Boundary Definition and 1000 Threads

Homework Statement Find the volume between z = x2 + y2 and z = 2 - (x2 + y2). Homework Equations The Attempt at a Solution if r2 = x2 + y2 then the lower part of the volume is defined by: r2 \leq z \leq 2 - r2 and: 0 \leq r \leq 1 the upper part by: 2 - r2 \leq z \leq r2...

I've read in several places that the boundary of the rational numbers is the empty set. I feel I must be misinterpreting the definition of a boundary, because this doesn't seem right to me. My understanding of the boundary of a set S is that it is the set of all elements which can be...

Hi, I need to know if the following statement is false or true. Given two topological spaces, X and Y, and an homeomorphism, F, between them, if bA is the boundary of the subset A of X, this implies that F(bA) is the boundary of the subset F(A) of Y?

Lets us say I have a cube and I apply to a face of the cube a heat flux of 100 watt/m^2. Lets us say i divide the face of the cube into say 10 elements (area of each face of the element is 1 m^2). What will be the flux on each element , will it also be 100 watt/m^2? Sorry for a...

Homework Statement Determine all the solutions, if any, to the given boundary value problem by first finding a general solution to the differential equation: y" + y = 0 ; 0<x<2π y(0)=0 , y(2π)=1 The attempt at a solution So the general solution is given by: y = c1sin(x) +...

Hello everyone :) I'm reading the book QFT - L. H. Ryder, and I don't understand clearly what are the generating functional Z[J] and vacuum-to-vacuum boundary conditions? Help me, please >"<

Homework Statement How do you I calculate the Laminar Boundary Layer Y in meters with Blasius Equation? I have an expression for U (m/s) and u/U and u(m/s) and also eta=0.1, 0.2...5.2. I am wondering how one could calculate the y laminar boundary layer (m), the y buffer boundary layer (m)...

Hi, All: Let S g,2 be the orientable genus-g surface with two boundary components, and let C be a simple-closed curve in S g,2 . If C is homologically non-trivial (i.e., C does not bound a subsurface of Sg,2), and C intersects one of the boundary components , must C also...

suppose g(r) is a scalar function which is constant inside the volume 'v' but discontinuous at the boundaries of 'v'. The magnitude of discontinuity is given by constant 'M' then can we write the following expression \int\nablag(r)dv=M\int\hat{n}\delta(r-rs)dv=M\hat{n}\intd\deltav where...

Hello everyone and greetings from my internship! It's weekend and I'm struggling with my numerical solution of a 1+1 wave equation. Now, since I'm eventually going to simulate a black hole ( :D ) I need a one-side open grid - using advection equation as my boundary condition on the end of my...

Hi, I was reading Lemon's Perfect Form and it talked about "natural boundary conditions". But I don't understand exactly how one determines them. It seems to me that one imposes some random condition then deduce stuff from it...?! Advanced thanks for any enlightenment!

I am trying to set up for an experiment, and I need to know the time dependence of the temperature of the front surface of an assembly of plates. The assembly has a heater on one side and is exposed to a gaseous environment (of constant known pressure and temperature) on the other. I am...

Homework Statement I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set U=\{x\in R^n : |x|\leq 1\}. While it is evident that...

Suppose I have a mass M_0 (here denoted with lowercase zero because of previous discussions on relativistic mass), and I have a gravitational field \phi which can under make a shift of 180^o between a negative plane and a positive plane. Assume also that the mass is considered as a charge...

Does anyone know any good references for a moving boundary problem? I am looking at a cylinder of charge being injected into a fluid, the PDE is: -\nabla^{2}\varphi +a\frac{\partial^{2}\varphi}{\partial t^{2}}+b\frac{\partial\varphi}{\partial t}=0 I want \varphi =\varphi_{0}, a constant on...

Is my claim correct?

Homework Statement n is given by: ∂2Θ/∂x2=1/α2 ∂Θ/∂t , where Θ(x, t) is the temperature as a function of time and position, and α2 is a constant characteristic for the material through which the heat is flowing. We have a plate of infinite area and thickness d that has a uniform...

I am trying to model the diffusion of fluorophores in a cell with a source in the middle by solving the appropriate differential equation. I can solve the PDE easily enough, however as I haven't done DE's in a while, I need a refresher on how to apply the appropriate boundary conditions for my...

Something has been bugging as of late: usually, derivatives (ordinary and partial) are defined for interior points. However, I often come across statements in which they seem to also be defined for boundary points. For example, Leibniz' rule of integration, as usually stated, assumes some...

I am trying to solve the following heat equation ODE: d^2T/dr^2+1/r*dT/dr=0 (steady state) or dT/dt=d^2T/dr^2+1/r*dT/dr (transient state) The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2. I have searched the analytical solution for this kind of ODEs in polar coordinate...

Hi I am studying magnetic vector potential from Griffiths book. The eq 5.76 in his book gives the boundary condition for the magnetic vector potential. \frac{\partial \vec{A_2} }{\partial n}- \frac{\partial \vec{A_1} }{\partial n}=-\mu_o \vec{K} where n is the vector perpendicular to the...

Hey guys, I'm having a conceptual problem implementing the Crank-Nicolson scheme to a PDE with nonlinear boundary conditions. The problem is the following: u_t + u_{xxxx} = 0, u(0,t) = 1,\quad u_x(0,t) = 0, \quad u_{xx}(1,t) = 0, u_t(1,t) - u_{xxx}(1,t) = f\bigl(u(1,t)\bigr). Taking m...

Hey guys, I'm confused as to what happens regarding the amplitude when a pulse moves across mediums. 1. Let's say a pulse is moving from a string of low density to a string of high density: I know that the reflected amplitude will be less than the incident's, and also the transmitted...

Homework Statement True or false: Let S be any set in R2. The boundary of S is the set of points contained in S which are not in the interior of S. Homework Equations The Attempt at a Solution Common sense tells me true. I don't really understand it though, if S is an open set...

Homework Statement Hi guys, I'm having trouble understanding the finite potential well, in particular the boundary conditions The well under scrutiny has potential V(x)= 0 for |x|<a and V(x)=V_0 for >a Homework Equations \frac{d^2\psi}{dx^2}=-\sqrt{\frac{2mE}{\hbar^2}}\psi=-\alpha^2\psi...

Homework Statement y^{(4)}+\lambda y=0 y(0)=y'(0)=0 y(L)=y'(L)=0 Homework Equations The hint says... let \lambda = -\mu ^4, \mu >0 or \lambda = 0The Attempt at a Solution Listening to the hint, I got r=\pm\mu With multiplicity 2 of each. So that means.. y=c_1 e^{\mu t}+c_2te^{\mu...

I wrote a program that uses the FEM to approximate a solution to the heat conduction equation. I was lazy and wanted to test it, so I only allowed Neumann boundary conditions (I will program in the Dirichlet conditions and the source terms later). When I input low values for the heat flux, I...

I am disappointed. This take on determinism comes from the Stanford Encyclopedia: "Determinism: The world is governed by (or is under the sway of) determinism if and only if, given a specified way things are at a time t, the way things go thereafter is fixed as a matter of natural law. The...

Okay, so I've read Max Born's book twice and believe I now have a reasonable, high-level understanding of general relativity. The thing I'm now trying to tease out is Mach's principle. One thought experiment that I've been falling asleep thinking about is how the presence of distant masses...

This is an example shown in "Introduction to Electrodynamics" by Griffiths. Page 226 example 5.8. Given a sheet of current K on the xy-plane where current traveling in +ve x direction. Find the magnetic field. I am confused on the way the book justify the z direction of B is zero. The...

Given a bounded domain with the homogeneous Neumann boundary condition, show that the Laplacian has an eigenvalue equal to zero (show that there is a nonzero function u such that ∆u = 0, with the homogeneous Neumann B.C.). I said: ∇•(u∇u)=u∆u+∇u2, since ∆u = 0, we have ∇•(u∇u)=∇u2 ∫...

Can't seem to work this out, any solutions would be greatly appreciated! Thanks in advance! Solve the boundary-value problem Uxx + Uyy + U = 0 , 0<x<1,0<y<1 U(0,y) = 0 , Ux(a,y)= f(y) U(x,0) = 0 , Uy(x,1)= sin(3*pi*x)

I am using Gaussian elimination to solve the airy stress function, but I am having difficulty implementing boundary conditions. A good synopsis on the problem of identifying boundary conditions is given here (section 5.2.1): http://solidmechanics.org/text/Chapter5_2/Chapter5_2.htm Given that...

Greetings, everyone! The problem below is actually a task on Numerical Methods. But I have difficulties making a mathematical model. Homework Statement Let us have a longitudinally homogeneous system of a pipe of radius R and a propeller of nearly the same radius inside it (we shall...

Hi, I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the Neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...

Hi, I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...

Homework Statement Consider \frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2} subject to u(0,t) = A(t),\ u(L,t) = 0,\ u(x,0) = g(x). Assume that u(x,t) has a Fourier sine series. Determine a differential equation for the Fourier coefficients (assume appropriate continuity)...

So, I do not think I did this properly, but if f(-x)=-f(x), then u(-x,0)=-u(x,0), and if g(-x)=-g(x), then ut(-x,0)=-ut(x,0). According to D`Alambert`s formula, u(x,t)=[f(x+t)+f(x-t)]/2 + 0.5∫g(s)ds (from x-t to x+t) so, u(0,t)=[f(t)+f(-t)]/2 + 0.5∫g(s)ds (from -t to t) f is odd, and so is...

I am sometimes just not sure how to go about solving magnetics problems and applying the right boundary conditions. I was hoping for a little advice. For example in an infinitely long cylinder (along z-axis) with radius a, and a permanent magnetization given by: \vec{M} =...

Hello, I'm having trouble getting started on this problem. Here's the question: [PLAIN]http://img810.imageshack.us/img810/2464/ee323assn3q3.jpg My issue is in setting up the governing partial differential equation in 3 dimensions. What I've tried so far is setting du/dt equal to the...

I understand application of Snell's law for transition from one medium to another but I have a question regarding this model. When an electromagnetic wave transitions from air into a conductive medium does the wavelength change instantaneously as the theory seems to imply or is there a boundary...

Hi there, I have completed my analysis in BEM. The results gives me velocity potential in form of 37954305E-04 +1.58625295E-06 +2.10275811E-07 +... I have real scattering values, imaginary and absolute scattering values in this form. I am new to the method and from a different...

Homework Statement A sphere under uniform rotation R, in a simple shear flow, given at infinity by ui = G(x2 + c)deltai1 The centre of sphere is fixed at x2 Boundary conditions are ui = EijkRjxk on sphere, and ui = G(x2 + c) at...

Homework Statement Find the solution of the equation v''- 4v'+5v=0,such that v=-1 and v'=1 when x=pi=3.14159Homework Equations ... The Attempt at a Solution I treat it as a polynomial=>r^2+4r+5=0 =>delta=-4=>r1=2+2i and r2=2-2i v=e^[x+2](A*cos[2]+B*i*sin[2]) v=-1=e^[pi+2](A*cos[2]+B*i*sin[2])...

Boundary conditions & time domain electromagnetic waves: does classical model fit? Consider two propagating media: a lossy dielectric medium and a lossless dielectric medium. Thus, the interface that separates them has two tangential components of electric field, one for each medium. One of...

I don't seem to grasp the meaning of boundary conditions for Laplace's equation. Consider the Lagendre expansion of the potential at x due to a unit charge 1/|x-x'|, where x' is the position of the unit point charge. To do the expansion, we need to consider a volume in space where the...

Homework Statement Solve the given boundary value problem or else show that it has no solutions: y'' + 4y = cos x, y'(0) = 0, y'(pi) = 0. Homework Equations N/A The Attempt at a Solution So I made it all the way through the problem I think, but I am not getting the correct answer...

When solving Schrodinger's eqn for a quantum ring, what would be the boundary conditions? The solution (polar) should be Ψ(Φ) = A exp(ikΦ) + B exp(-ikΦ) And I believe the boundary conditions are Ψ(0) = Ψ(2pi) Ψ(0) = A + B Ψ(2pi) = A exp(ik*2π) + B exp(ik*2π) and I suppose I can...

Hi, I am trying to work this problem out but I don't know how to solve the boundary value. here is the problem statement thanks in advance

Hi Everyone, Apologies if this is posted in the wrong place. I am trying to produce numerical solutions for the following initial value partial differential equations, namely the overdamped Schmoluchowski equation \frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial...