# What is Euler's equation: Definition and 21 Discussions

In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula.
Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler.

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1. ### Euler's equation of thermodynamics in free expansion (Joule expansion)

Hi everyone, I am confused when I apply Euler's equation on the free expansion of an ideal gas. Consider a free expansion (expansion of gas in vaccum) where the volume is doubled (V->2V) The classical free expansion of an ideal gas results in increase in entropy by an amount of nR ln(2), a...
2. ### Fundamental equation (thermodynamics) from Euler's equation

by substituting all values in the euler equation you get most of the terms in the fundamental equation but not (N/No)^-(c+1) How do you get this term?
3. ### 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...
4. ### Euler's equation pressure difference

Homework Statement I am after PC - PA However I must do so without breaking into components. My problem has different values L=3 H=4 SG=1.2 downward a = 1.5g horizontal a = 0.9g and my coordinate is conventional positive y up and positive x to the right cos##\theta## = 3/5 sin##\theta## =...
5. ### I Euler's equation not making sense

Given: e^(i*pi) = -1 and e^(2*i*pi)=1 Adding we get: e^(i*pi) + e^(2*i*pi) = (-1+1) = 0 Factoring gives e^(i*pi) * [ 1 + e^(i*pi) ] = 0 so setting the second factor = to 0 gives 1 + e^(i*pi) = 0 which gives e^(i*pi)=-1 Okay so far, but setting the first...
6. ### I Proof of double angle formulas using Euler's equation

Hi all, I'm slowly working through "Mathematical Methods in the Physical Sciences" by Mary Boas, which I highly recommend, and I'm stumped on one of the questions. The problem is to prove the double angle formulas sin (2Θ)=2sinΘcosΘ and cos(2Θ)=cos2Θ-sin2Θ by using Euler's formula (raised to...
7. ### Euler Representation of complex numbers

I am bit confused with the Eueler representation of Complex Numbers. For instance, we say that e^{i\pi}=cos(\pi)+isin(\pi)=-1+i0=-1. The derivation of e^{i\theta}=cos(\theta)+isin(\theta) is carried out using the Taylor series. I quite understand how ##e^{i\pi}## turns out to be ##-1## using...
8. ### What is the result of using Euler's equation for Fourier transform integrals?

when I am using Euler equation for Fourier transform integrals of type \int_{-\infty}^{\infty} dx f(x) exp[ikx] I am getting following integrals: \int_{-\infty}^{\infty} dx f(x) cos(kx) (for the real part) and i* \int_{-\infty}^{\infty} dx f(x) sin(kx) (for its imaginary part) I am...
9. ### Why y, y' (derivative of y), x are independent?

In calculus of variations when we solve Euler's equation we always do think of y, x and y' as independent variables. In thermodynamics we think that different potentials have totally different variables I don't understand why the slope of the function is not directly dependent on function itself.
10. ### MHB Prove Euler's Equation for Functional $J(y)$

Hello! (Wave) According to my notes, the following theorem holds: If $y$ is a local extremum for the functional $J(y)= \int_a^b L(x,y,y') dx$ with $y \in C^2([a,b]), \ y(a)=y_0, \ y(b)=y_1$ then the extremum $y$ satisfies the ordinary differential equation of second order \$L_y(x,y,y')-...
11. ### Euler's equation for one-dimensional flow (Landau Lifshitz)

One page 5 in Landau & Lifshitz Fluid Mechanics (2nd edition), the authors pose the following problem: The authors then go on to give their solutions and assumptions. Here are the important parts: For the condition of mass conversation the authors arrive at (where ρ_0=ρ(a) is the given...
12. ### Solving Euler's equation

How do I solve the following Euler's equation: r^2 B_n'' + r B_n' - n^2 B_n = 3 \delta_{n1} r^2 Such that the solution is: B_n(r) = \beta_n r^n + \delta_{n1}r^2, \forall n \ge 1 where βn is a free coefficient, δ is the Kronecker delta function, and the solutions unbounded at r=0 are discarded.
13. ### Can you explain Euler's Equation and KVL in engineering?

In one of my engineering classes we discussed these two topics and I have two questions about this stuff. First question is how does euler's equation work exactly.. e^{j\varphi}=cos\varphi+jsin\varphi Second question is how do you solve this: V_{M}cos\omega t=Ri(t)+L\frac{di(t)}{dt}...
14. ### Showing this Euler's equation with a homogeneous function via the chain rule

Homework Statement Ok I have this general homogeneous function, which is a C^1 function: f(tx,ty)=t^k f(x,y) And then I have to show that this function satisfies this Euler equation: x\frac{\partial f}{\partial x}(x,y)+y\frac{\partial f}{\partial y}(x,y)=k\cdot f(x,y) Homework...
15. ### Derivations for Continuity equation of Fluid & Euler's Equation of Fluid Motion

Will anyone give me the derivations for continuty equation of fluid and euler's equation of fluid motion .
16. ### Zeros of Euler's equation, y''+(k/x^2)y=0

Homework Statement Show that every nontrivial solution of y''+\frac{k}{x^2}y=0 (with k being a constant) has an infinite number of positive zeros if k>1/4 and only finitely many positive zeros if k\le 1/4. Homework Equations The Attempt at a Solution I set y=x^M = e^{M \log{x}} (for some...
17. ### Transforming Euler's Equation to Constant Coefficients

Homework Statement Hi. I have this problem, which says: The equation x^2y''+pxy'+qy=0 (p and q constants) is called Euler equation. Demonstrate that the change of variable u=\ln (x) transforms the equation to one at constant coefficients. I haven't done much. I just normalized the equation...
18. ### A solution of Euler's equation

Homework Statement The velocity vector for a flow is u = (xt, yt, -2zt). Given that the density is constant and that the body force is F = (0,0,-g) find the pressure, P(x,t) in the fluid which satisfies P = P_0(t) at x = 0 Homework Equations Euler's equation...
19. ### Euler's Equations for Extremas of J: y=C*e^x

Homework Statement For the functional J(y(x))=\int^{x1}_{x2}F(x,y,y')dx, write out the curve y=y(x) for finding the extremas of J where F(x,y,y')=y'^2+y^2. Homework Equations Euler's Equations: \frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'}=0 \frac{\partial...
20. ### Principle Axes and Euler's Equation

A flat rectangular plate of Mass M and sides a and 2a rotates with angular velocity w about an axle through two diagonal corners. The bearings supporting the plate are mounted just at the corners. Find the force on each bearing. I am not sure how to find force using Euler's equations since...
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### Euler's Equation: A sign from god?

The first time I saw Euler's equation, it blew my mind. e^{i\pi}+1 \equal 0 Here, we have three of the most important numbers in math, all related to each other in such a remarkably compact equation. Does anyone know what this means? I think you can prove this through Taylor Series...