Conditions Definition and 978 Threads

Because of the wind, airplane was shifted to the east. Assume airplane is shifted D units long from B. When airplane turnaround, the wind shifted airplane to the east again as far D and to the south as far as L to the A'. Therefore, $$2D = (v - v') t_{AA'}$$ But, $$D = v'(t_{a}/2)$$ Thus...

Hello! The integral in equation (16), at the paper, is: ##I = r \int_{-\pi}^{\pi} e^{-2kr\phi} ~d\phi ## My integration is as the following : ## I = - \frac{1}{2 k} e^{-2kr\phi} ~|_{-\pi}^{\pi} + C ##, so ## I = - \frac{1}{2 k} ( e^{-2kr\pi} -e^{2kr\pi})+ C ## Now how to use the initial...

$\tiny{7.68}$ Normal curve $\quad\mu=12\quad\sigma = 2$ Let z denote a variable that a standard normal distribution. Determintie the value $z*$ to satisfy the following conditions: a. $P(z<z*)=.025$ b. $P(z<z*)=.01$ c. $P(z<z*)=.05$ d. $P(z<z*)=.02$ e. $P(z>z*)=.01$ f. $P(z>z*)\textbf{ or...

[Mentor Note -- Thread moved to the ME forum to get better views] Let's consider an incompressible block of Neo-Hookean material. Let the initial reference geometry be described by ##B=[0,b] \times [0,b] \times [0,h]##. The professor gave me the following task: Of course there can be many...

Each operator has a domain, so for a power of an operator to exist, the domain of the operator must remain invariant under the operation. Is that correct? mentor note: edited for future clarity

I am studying the real scalar field theory in ##d## spacetime dimensions as beautifully presented by M. Srednicki QFT's draft book, chapter 18 (actually, for the sake of simplicity, let us include polynomial interactions of degree less than or equal to 6 only) \begin{equation*} \mathcal{L}...

One dimensional Ising model is often treated as open chain system with free ends. Then when external field is added it is treated with cyclic boundary condition. Can someone explain me are those methods equivalent, or not?

Hi all, I was hoping someone could check whether I computed part (4) correctly, where i find the solution u(t,x) using dAlembert's formula: $$\boxed{\tilde{u}(t,x)=\frac{1}{2}\Big[\tilde{g}(x+t)+\tilde{g}(x-t)\Big]+\frac{1}{2}\int^{x+t}_{x-t}\tilde{h}(y)dy}$$ Does the graph of the solution look...

Assume that a contraption with loosely suspended internal weights is fixed to the body of a vehicle that is moving in free 3D space without gravitational sources nearby. Given the position and orientation of the vehicle as functions of time, how can one tell whether it is possible for the...

in Finite Difference Method (FDM), the boundary conditions can be implemented by applying the continuity of parallel component of magnetic field intensity. when it comes to the interface of two areas, it is done at ease, but consider this case at the red point: in FDM we exactly require on...

ial this is the question.

How to run a numerical simulation of Laplace equation if one of the boundary condition is like this: $$V(x,y) = 0 \text{ when } x \to \infty$$ I am trying to use Python to plot the solution of this Example 3.5. in Griffins EM

The problem states: Two parallel plates separated by distance h, the plate at the top moves with velocity V, while the one at the bottom remains stationary. My initial approach was: I considered, ##du/dy = V/h## and for the shear stress ##\tau = \mu \frac{\partial u}{\partial y}## For...

Hi everyone. I'm a new member, great to be here:) I have a few questions that I wanted to ask you guys regarding the method by which we implement the Runge-Kutta approximation of Projectile Motion if we should do it using a numerical iterative method with a Spreadsheet like Excel. I have...

The question is as follows: Solutions given only contain part a) to c), which is as follows: So I now try to attemp d), e) and f). d) The magnetic field of a uniformly magnetized sphere is: $$ \vec B =\frac{2}{3}\mu_{o} \vec M = \mu_{o}\vec H$$ $$\frac{2}{3}\vec M = \vec H$$ The perpendicular...

Theta in the incident angle Phi is the refraction angle '' denotes everything that propagates to the other medium, that is, everything related to refraction ' denotes the reflection in the original medium I am rather confused, would appreciate any help. I see the second equation of TE is...

The derivative of a point of maximum must be zero, and since $$y'=3ax^2+2bx+2 \rightarrow y'(-1)=3a-2b+2 \rightarrow 3a-2b+2=0$$ we get the first condition for ##a## and ##b##. Now, since we want ##x=-1## to be a local maximum, the derivative of the function must be positive when tending to...

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Are there any models, theories or physicists who propose that the fundamental laws of nature come from the initial conditions? Are there any physicists who propose that the most fundamental laws of physics emerged from initial conditions at the origin of the universe? And according to this view...

Let a continuous function f: R -> R such that f(x)f(f(x)) = 1 and f(2020) = 2019. What is the value of f(2018)? I am having problem with this question. I already tried, through various attempts, to find the function explicitly which satisfy this condition and f(0) different of 0. But it leads...

Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##: $$ N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$ $$ \frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$ What freedom...

Hi, In my course in analytical mechanics, it is said that for a system of n particles subjected to r constraint equations, it is necessary to impose regularity conditions on the constraint surface defined by G = 0 where G is a function of the position of the position of the particles and time...

I think in solid or liquid phase, there are many molecule having a very large speed due to random character in moving.So the liquid or solid matter must co-exist with other phase because some molecules escape from surface of solid or liquid matter.Then is there any condition for existing only...

I've attempted this questions after understanding the theory from my lecture notes, but my equation is looking bizarre here. Is this correct so far? Are my mesh equations correct here? Thanks.

Let's present two examples $$-\frac 1 2 \int d^3x'\big (-i \phi(x', t)\nabla^2\delta^3(x-x') \big )$$ Explicit evaluation of this integral yields $$-\frac 1 2 \int d^3x'\big (-i \phi( \vec x', t)\nabla'^2\delta^3(\vec x-\vec x') \big ) =\frac{i}{2}\phi(\vec x', t) \nabla' \delta^3(\vec...

Let us consider two sequences: $$n_k \in \Omega,\,k=1,2,...K,$$ $$m_k \in \Omega,\,k=1,2,...K,$$ where $$\Omega:=\{n\in\mathbb{N}|\,n\leq K\}.$$ Let us define $$\sigma_n:=\sum_{k=1}^K k\, n_k,\,\sigma_m:=\sum_{k=1}^K k\,m_k$$ The task is to find all possible ##(n_k,\,m_k)## pairs such that...

My understanding in modal analysis is very limited. All I know is it helps to find a specific mode of vibration and the natural frequency corresponding to it. While I was discussing about this with my NVH team colleague, he told me that there is no force input or excitation input given to a...

Hi PF! Can someone explain to me why in math/physics the frequencies associated with waves (or say drum heads) tend to be larger when the boundaries are pinned as opposed to free? If possible, do you know any published literature on this? Thanks!

Hello, I was going to solve numerically the eigenfunctions and eigenvalues problem of the schrödinger equation with Yukawa Potential. I thought that the Boundary condition of the eigenfunctions could be the same as in the case of Coulomb potential. Am I wrong? In that case, do you know some...

I'm struggling with a few steps of this argument. It's given that we have a surface ##S## bounding a volume ##V##, and a scalar field ##\phi## such that ##\nabla^2 \phi = 0## everywhere inside ##S##, and that ##\nabla \phi## is orthogonal to ##S## at all points on the surface. They say this is...

I create an algorithm that can solve [K]{u}={F} for atomic structure, but the results are not converge Do the boundary conditions affect the convergence of the resolution of a system of nonlinear partial equations? And how to know if the solution is diverged because of the boundary conditions...

Hi everyone! I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example, $$ u_t=k u_{xx}, $$ $$ u(t,-1)=\alpha, $$ $$ u(t,1)=\beta, $$ $$ u(0,x)=f(x), $$ $$...

I was studying how to compute an unpolarized cross-section (QFT Mandl & Shaw, second edition,https://ia800108.us.archive.org/32/items/FranzMandlGrahamShawQuantumFieldTheoryWiley2010/Franz%20Mandl%2C%20Graham%20Shaw-Quantum%20Field%20Theory-Wiley%20%282010%29.pdf) and came across the following...

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...

I have an equation that comes from an especific topic of cam mechanisms and it goes like this: $$ 2M[tan(B)-B] - \beta Ntan(B) - 2\pi\sqrt{1 - N^2} = 0 \ \ \ \ \ \ \ \ \ (1) $$ For this it doesn't matter what each variable means. I'm trying to create a 3x3 matrix with a determinant equal to...

I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...

Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk...

I found a theorem that states that if A and B are 2 endomorphism that satisfies $$[A,[A,B]]=[B,[A,B]]=0$$ then $$[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$$. Now I'm trying to apply this result using the creation and annihilation fermionics operators $$B=C_k^+$$ and $$A=C_k$$...

Hi I have a project regarding micromechanics of composites. I'm starting my analysis on the Fiber Matrix RVE. Right now I'm trying to find the natural frequency of the unit cell. The Unit cell has some unique geometry which I will keep on changing to see how natural frequency changes. I have...

I am seeing the heat conduction differential equation, and I was wondering about a boundary condition when the equation is of transient (unsteady) nature. When analyzing boundary conditions at the surface of say, a sphere, the temperature does not depend on time. For example, if you have...

Hello, I'm studying the renormalization of QED. I have the Lagrangian $$\mathscr{L}_{QED}=\mathscr{L}_{physical}+\mathscr{L}_{counterterms}$$ where $$\mathscr{L}_{physical}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi - e \bar{\psi}\gamma^\mu\psi A_\mu$$...

Ok so here are a few multiple choice questions that I have been given to me and these are what my selected options turned out to be Do they seem right? I am rather confused on the wording of the first question? Is it asking to state the conditions of SHM for it be in SHM? I know...

Why do we take ISA conditions(Atm. Temperature, Atm. Pressure) as static and not stagnation? Thanks

Hi all, Kirchhoff's equation for this simple circuit is equivalent to \dot I=\frac{V}{L} Where V=V_0 \sin(\omega t). Integrating both sides should give I(t) = -\frac{V_0}{L\omega} \cos(\omega t)+c where c is an arbitrary constant (current). Here, most of the derivations I've found simply drop...

Hello guys. I am studying the heat equation in polar coordinates $$ u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}) $$ via separation of variables. $$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$ which gives the ODEs $$T''+k \lambda^2 T=0$$ $$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...

Other than for null-homotopic maps, which continuous maps defined on ##D^1 \rightarrow D^1## (Open disk)extend continuously to maps ##B^1 \rightarrow B^1 ## ,(##B^1## the closed disk) which maps can be extended in opposite direction, i.e., continuous maps ## f: S^1 \rightarrow S^1 ## that...

The book of Balanis solves the field patterns from the potential functions. Let say for TE modes, it is: F_z(\rho, \phi, z) = A_{mn} J_m(\beta_{\rho}\rho) [C_2 \cos(m\phi) + D_2 \sin(m\phi)] e^{-j\beta_z z} There is no mention of how to solve for the constant A_{mn} . Then, from a paper...

Hi everyone! This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent! I am working on infinite square well problems, and in the example problem: V(x) = 0 if: 0 ≤ x ≤ a...

Hello, I am trying to solve a differential equation corresponding to a visco-elastic material model consisting of 5 units of springs and dashpots connected in parallel as can be seen in the image below. I am able to come-up with a single fifth order ODE, however I am struggling to find the...