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. K

    Question on the 3D Kinematics equation.

    How exactly did the terms in parenthesis of the equation (14) become (15)? I don't even know what the Transport Theorem is about?
  2. M

    Euler- Lagrange equation proof

    For this problem, The solution is, However, I have a question about the solution. Does someone please know why they write out ##\frac{dF}{dx} = \frac{\partial F}{\partial y}y' + \frac{\partial F}{\partial y'}y''## since we already know that ##\frac{dF}{dx} = 0##? Thanks!
  3. L

    Why Does the Multidimensional Chain Rule Lead to Euler's Theorem Proof?

    Hi, I am having problems with the following task: My lecturer gave me the tip that I should apply the multidimensional chain rule to obtain the following transformation ##\sum\limits_{i=1}^{n} \frac{\partial f}{\partial x_i}(x) \cdot x_i= \frac{d f(tx)}{dt} \big|_{t=1}## Unfortunately, I...
  4. L

    Spring pendulum Lagrangian has any ignorable coordinates?

    Are there any ignorable coordinates in this scenario? I don’t think so right, because the lagrangian has explicit dependency on both x and theta. Ignorable coordinates means there is no explicit dependence of that coordinate right? If there are no ignorable coordinates that also means there are...
  5. jonagad

    I Error in Euler angles and quaternions

    Hi, I got a set of Euler angles and a set of quaternions, and I wanted to compare each set against its corresponding set obtained from STK, and I was wondering what would be a good indicator to measure the error between the Euler angles I got and those from stk , and the same for quaternions...
  6. T

    A Euler, Tait-Bryan, Tait, proper, Improper

    Can I try again? I have seen (on the web), all these names, DISTINCTLY: Euler, Tait-Bryan, Tait, proper, Improper I am still trying to make sense of this and am facing some strange naming conventions. I now can see this (the actual math does not concern me--it is only the names that cause me...
  7. Reuben_Leib

    I Help with Euler Lagrange equations: neighboring curves of the extremum

    I tried writing this out but I think there is a bug or something as its not always displaying the latex, so sorry for the image. I have gone through various sources and it seems that the reason for u being small varies. Sometimes it is needed because of the taylor expansion, this time (below) is...
  8. T

    A Euler vs. Tait (steady precession vs... what?)

    Good Morning When one studies body rotations, there are two general approaches one uses: Euler Angles vs. Tait-Bryan Angles. The significant difference is that: Tait–Bryan angles represent rotations about three distinct axes (e.g. x-y-z, or x-y′-z″): Yaw, Pitch, Roll Euler angles use the same...
  9. chwala

    I Understanding Euler Method: Finding Initial Condition of y(0)=1

    The Euler method is straightforward to me; i.e ##y_{n+1}=y_n+ hf(t_0, y_0)## where the smaller the steps i.e ##h## size the better the approximation. My question is 'how does one go about in determining the initial condition ##y(0)=1## in this problem? am assuming that this has to be a point...
  10. D

    I How can Euler angles be visualized using a polar plot?

    Dear Forum, say I am projecting an ellipsoid along the z-axis to the xy-Plane. The resulting ellipsis is rotated around the z-axis by the angle gamma until the principal axes coincide with the x- and y axis. Now before projecting, I rotate the ellipsoid first around the z- and then around the...
  11. ke7ijo

    B Differentiating Euler formula vs. multiplying by i

    I differentiated both sides of Euler's formula with respect to x : e^ix = sin x + i cos x => ie^ix = cos x - i sin x Then for comparison I multiplied both sides of Euler's formula by i: e^ix = sin x + i cos x => ie^ix = i sin x - cos x Each of these two procedures seems to yield the...
  12. T

    Understanding the Differences between Euler Versus Tait Angles

    Good Morning! I understand that the definitions and notations used for Tait–Bryan angles are similar to those described above for proper Euler angles, and I can work problems in either. However, I lack the ability to "rise above both" and categorize them. I do understand that the only...
  13. MidgetDwarf

    Studying Book recommendations to start learning programming for project Euler

    I do not know much about programming. I have used Mathematica, and some Python in the past for very specific problems. Any book recommendations for one wanting to learn programming in order to solve problems from Project Euler. I am familiar with number theory, but not with programming.
  14. D

    Solve Euler Method in C++ for Beginners

    Summary: Problem with Euler Method in C++ Hello, I have a very difficult problem for me (a beginner in programming) how to make the version of the euler method presented in c ++ with the void, float functions, so that the program will calculate from the data that I enter during the program...
  15. M

    In 1752, Goldbach submitted the following conjecture to Euler?

    Proof: Suppose ## 5777=p+2a^2 ##, where ## p ## is either a prime or ## 1 ## and ## a\geq 0 ##. Now we consider two cases. Case #1: Suppose ## p ## is a prime and ## a\geq 0 ##. Let ## p=2 ##. Then ## 5775=2a^2 ##. Thus ## a=\pm \sqrt{2887.5} ##, which contradicts the fact that ## a\geq 0 ##...
  16. T

    A Euler Lagrange and the Calculus of Variations

    Good Morning all Yesterday, as I was trying to formulate my confusion properly, I had a series of posts as I circled around the issue. I can now state it clearly: something is wrong :-) and I am so confused :-( Here is the issue: I formulate the Lagrangian for a simple mechanical system...
  17. Svein

    Insights Investigating Some Euler Sums

    Continue reading...
  18. maistral

    A Backward Euler technique vs. periodic function: Damping out?

    So I've been programming the BDF methods and for some reason I have an issue with the Backward Euler technique. Given the differential equation y" + y = 0 (with y(0) = 2, y'(0) = 0), my backward Euler solution goes like this: Obviously this is not possible as the function should be a...
  19. alpha_michi

    A Could this be a Perfect Euler Brick?

    so here is the Math: for a²+b²: for b²+c²: https://www.physicsforums.com/attachments/284600 for a²+b²+c²:
  20. H

    Question about the argument in a Complex Exponential

    I know that e^-ix = cos(-x)-isin(x), but if we have e^-iwx does that equal cos(-wx) - isin(wx)? Thanks
  21. T

    The order of Euler Angle rotations for a top

    Good Morning All. I have asked this before, but my post was not clear (my fault: I apologize). I hope this is more clear (please be patient as I try to get to the core of my confusion). In the first figure, below, the spinning top precesses as shown (well, it is not a animated jpg, but it...
  22. L

    I'm trying to understand compressive strength and Euler stress for columns

    A cloumn has a compressive strength of 220MPa, but its Euler yeild stress is 350MPa. its compressive strength is less than its euler stress. what does this mean?
  23. A

    The Euler Equation and Incompressible Fluid Theorems

    $$\frac{Du}{Dt} = -\frac{\nabla p}{\rho} - \nabla \chi$$I re-write the Euler equation for incompressible fluid using suffix notation $$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} + \frac{\partial}{\partial x_i} \left(\frac{p}{\rho} + \chi \right) = 0$$what theorems...
  24. scottdave

    Leonhard Euler was a nice guy....

    Here's an interesting article that I came across: Euler: a mathematician without equal and an overall nice guy https://www.irishtimes.com/news/science/euler-a-mathematician-without-equal-and-an-overall-nice-guy-1.4455424 I'm also posting a link to the article on the author's blog -...
  25. greg_rack

    Derivation of the Euler equation for a streamline

    Hi guys, in the derivation of the Euler equation we apply Newton's 2nd law to a gas flowing through a streamline. To do so, we consider a "box" with sides ##dx## ##dy## and ##dz##: as such; Here, with reference to the image, I can't understand where does that '##+dp##' comes from, and hence...
  26. T

    I Euler, Calculus of Variations and Mast on a ship

    From Wikipedia: "In 1727, [Euler] first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship." Does anyone know how he did this? Is there an on-line paper? (But what that is accessible with today's knowledge). And by...
  27. mcastillo356

    Why Euler spoke of them as "complex" numbers?

    Hi PF, this is just for fun...Or not; I don't know. In 1777 Euler set up the notation ##i## to identify any roots of ##x^2-1##, which are indistinguishable, and verified ##i^2=-1##. This way, the set of real numbers grew larger, to a bigger set called complex numbers. This is a translation made...
  28. LucaC

    A Invariance of ##SO(3)## Lie group when expressed via Euler angles

    I am trying to understand the properties of the ##SO(3)## Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of ##SO(3)## dynamics ##\mathbf{\dot{R}} =...
  29. K

    I Help - Derivation of Pulsating Star Euler ODE

    to I am a bit clueless on how to get break the ##r X(r)## from inside the derivative.P.S. I tried to copy from Symbolab instead of pasting the picture, but it didn't let me.
  30. JD_PM

    Euler Lagrange equations in continuum

    OK I've been stuck for a while in how to derive ##(1)##, so I better solve a simplified problem first: We work with Where $$\mathscr{L} = \mathscr{L}(\phi_a (\vec x, t), \partial_{\mu} \phi_a (\vec x, t)) \tag{3}$$ And ##(3)## implies that ##\mathscr{L}(\vec x, t)## We know that...
  31. A

    Angular velocity in terms of Euler angles

    In Chapter 4, derivation 15 of Goldstein reads: "Show that the components of the angular velocity along the space set of axes are given in terms of the Euler angles by $$\omega_x = \dot{\theta} \cos \phi + \dot{\psi} \sin \theta \sin \phi, \omega_y = \dot{\theta} \sin \phi - \dot{\psi} \sin...
  32. O

    I Derive local truncation error for the Improved Euler Method

    I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x). I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3). Any help would be greatly appreciated!
  33. e_mts

    Real and Complex representations of an oscillation equation

    I've been trying to continue my education by self-teaching during quarantine (since I can't really go to college right now) with the MIT Opencourseware courses. I landed on one section that's got me stuck for a while which is the second part of this problem (I managed to finish the first part...
  34. Hamiltonian

    I Lagrangian and the Euler Lagrange equation

    I am new to Lagrangian mechanics and I am unable to comprehend why the Euler Lagrange equation works, and also what really is the significance of the lagrangian.
  35. nomadreid

    I Connection of zeta to primes: less Euler than error or L-functions?

    Often I read that the Riemann Hypothesis (RH) is related to prime numbers because of the equivalence on Re(s)>1 of the zeta function and Eurler's product formula , but is it more accurate that the relevance of the RH to primes (or vice-versa) is either that the RH implies formulas for the...
  36. T

    Euler Lagrange equation and a varying Lagrangian

    Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it Here is the Lagrangian The first variation...
  37. M

    I How could Euler have gone about creating his buckling formula?

    Admittedly Euler was a genius and I am a noob, but I sometimes feel that there must have been a method or process that he followed to go about such problems and came up with these elegant solutions. It couldn't have been just a sudden flash of genius. For example, I wonder how he could have come...
  38. person123

    I Boundary Conditions For Modelling of a Fluid Using Euler's Equations

    Hi! I want to use Euler's equations to model a 2 dimensional, incompressible, non-viscous fluid under steady flow (essentially the simplest case I can think of). I'm trying to use the finite difference method and convert the differential equations into matrices to be solved using MATLAB. I set...
  39. benorin

    I Extend Euler Product Convergence Over Primes: Basic Qs

    I would like to extend the convergence of the Euler product over primes, and I tried to do so in the exact manor it was done for the Dirichlet series, namely, given a completely multiplicative sequence ##a( {kj} ) =a(k) \cdot a(j)\text{ and }a(1)=1##, the Dirichlet series ##\xi (s) :=...
  40. T

    A Algebraic proof that Euler angles define a proper rotation matrix

    I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...
  41. T

    A Are Tait and Euler angles a complete parametrization of 3D space?

    Hi With displacements, I KNOW that three orthogonal axes cover all of 3D Space. What about rotations? How do I KNOW that the Tait or Euler angles cover all orientations? For Tait, I would almost "expect" it. The object rotates about the local body axes in order of: one axis, then a second...
  42. A

    Introductory application of the Newton Euler equations to a composite body

    α is the second derivative of angle and w is the first derivative In the free body diagrams the only force on A is the normal force since it is only constrained not to move vertically. Have I drawn the free body diagram and kinetic diagram correctly? By relating the accelerations of the...
  43. C

    I Does the Euler method give a better insight into adiabatic processess?

    I verified with others the equation below is an Euler method as well with ##a## can be any value such that it give the same ##\frac{dE}{dv}=-1.4\frac{p}{v}## but with ##a## other than one, it have no meaning in physics. For anyone that already understand Euler method can omit the part i have...
  44. J

    Why does the Euler approximation fail for the Airy or Stokes equation?

    I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not...
  45. Avatrin

    Difficulty of Project Euler problems 1-100; now versus then

    Hi I recently noticed that the first Project Euler problem was published in 2001! The first 100 problems were all published before 2006. That made me curious; What I wonder is how different the difficulty is now versus then... I mean, I can think of two factors that make things easier: 1. The...
  46. M

    How did Euler come up with the value of e?

    Homework Statement: This is not a homework problem. I am trying to imagine how Euler would have gone about getting the value of e, while he was trying to figure out the case of continuous compounding. Homework Equations: I know how he reached up to the equation given below. I am not sure how...
  47. Z

    Difference between gyroscope angular displacement and Euler angles

    Hi guys, I'm trying to understand between gyroscope angular displacement and euler angles? for example { Δx = Δx + h * Rx * SCx);} this is gyroscope output about anguler displacement.This value can be used to determine angle that device created.Why we should euler angles to fly.(I know...
  48. P

    How can I rotate Euler angles through a specific angle?

    Summary: I found an old topic on this forum that describes a similar problem to what I am facing currently. I want to rotate an object with euler angle values, but the rotation has to be translated based around a specific angle. However, I do not fully understand the solution provided in that...
  49. Arman777

    Sum of All Positive Integers n where d+n/d is Prime to 100M

    [Project Euler Problem 357] Consider the divisors of 30: 1,2,3,5,6,10,15,30. It can be seen that for every divisor d of 30, $$d+30/d$$ is prime. Find the sum of all positive integers n not exceeding 100 000 000 such that for every divisor d of n, $$d+n/d$$ is prime. import math import time...
  50. Avatrin

    A Book to understand Euler angles (and Tait–Bryan )

    Hi I am trying to understand Euler angles and why they work. What are some great books to use as a resource to build a deeper understanding of them? I want to know why 3-2-1 Euler angles are so commonly used, and proofs regarding the mathematical properties of this method..
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