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.
I have multiple robot cells that i would like to move fixtures from robot to robot with the out come of the part to be perfect. in order to do that i need to make sure my work object is in the same spot each time. so i need to find the yaw pitch and roll and convert them into Euler angles.
On...
I have derived an explicit formula for the http://mathworld.wolfram.com/EulerZigzagNumber.html" , the number of alternating permutations for n elements:
A_j=i^{j+1}\sum _{n=1}^{j+1} \sum _{k=0}^n \frac{C_k^n(n-2k)^{j+1}(-1)^k}{2^ni^nn}
For details, please refer to my article in...
HI there,
I am having problems understanding something.
If I have an axis pointing towards the galactic north pole, and I rotate it using an Euler rotations how can I can I establish the rotation angles needed so that it will be pointing to the equatorial North pole.
I am looking for...
Hello guys,
I'm trying to find an algorithm to solve an overlap problem between many polyhedra in space, so here this question appears.
Imagine an Ellipsoid rotated by Euler angles in 3D space. This Ellipsoid could be characterised by its 3 radii Rx,Ry and Rz, and by 3 Euler angles in...
I was wondering if the Euler buckling method is simply theoretical or actually accurate in the real world. I am asking this both for complex structures and a simply compressed rod or beam.
Thanks
I'm looking at the following example (http://www.cyclismo.org/tutorial/matlab/control.html):
y'=x^2-y^2, y(0)=1, where they use the Euler method to approximate numerical solutions.
This is the code:
I don't understand the third and fourth line at the top. What is y = 0*x and how did they...
Every textbook i find breezes over the following point:
\delta\partial (x) =\partial \delta (x)
where delta is just the variation. Someone asked me why that's true and i guessed the only thing i could say was that delta is an operation not a variable so this is more like an algebraric...
hi guys,
can someone give me a quick tutorial on how to solve and explain to me the concept of slope field of the following differential equation:
sketch the slope field for dy/dt = 2t+1
showing the solution y=t^2+t-4, which satisfies the initial condition y(-2)= -2
Also how to use...
Hello every one, I am a new comer.
During my research of ionization rate of molecule using ADK method, I meet a question.
What is the rotation matrix and R the Euler angles between the molecular axis (in Eq. (8) of reference PHYSICAL REVIEW A, 66, 033402 (2002)) and what form is the...
How do you guys deal with the fact that even though you enjoy Mathematics or Physics, you will probably never be the next Euler, Gauss, Nash, Einstein, Feynman etc? It really puts me off reading about how talented these people were.
Sometimes i just feel redundent. I will get frustrated when...
Homework Statement
determine the gen sol of the DE
x^2y'' - 5xy' + 9y = 0
Q/P = A = -5 and R/P = B = 9
then use y = x^r in original DE...
y' = rx^(r-1)
y'' = (r-1)rx^(r-2)
insert into DE
x^2 (r-1)rx^{r-2} - 5x(rx^{r-1})+ 9x^r = 0
i am confused as to how to proceed...
The Euler Characterisitc can be computed from the dimensions of the Z homology of a space.
Are there any other invariants that can be computed from the dimensions of the Z homology?
I should probably know the answer to this but anyway...
I want to compute the critical buckling load for a simple 3D rectangular cross-section beam, though the thickness in the z-direction is small.
If a load is applied in the negative y direction, which direction do I use for my...
I am having trouble with this proof: show that if d|n then phi(d)|phi(n). I know that if d|n, then ad=n and that phi(ad)=(a,d)*phi(a)&phi(d)/phi((a,d)), but I can't seem to get anywhere with this info. Thanks for your help.
Matlab Help! Euler method
Use Euler's method:
cooling constant r = 0.2 , initial temperature of 86 C, and room temperature at 17 C. Choose your time step h and integrate to 5 minutes.
You can add some cream which will drop the temperature instantly by 5 C.
(a) Now, add the cream at the...
Is it possible for an Euler equation to satisfy the boundary conditions Y(1)=0, Y(2)=0?
I have considered the three possibilities, distinct real roots, repeated roots and conjugate complex roots and cannot find any solutions.
Are there any other possibilities to consider?
Thanks
Dear everyone
can anyone help me with the euler lagrange equation which is stated in d'inverno chapter 11?
in equation (11.26) it is said that when we use the hilbert-einstein lagrangian we can have:
∂L/(∂g_(ab,cd) )=(g^(-1/2) )[(1/2)(g^ac g^bd+g^ad g^bc )-g^ab g^cd ]
haw can we derive...
Hi,
I would like to start from the stress energy tensor for the perfect fluid:
T^{\mu\nu}=\begin{pmatrix} \rho c^2 & 0 & 0 & 0\cr 0 & p & 0 & 0\cr 0 & 0 & p & 0\cr 0 & 0 & 0 & p\cr\end{pmatrix}
where \rho is the mass density and p is the pressure, and I would like to derive the...
Given the homogeneous equation:
x3y''' + 15x2y'' + 61xy' + 64y = 0
I get 3 solutions which are all -4.
Does this mean that the solution for y should be:
y = c1x-4 + c2x-4lnx + c3x-4ln2x
?
What would be the euler angles of rotation 2pi/3 about the line x=y=z? If something were in the xy plane and it underwent that rotation, would it end up in the yz plane?
Homework Statement
This is a problem that the book uses as an example and I've been trying since friday to get the same answer but i have not had any success, i need help :(
Well in this problem, F = y(1+(y')^2)^1/2
Homework Equations
eulers eqn
d/dx(partial(F)/partial(y')) -...
I am a self studier, there might be things seems obvious to you guys that are not obvious for me. I have a few questions here, I write down the equation first and present the question in the last part. Please bare with me. Thanks.
Eulers equation of 2nd order: L[y]=x^{2} y''+ \alpha xy'+...
Greetings, I am trying to implement backward euler implicit method by Newton-raphson iteration. The differential equation is for a simple planar pendulum. So the function for the pendulum is :
(1) angularAcceleration (angle) = ( -gravity/length ) * sin(angle);
and the update function for...
Homework Statement
in fortran 90 write a program to using the euler scheme to solve the differential equation for decay of a radioactive substance
dx/dt = -j*x
solving the equation we get
x = e^(-j*t)
following info is given
2. Homework Equations and values
t0 = 0.0
tF =...
I am confused by both of these. I get what it does, and how it does it, but when using it, somehow I ultimately screw up.
Could someone just post a quick description of what to do in conversational English, please? I am pretty sure that would solve this random mental block I have...
Hi all,
I am trying to implement the backward euler integration (in c++) for the pendulum problem. I have the forward euler implemented, but frankly I don't know where to even start from for the implicit integration. I understand the update expressions for implicit, and of course the...
Hello,
I am getting very confused.
I have a list of supernovae with galactic coordinates l (from 0->360) and b (from -180->180).
I am trying to write a program in Matlab that rotates the coordinates by 2 Euler angles. Firstly I change l and b into X,Y,Z i.e change from spherical to cartesian...
So I registered on http://projecteuler.net/" and I must say: it's quite addictive!
I didn't expect it to be that much fun.
I'd like to share what things I've learned from solving the problems:
- When solving a problem you get access to a thread. I was impressed
how different the...
Tetrahedron is the simplest polyhedron, it is just formed from four triangles, so it has V0=E0=4*3=12, and F0=F=4, of course. Then we find that V0+E0+F0=2(V+E+F) or V0+F0=2V+2F (Because of E0 is always equal to 2E)=4F.
Assume that other polyhedrons are the "evolution" product of a tetrahedron...
What is the largest prime factor of the number 600851475143 ?
My attempt:
#include <stdafx.h>
int main()
{
int num1, num2, num3, num4, lrgprime, sum;
lrgprime=num2=num4=sum=0;
for (num1=2; num1<300425737572; num1++) {
num2=600851475143%num1;
if (num2==0) {
for...
im trying to do the Euler problem #14 - to determine which starting value under one million produces the longest chain
and here's my code in Python
Count=[]
Count2=[]
List=[]
def seq(n):
if n%2==0:
return n/2
else:
return 3*n+1
m=1
for i in range(1,1000000):
j=i
List.append(j)
while...
Hi all, I'm having trouble understanding a basic concept introduced in one of my lectures. It says that:
To solve the DE
y(t) + \frac{dy(t)}{dt} = 1 where y(t) = 0,
using the Euler (forward) method, we can approximate to:
y[n+1] = T + (1-T)y[n] where T is step size and y[0] = 0...
Homework Statement
Solve the euler differential equation
\x^{2}y^{''}+3xy'-3y=0
\int_X f = \lim\int_X f_n < \infty
by making the ansatz [tex]y(x)=cx^{m}[tex], where c and m are constants.
The Attempt at a Solution
[tex]y(x)0=c^{m}[tex]
[tex]y^{'}(x)=cm^{m-1}[tex]...
I'm reading a book called The Road to Reality by Roger Penrose, and I'm on the chapter for complex logarithms. What I don't understand is how the identity e\theta i = cos \theta + i sin \theta is found through the use of complex logarithms. I also don't understand how if w = ez, z = log r +...
Homework Statement
i need to obtain the remainder of the divison 2^{1000005} divided by 55
Homework Equations
Euler theorem 2^{\phi (55)}=1 mod(55)
The Attempt at a Solution
my problem is that applying Euler theorem i reach to the conclusion that the remainder is the...
Homework Statement
Hi, I'm trying to compute the equations of motion for a car as shown
in the attached image.
α = steering angle
θ = orientation of the car relative to the world coordinate system
Say you're given the linear velocity v and the steering
angle α. How do you compute...
Hi, I'm trying to compute the equations of motion for a car as shown
in the attached image.
α = steering angle
θ = orientation of the car relative to the world coordinate system
Say you're given the linear velocity v and the steering
angle α. How do you compute the position and angle...
Hi,
I need to solve the following simple ODE with both the Euler Forward and Euler Backward numerical methods. I also need to answer for which values of T this can still be calculated:
y'(t) = \frac{-1}{2y(t)}
y(0) = 2
t \in (0, T]
Obviously the analytical solution is
y(t) = \sqrt{4 - t}
So it...
Hello,
I am trying to extract the 3 angles from a unit vector to position a part on a 5 axis milling machine:
This machine has 3 linear axes (x,y,z), direct coordinate system
It has 2 rotary axes:
A turns about the x (called 'tilt')
C is carried on the A axis (called 'rotary')
See...
Homework Statement
I want to find the euler buckling for a open I beam( middle section like a box) that is subjected to three point bending with a point load on top.
I am not so interest in a solution but explanation on how this could be worked out using the euler equation for buckling...
Homework Statement
Prove that the rate of convergence for Euler's method (in solving the problem y'=y) is comparable to 1/n by showing that lim_{n\rightarrow infinity} \frac{e-(1+\frac{1}{n})^n}{1/n}.
Homework Equations
Hint: Use L'Hopital's Rule
If lim x-> infinity f(x)/g(x) is in...
Could somebody explain to me how lagrange multipliers works in finding extrema of constrained functions? also, what is calculus of variations and lagrangian mechanics, and can somebody explain to me what the lagrangian function is and the euler-lagrange equation. And, i read something about...
1. I use Euler formula of arctan to calculate arcsine
2. This equation\arctan x = \frac{x}{1+x^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k x^2}{(2k+1)(1+x^2)}.
If I input 0.9999999999, I will not be able to get the expected result. If input = 0.9 then it is pretty correct, but 0.99 is definitely...
Homework Statement
Well, the question is to find a formula for \phi(p^n) where \phi is the Euler's totient, and p is a positive prime.
Homework Equations
\phi(2^n) well in this case here is what i did, i first took n=1, n=2, n=3, and it looks like
\phi(2^n)=2^{n-1} but i am...
How do i comput the euler phi function of a large interger?
i know that if p is prime then phi(p)=p-1 and I've found a formula for computing non primes but i don't know how to implement in something like Matlab.
Does anyone know how?
How do i comput the euler phi function of a large interger?
i know that if p is prime then phi(p)=p-1 and I've found a formula for computing non primes but i don't know how to implement in something like Matlab.
Does anyone know how?
Hello, can anyone help with this question? Thank you.
Let n even perfect number and q prime. Show that n/Φ(n)=2q/q-1.
Φ(n) is the Euler function-totient (the number of positive integers less than or equal to n that are coprime to n)
I have tried euler-euclid theorem but could not...