Boundary Definition and 900 Threads

What does it mean to say something "extends smoothly" to a boundary in C (or R^2)? I'm studying Cauchy's integral formula, and one of the assumptions of the theorem is that a function be analytic on a domain D and extend smoothly to the boundary of D. What does that mean, exactly?

Homework Statement Let W\subset S \subset \mathbb{R}^n. Show that the following are equivalent: (i) W is relatively closed in S, (ii) W = \bar{W}\cap S and (iii) (\partial W)\cap S \subset W. Homework Equations The only thing we have to work with is the definitions of open and closed sets...

Hi. I have got two questions... 1. How can we determine the pressure at the boundary of two gas phases (say, compressed carbon dioxide released from a can and the atmosphere) which are originally separated? If we apply the ideal gas equation on each of these two phases, different pressures...

Zhonghui Liu of the University of Hong Kong has made the most comprehensive deep-sea core research to date http://www.sciencemag.org/cgi/content/abstract/323/5918/1187?ck=nck. Global SST's fell by an average of 4.5 to 6 degrees F at the E-O boundary, with temps near the South Pole and North Pole...

Hello all, I have a small problem with COMSOL. I am trying to use the particle tracing feature, more precisely I want to use the boundary coordinates to specify where are the points from which to trace the electrons. From what I understand in the User's Giude, the Boundary Coordinates feature...

Hi guys, i need some help on this topic. Basically, i need to calculate the angle of reflection of a oblique shock on a solid boundary. Here is the description of the question i have: an incident supersonic flow reaches a compression corner, with a deflection angle of 15 degrees...

Homework Statement Given a BVP: \Delta(u)+u=1 in \Omega u=0 on \partial\Omega using linear piecewise functions, calculate the corresponding local stiffness matrix on the reference triangle : {(x,y); 0<=x<=1, 0<=y<=1-x}. The domain is a square with one point in the middle (at...

What methods are there to figure out if our universe has a boundary or not? And if it does are we living on the boundary (like in the case of our Earth that we live on a sphere and not inside the earth)?

A partial differential equation requires boundary conditions. Consider a 2-dimensional problem, where the variables are 'x' and 'y'. The boundary is the line x=0 and you are given all sorts of information about the function on that line. If you are given just the values of the function on the...

While reviewing the Archimedes Principle with my AP students using ConcepTest questions from the Wilson, Buffa, Lou text I came across an answer that I believe is incorrect. The question was the second question in a series of questions about a helium filled balloon. The first question asked...

Homework Statement Solve, u_{t} = u_{xx}c^{2} given the following boundary and initial conditions u_{x}(0,t) = 0, u(L,t) = 0 u(x,0) = f(x) , u_{t}(x,0) = g(x)Homework Equations u(x,t) = F(x)G(t) The Attempt at a Solution I solved it, I am just not sure if it is right. u(x,t) =...

What would the boundary conditions be for a fourth order differential equation describing the deflection (elastic curve) of a propped cantilever beam with a uniform distributed load applied? i.e. a beam with a built in support on the left and a simple support on the right. I need 4 obviously but...

Homework Statement a) Solve for the BVP. Where A is a real number. b) For what values of A does there exist a unique solutions? What is the solution? c) For what values of A do there exist infinitely many solutions? d) For what values of A do there exist no solutions? Homework...

{\frac {{\it du}}{{\it dx}}}=998\,u+1998\,v {\frac {{\it dv}}{{\it dx}}}=-999\,u-1999\,v u \left( 0 \right) =1 v \left( 0 \right) =0 0<x<10 Second Order Backward Difference formula {\frac {f_{{k-2}}-4\,f_{{k-1}}+3f_{{k}}}{h}} I'm trying solve this numerically in matlab, but can't seem to...

Dear all, I have system(4 equations) of BVPs. Could anybody recommend me, how to solve this system(whatever numericaly or analytical): x'=-y/sqrt(x^2+y^2) + u y'=x/sqrt(x^2+y^2) + v u' = -xy/(x^2+y^2)^3/2 u - [1/sqrt(x^2+y^2) - x^2 /(x^2+y^2)^3/2] v v' = xy/(x^2+y^2)^3/2 v -...

Homework Statement I'm trying to figure out the the boundary of the set of all 1/n, where n is a natural number. Consider this as a subset of R with its usual metric, nothing fancy. Homework Equations There are at least two "equivalent" definitions of the boundary of a set: 1. the boundary of...

If you have the value of a function of many variables, and its 1st-derivatives, at a single point, and a 2nd-order partial differential equation, then haven't you determined the entire function? You can use a Taylor expansion about that point to build the entire function because you have the...

Dear everybody, I am new here. I need your suggesstion and helping for solving the next nonlinear boundary value problem: x'=-y/sqrt(x^2+y^2) y'=x/sqrt(x^2+y^2) with x(0)=y(0)=-1 and x(pi/4)=y(pi/4)=1 so is it easy to get the analytical solution to the problem above or I have to solve this...

Homework Statement A dielectric interface is defined as 4x + 3y = 10 m. The region including the origin is free space, where D1 = 2ax - 4ay + 6.5az nC/m2. In the other region, εr2 = 2.5. Find D2 given the previous conditions. Homework Equations an12 = ± grad(f)/|grad(f)| D2n = D1n =...

Hello, I'm trying to find an expression for the work done by a gas that expands adiabatically from one chamber to another through a controlled opening and closing of a valve. The setup is the following. There is spherical bulb divided inside into to chambers. One of the chambers (A)...

Homework Statement Use a Gaussian surface and an Amperian loop to derive the electrostatic boundary conditions for the polarisation field P at an interface between electric media 1 and 2 of relative permittivities e1 and e2. (Hint: determine results for D and E first) Homework Equations...

Homework Statement Calculate the outward flux of the two dimensional vector field f:\Re^{2}\rightarrow\Re^{2} , f(x,y)=(x/2 + y\sqrt{x^{2}+y^{2}},y/2 + x\sqrt{x^{2}+y^{2}}) through the boundary of the ball \Omega = {(x,y)\in\Re^{2} \left| x^{2}+y^{2} \leq R^{2}} \subset\Re^{2}, R>0...

Hi all, I am facing difficulties about boundary condition in Hollow Cylinder. its like wave diffusion equation in hollow cylinder. can anyone help me out to solve this problem. I need some good reverences. Thank you

u_t=u_{xx}+2u_x 0<=x<=L, t>=0, u(x,0)=f(x), u_x(0,t)=u_x(L,t)=0 How to do this?

I am stuck at the problems of Boundary conditions for two dimensional problem in QM. iIf we have a two-dimensional domain, along the boundary, we can define two directions, one is tangential, the other is normal, assuming that there is no current flowing in and out along the normal direction...

I'm looking to get a full solution to the Navier-Stokes equation to describe fluid flow through a pipe with moving surfaces. For now I am just concerned with a two dimensional system. The upper and lower boundaries are parallel to the x-axis. The surfaces of the boundaries move sinusoidally...

y"+by'+c=0 y(0)=y(1)=0

I am reading both Griffiths and Gasiorowicz and I can't get either of them to tell me why the continuity of the derivative of the natural log of the amplitude \frac{d(ln(u(x)))}{dx}=\frac{1}{u(x)}\frac{du(x)}{dx} or put a different way \frac{1}{u(a^{-})}\frac{du(a^{-})}{dx}=...

Is there a firm boundary? For example, if a scientist makes an observation which is probabilistically determined, his actions (for example recording data) are altered, although we would generally think of a scientist as a deterministic system. How do physicists deal with this conceptualy?

How close to 4 do we have to take x so that 3x + 2 is within a distance of (a) 0.1 and (b) 0.01 from 14? http://i.imagehost.org/0222/Picture_1.png a) Let k be any integer. Prove that if k^3 is even, then k is even. b) Prove that 3√2 is irrational. Hint: Mimic the proof that √2 is...

Homework Statement Show that the conditions for a bound state, Eqn1 and Eqn2, can be obtained by requiring the vanquishing of the denominators in Eqn3 at k=i\kappa. Can you give the argument for why this is not an accident? Homework Equations Eqn1: \alpha=q*tan(qa) Eqn2...

Homework Statement \sum^{infinity}_{n=0}\frac{cos(x)^n}{3^n} Find the values of x for which the series converges. (z,p) Find the sum of the series for those values of x. Homework Equations Geometric Series Sum (a/(1-r)) Geometric Series a*r^n The Attempt at a Solution I...

Hey PhysicsForums, This is my first time here but I have seen many knowledgeable responses to tough questions and I truly am stumped. My question is as follows (This a third year Analysis course in my Mathematics undergrad degree): 1. Let A and B be Subsets of RN with Boundaries B(A) and...

Homework Statement Find the temperature distribution in the long thin bar −a ≤ x ≤ a with a given initial temperature u(x,0) = f(x). The side walls of the bar are insulated, while heat radiates from the ends into the surrounding medium whose temperature is u = 0. The radiation is taken...

Use Green's Functions to solve: \frac{d^{2}y}{dx^{2}} + y = cosec x Subject to the boundary conditions: y\left(0\right) = y\left(\frac{\pi}{2}\right) = 0 Attempt: \frac{d^{2}G\left(x,z\right)}{dx^{2}} + G\left(x,z\right) = \delta\left(x-z\right) For x\neq z the RHS is zero...

Dear all, How can I show that: The boundary of a set S is equal to the intersection of the closure and the closure of the complement of S ? Thanks a lot in advance

Question on Boundary Points! Determine all boundary points: S={(x,y); 0 < x< 1 and y= sin(1/(1-x)} I'm really confused! I don't understand how I can actually find all the points.. can anyone get me started?? The help is appreciated!

Homework Statement A thin conductor plate is in free space. Its conductivity is finite and thickness is approaching zero. Relate the tangential electric field in either side of the conductor. Repeat for tangential magnetic field. How are electric and magnetic fields related. Homework...

Hi everyone, I would like to simulate the flow inside a tube having one closed end and the other end is driven by a loud speaker. The speaker creates an oscillating pressure (and velocity) which will have an effect on the acoustic field inside the tube. so to be clear: which boundary...

For a string fixed at x=0 and free at x=l I know \frac{dy}{dx}(l,t)=0 (d's are meant to be partials) but what is the other boundry associated with the end of the string? Is the second derivative also equal to 0?

I am trying to calculate the thickness of the boundary layer for a NACA 2412 airfoil. I am working on constructing some wind tunnel tests that simulate a Cessna 172 (NACA 2412 airfoil). I am trying to determine the thickness of the boundary layer (distance from surface to 99% of free-stream...

I hope this is the right place to post this question. I'm trying to figure out how to run a numeric integration for a nonlinear second order ODE with Neumann B.C. I've started programming up Runge Kutta 4, but clearly without a boundary condition on the function, but only on its derivative...

Hi everybody! I'm having some problem with the following system... I have a 2D problem, a disk is rotating with constant velocity, but there is a friction on the boundary due to the contact between the disk and the surrounding material. I'd like to find the analytical formula of the...

Hey all, I finally figured out how to solve the integral: \int{dp} = \int{6U\eta(\frac{h-\overline{h}}{h^{3}})}{dx} + C using maple and have it export to MATLAB where: h=R+h0-\sqrt{R+x}\sqrt{R-x} \overline{h}=R+h0-\sqrt{R+\overline{x}}\sqrt{R-\overline{x}} how do i find the...

Could someone plase hep me with normal coordinate substitutions with periodic boundary conditions, I can't see where the 1/N cancels in the attached file Thanks Doug

Using perturbation theory, I'm trying to solve the following problem \frac{\partial P}{\partial \tau} = \frac{1}{2}\varepsilon^2 \alpha^2 \frac{\partial^2 P}{\partial f^2} + \rho \varepsilon^2 \nu \alpha^2 \frac{\partial^2 P}{\partial f \partial \alpha} + \frac{1}{2}\varepsilon^2 \nu^2...

Homework Statement Insulated piston-cylinder device contains 5L of saturated liquid water at constant pressure of 175kPa. Water is stirred by a paddle wheel. A current of 8A flows for 45 minutes through a resistor in the water. If 1/2 of the liquid is evaporated during the constant...

Hi friends, I'm new to CST microwave studio. Just finish constructed a structure of an L-probe patch antenna (from IEEE paper) and just run the simulation by transient time solver, the curve of the return loss(S11) against frequency that i get is different from what showing on the IEEE paper...

Hallo, Why does dp*dx>>h represents classical mechanics state? Thanks, Omri

Hello, You've all probably seen the classic Blasius solution concerning the thickness of a boundary over a flat plate. This problem though assumes that the free stream velocity is constant. http://www.see.ed.ac.uk/~johnc/teaching/fluidmechanics4/2003-04/fluids9/2-dboundary.html I am...