have tried this question and came up with strange numbers, my fortran is definitely not correct. Please help! When the effect of the air resistance is taken into account, the equation of motion for a particle of mass m falling vertically in a...
  13. B

    Timoshenko - Euler Bernoulli In Plane Curved Beams

    Folks, Searches of Timoshenko and Euler Bernoulli Beam Theory show differential equations for straight beams. Is there any material out there illustrating differential equations for "curved in plane beams"..? Thanks
  14. H

    Non-homogenous secx ODE's and Euler eq's

    Suppose we have Asec(x) on the right hand side in a non-homogenous ODE and in a Euler equation. How do we solve it? ( I know how to solve for cos and sin on the right hand side but not for any other trig function).
  15. H

    Solving ODE's or Euler second order diff. eq's containing Asecx?

    I know how we solve ODE's and euler equations in which we have cos and/or sin terms on the right. We take the particular solution to be Acos(x) + Bsin(x). But what if we have secant or cosecant terms on the right or tan and/or cotangent terms? Qno. 1 Are these 4 terms possible i.e. can they...
  16. alane1994

    MHB Solving an IVP: Finding Euler Approximations & Errors

    Here is my problem, I have been trying this for a couple of hours. I have sought help with a professor, and yet we still couldn't get it. Here is the question in full. Consider the initial value problem below to answer the following. a)Find the approximations to y(0.2) and y(0.4) using Euler's...
  17. M

    Is a Euler or an Euler correct?

    Is "a Euler" or "an Euler" correct? Given the pronunciation sounds like "oiler", which article do we use? Couldn't find the grammar forum!:redface:
  18. B

    Kinematics of Euler Bernoulli and Timoshenko Beam Elements

    Folks, Trying to get some appreciation for what is going on in the attached schematic of 1)Euler bernoulli and 2) Timoshenko beam elements. For the first one, ie the top picture, how was ##u- z \frac{dw}{dx}## arrived at? thanks
  19. B

    Euler Bernoulli Beam 4th order ODE -Balance of Units

    Folks, I am trying to understand the balance of units for this eqn ## \displaystyle \frac{d^2}{dx^2}(E(x)I(x) \frac{d^2 w(x)}{dx^2})+c_f(x)w(x)=q(x)## where ##E## is the modulus of Elasticity, ##I## is the second moment of area, ##c_f## is the elastic foundation modulus, ##w## is...
  20. M

    Obtaining the Green Function for Euler Beam with Specific Boundary Conditions

    Hello there, I would like to obtain the Green function for the operator F, F [u(x)] = u '''', and the boundary conditions u(0) = u'(0) = u (1) = u' (1) = 0. I am looking for a function G ( x, s ) such that G'''' (x,s) = delta (x-s) (the apecis referring to differentiation w.r.t. x, and...
  21. A

    Angle projections to Euler angles

    Consider a vector in 3D. Its projections on two planes, say YX and YZ planes, makes some angle with the vertical axis ( the y-axis in this case). I know these two angles (I call them projected angles). This is the only information I have about the vector. I need Euler angles which when...
  22. P

    Application of the euler equations using fermats principle

    Homework Statement The speed of light in a medium with index of refraction n is c/n, where c is the speed of light in vacuum. Notice that n ≥1: Suppose a light ray travels in the xy-plane between (x1; y1) and (x2; y2) in a non-uniform material so that n(x) is the refractive index of the...
  23. B

    Euler Equation to Compute Extreme?

    A problem on my homework: We learn early on that "the shortest distance between two points is a straight line." Let's prove it...Using the Euler equation, compute the extrema of ∫sqrt(1 + (dy/dx)2)dx from x1 to x2 ...show that this corresponds to lines "y = mx +b". Euler had a lot of...
  24. P

    Back Euler method for 2nd order d.e

    Hi, How can one use back Euler method for 2nd order d.e? Is it possible this method to be expanded for a system of 4 odes? Thanks
  25. M

    Deriving equation from 3D Euler Equations.

    Homework Statement I've got the 3D Euler equations \frac{\delta u}{\delta t} + (u\cdot \nabla)u = -\nabla p \nabla \cdot u = 0 I've been given that the impulse is \gamma = u + \nabla\phi Homework Equations And I need to derive \frac{D\gamma}{Dt} = -(\nabla u)^T \gamma + \nabla...
  26. T

    Verifying an Integral Representation of the Euler Constant

    Homework Statement I need to verify an integral representation of the Euler constant: \int^{1}_{0}\frac{1-e^{-t}}{t}dt-\int^{\infty}_{1}[\frac{e^{-t}}{t}dt=\gammaHomework Equations The Attempt at a Solution OK, I'm supposed to use this fact (which I have already proved)...
  27. S

    Euler equation in Polar coordinates

    Hello. I have 2D Euler equation for fluids. I can't derive it in polar coordinates. I defined functions u(x,y,t) = u'(r, theta, t) and v(x,y,t) = v'(r, theta, t). I started by computing derivatives \frac{\partial u'}{\partial r}=\cos\theta\frac{\partial u}{\partial...
  28. S

    Euler angles equivalence with a single rotation

    imagine we rotate a vector centered at the origin with euler angles alpha,beta,gamma. now the question is, can we do this rotation by the means of defining a vector N(which its length is 1).and rotating the vector zeta radians counter clockwise around N? I think it must be possible and I want...
  29. E

    How to determine particular solutions for cauchy euler

    If given a cauchy euler equation (non-homogeneous) equation, does the approach in looking for a particular solution (in order to solve the non-homogeneous part), differ from normal? I am also in general confused about how to assign a particular solution form, in many cases. I have yet to find...
  30. E

    Cauchy Euler, non-homogeneous, weird condition

    xy''+y'=-x y(1)=0, y(0) bounded (so the natural log, 1/x etc. terms drop out) homogeneous, cauchy euler: y=a+bx variation of parameters, and using the conditions gives y=1-x, I think (i tried this previously and I think this is what I got, I didnt write it down). Very different from what I...
  31. S

    Understanding the Limitations of the Euler Method in Computational Physics

    This is an extract from my third year notes on 'Computational Physics': The Euler method is inaccurate because it uses the gradient evaluated at the initial point to calculate the next point. This only gives a good estimate if the function is linear since the truncation error is quadratic in...
  32. O

    Disproving Euler's Identity: Check My Math

    I was messing around with Euler’s Identity and I think I accidently disproved it. I would like someone to check my math to make sure I didn’t make any rookie mistakes. \begin{array}{l} e^{\pi i} + 1 = 0 \\ e^{\pi i} = - 1 \\ \left( {e^{\pi i} } \right)^2 = \left( { - 1}...
  33. W

    Help with Derivation of Euler Lagrange Equation

    Hello all, I am having some frustration understanding one derivation of the Euler Lagrange Equation. I think it most efficient if I provide a link to the derivation I am following (in wikipedia) and then highlight the portion that is giving me trouble. The link is here If you scroll...
  34. H

    How Does the Euler Totient Function Apply to Multiplicative Proofs?

    Hello, I am looking at the proof (Theorem 2.5 (b) Apostol) of $$ \phi (mn) = \phi(m) \phi(n) \frac{d}{\phi(d)} $$ where $$ d = (m, n) $$. Can someone explain how they go from $$ \prod_{p|mn} \left( 1 - \frac{1}{p} \right) = \frac{\prod_{p|m} \left( 1 - \frac{1}{p} \right) \prod_{p|n}...
  35. W

    Euler transform matrix multiplication help

    !Euler transform matrix multiplication help! Homework Statement This may be rather simple but i am really struggling to complete a 3 3x3 matrix multiplication. I NEED STEP BY STEP WORKING!. This would really help me I understand the theory. Basically I have three matrices T1= cosψ sin ψ...
  36. P

    Euler Angles - Why Post Multiplication

    In robotics, you have a co-ordinate frame which is at the base & another which is at the next joint. You want to describe the position and orientation of the 2nd frame with respect to the first in terms of position and orientation. For the position, you use a translation matrix. The...
  37. P

    Euler Angles - Why Post Multiplication

    Normally with column vectors, we premultiply rotation matrices if the angles are with respect to fixed axis. Why then do we post multiply if the angles are Euler Angles, angles with respect to the mobile axis?
  38. P

    Topological sigma model, Euler Lagrange equations

    Homework Statement My question refers to the paper "Topological Sigma Models" by Edward Witten, which is available on the web after a quick google search. I am not allowed to include links in my posts, yet. I want to know how to get from equation (2.14) to (2.15). We consider a theory of maps...
  39. R

    Implicit Methods for Drag-Dependent Acceleration in Euler Integration

    I'm trying to write a code to implement he backwards Euler method to integrate the equation of motion. The sticking point seems to be that the acceleration is due to drag, and thus is dependent on the new position and velocity. I understand the method to be: v_{i+1}=v_{i}+a_{i+1}δ...
  40. J

    Can Euler's Differential Equation Be Solved Using Initial Values?

    Problem: Solve the initial Value: when x=1, y=0 dy/dx = 1 2x^2(d^2y/dx^2) + 3x (dy/dx) - 15y = 0 My attempt: x = e^t dx/dt = e ^t dy/dt = dy/dx * dx/dt dy/dt = x*dy/dx d^2y/dt^2 = d/dt(dy/dt) = d/dt(x*dy/dx) =d/dx(x*dy/dx)*(dx/dt) since dx/dt = x =(x^2*d^2y/dx^2) + (x*dy/dx)...
  41. J

    Euler characteristic of complex projective plane

    How to compute \chi(\mathbb{C}\mathrm{P}^2)? This problem is from a class on differential topology, so we have defined the Euler characteristic as the sum of the indices of isolated zeros on a non-vanishing vector field. Off the top of my head, I cannot think of any theorems which really help...
  42. N

    Comp Sci Project Euler Problem 002: Fibonacci sequence (in C++)

    Homework Statement Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... By considering the terms in the Fibonacci sequence whose values do not exceed four million...
  43. R

    Solving Laguerre DEby translating it into an Euler equation

    Homework Statement Find the indicial equation and all power series solutions around 0 of the form xr Ʃan xn for: x y'' -(4+x)y'+2y=0 - apparently one of these solutions is a laguerre pilynomial Homework Equations the indicial equation is the roots of r(r-1) +p0r+q0 where p0=lim(x->0)(...
  44. L

    Fluid mechanics Lagrange & Euler formalism

    Lagrange & Euler formalism How we get relation (\frac{\partial T^{(L)}}{\partial t})_{r_0}=(\frac{\partial T^{(E)}}{\partial t})_{r}+\frac{\partial T^{(E)}}{\partial x}(\frac{\partial x}{\partial t})_{r_0}+\frac{\partial T^{(E)}}{\partial y}(\frac{\partial y}{\partial...
  45. A

    Taylor and Euler Matlab Comparison for Numerical Analysis.

    1. Solve y'=3t^2y^2 on [0, 3] , y0 = −1, using Euler method and Taylor method of order 3. Compare your solutions to the exact solution. y(t)=(-1/((t^3)+1)) I DONT KNOW WHAT IS WRONG WITH MY PROGRAM! PLEASE HELP =D Homework Equations http://en.wikipedia.org/wiki/Euler_method...
  46. M

    How to Compute Euler Angles for Rotating Reference Frames

    I know this is rather trivial question but it's not homework! I need this as part of a bigger project. What I have to do is rotate my reference frame to another one. I want my new z-axis to be a vector Z=(Z1,Z2,Z3) Following the notation of this wikipage...
  47. V

    Attitude quaternion derivatives from Euler angular velocities

    I'm struggling to understand what the derivative of an attitude quaternion really is and how to use it. I need it to solve a problem relating to a rotating frame of reference relative an inertial frame. The information I have is a vector of Euler angular velocities (i.e for roll \phi, pitch...
  48. B

    Euler Lagrange equation as Einstein Field Equation

    I want to prove that Euler Lagrange equation and Einstein Field equation (and Geodesic equation) are the same thing so I made this calculation. First, I modified Energy-momentum Tensor (talking about 2 dimension; space+time) : T_{\mu\nu}=\begin{pmatrix} \nabla E& \dot{E}\\ \nabla p &...
  49. H

    Calc Euler Rotation Vector: Eurasia-North America | Lat/Long Pole

    Calculate the Euler rotation vector for Eurasia relative to North America. The rotation vectors for Eurasia and North America relative to the Pacific are pacωeur = [0.000529,-0.007235, 0.013123] and pacωNA = [0.001768 -0.008439 0.009817]. Also, give the latitude and longitude of the pole of...
  50. C

    MHB Pharaoh's modified Euler method question from Yahoo Answers

    Part 2 of Pharaoh's Taylor series and modified Euler question from Yahoo Answers There is a standard method of converting higher order ODEs into first order systems, in this case it is to introduce the state vector:\(Y(t)=\left[ \begin{array}{c} y(t) \\ y'(t) \end{array} \right] \)Then the ODE...
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