# Euler Definition and 30 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. ### Euler Method in C++

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...
2. ### 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
3. ### 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...
4. ### 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...
5. ### 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...
6. ### Euler Approximation Failure

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...
7. ### 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)...
8. ### I Core of Euler's equation

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...
9. ### 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?
10. ### 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. ### 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...
12. ### 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...
13. ### Complex Numbers: Euler's formula problem

Homework Statement Homework Equations The 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...
14. 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 Equations The 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...
15. ### 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...
16. ### 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 Equations The 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...
17. ### 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...
18. ### 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...
19. ### 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...
20. ### 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...
21. ### 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...
22. ### Using Euler Approximation

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. ### 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...
24. ### 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...
25. ### 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...
26. ### Bending of a Free Beam

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...
27. ### 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...
28. ### 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...
29. ### Why does (p*q+2)-(p+q) always give a prime number?

Homework Statement Why does (p*q+2)-(p+q) always give a prime number when p and q are prime? Is there a similar formula that would prove this Homework Equations That's what I'm looking for. It might have something to do with Eulers formula The Attempt at a Solution I tried to find online a...
30. ### FORTRN90: Euler Midpoint Method for SHO

Homework Statement Write a program to simulate motion of simple harmonic oscillator. Initial conditions: Let ω = 1, x(t=0) = 1, v(t=0) = 0. Integrate over 30 seconds in intervals of 0.05s. Homework Equations δ2x / δt2 = -ω2x As set of 2 coupled ODE's; x' = v, v' = -w2x The...