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. A

    Where Is Earth's Nutation and Does the 23.5° Axial Tilt Ever Change?

    Hi, I would like to see the resemblance between planet Earth and a spherical top. I draw here the x-convention Euler angles as I know them: 24 hours to complete one complete one revolution around it's own z' axes. And the 23.5° Earth axes revolves around the "sun's z axes" - the line...
  2. V

    Proving Euler Line Parallel to Side BC in Triangle ABC

    Can anyone help me solve the following question. ABC is any triangle. XAB, YBC and ZAC are equilateral triangles formed on this triangle. Prove that AY, BZ and CX are concurrent. In a triangle ABC, the Euler line is parallel to side BC. Prove that tanB*tanC = 3. I just need a hint. I...
  3. M

    Solving Classic Euler Question on Buying Rice and Corn with 1770 Dollars

    Hi I'm told that the the following can be presented as a classic euler question: A farmer goes to marked with 1770 dollars to buy rice and corn. One bag of corn costs 10 dollars and one of rice cost 20 dollar. How much corn and rice can the farmer buy? This can be presented in the following...
  4. K

    Generating function for terms of Euler triangle?

    I'm sure this is relatively easy, but after an hour or so googling, I can't seem to find the formula for generating terms of the http://steiner.math.nthu.edu.tw/chuan/123/test/euler.htm Is this known by some other name? Maybe that's why I can't find it? Thanks
  5. D

    Please help me on calculating orbifold euler characteristics

    how could v 'calculate' the orbifold euler char. using String theorists' formula for the same ? For example, i know the Euler char. of S^1/Z_2 is 1, when we identify the x to -x points of S^1(not the antipodal points i.e. dimetrically opposite points, but putting a mirror along the horizontal...
  6. B

    Solving Euler Equation for X^m (m^2 + 1)^2 = 0

    I did solve this differential equation (x^4)y'''' + 6x^3y''' + 9x^2 + 3xy' + y = 0 using Cauchy Euler Equation. I got X^m (m^2 + 1)^2 = 0 I'm not sure how to get the roots of (m^2 + 1)^2. In my calculation I got m = -i, +i, -i, +i when I put m^2 = -1. In the book they have m = (+-)...
  7. M

    Euler Circuits and Vertices of Even Degree: Proof and Counter-Proof

    If a connected graph has a Euler circuit then this implies that all the vertices of the graph have even degree. Is the converse of this argument true? i.e. If a connected graph only contains vertices of even degree does this imply it contains an Euler Circuit? Could somebody please show me a...
  8. P

    Improving Euler's Approximation for Differential Equations: Tips and Resources

    Would you please tell me how to improve Euler's approximation to be better in solving differerential equations ? Can you give me some links to this? Thank you,
  9. honestrosewater

    Creating a Sequence Without Gauss: The Euler Challenge

    I think this will be my new mantra ;) but it is actually related to the question. I want to define the sequence (s_n) where n is natural and whose nth term, t_n, is the sum of the digits of n. I want to do this without using Gauss's modular arithmetic. This may be one of those 'easier said...
  10. S

    Euler' s equations and tau = zero?

    A uniform circular disk of mass m and radius a is constrained to rotate with constant angular speed omega abotu an axis making an angle theta with the disk' s axis of symmetry. Find the magnitude and direction of the angular momentum L and the torque tau exerted on the disk by its supporting...
  11. E

    Is There an Identity for the Infinite Product 1/1-exp(-sp) Over All Primes?

    Euler proved that the infinite product 1/1-p**(-s) with p running over all primes was equal to R(s) where R(s) is teh Riemman z function my question is if the product: 1/1-exp(-sp) over all primes would be hte same as summing the series 1+exp(-s)+exp(-2s)+..=1/1+exp(-s) or what would...
  12. MathematicalPhysicist

    Taking the logarithm of the euler product

    can some one explain to me how is taking the logarithm of euler product gives you -sum(p)[log(1-p^s)]+log(s-1)=log[(s-1)z(s)]? my question is coming after encoutering this equation in this text in page number 2...
  13. G

    The Euler 9 Point Circle: Solving the Mystery of its Triangle Compatibility

    [SOLVED] Euler 9 point circle I'm doing a project on the nine point circle and i need to know what type of triangle it works with. I tried constructing it but it didn't work with an isoscoles or a obtuse triangle, but a website said it works with all triangles, can anyone help?
  14. J

    Solve Euler Integration for dx/dt = (x-at)/(x/a+t)

    Hi Im a bit stuck on the method for Euler Integration. I have the following first order differential equation: dx/dt = (x-at) / (x / a+t) where constant a = 1.0V/s, and initial condition x = 1.0V at t=0s I have a time step of 0.02 and I need to calculate the output voltage at a time...
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