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. 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...
2. 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...
3. 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...
4. 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...
5. 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...
6. 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...
7. 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...
8. 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...
9. 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.
10. 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...
11. 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 ##...
12. 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...

14. 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...
15. 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:
16. 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
17. 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...
18. 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?
19. 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...
20. 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 -...
21. 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...
22. 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...
23. 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...
24. 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}} =...
25. 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.
26. 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...