Boundary Definition and 1000 Threads

this is a rather stupid question regarding preliminaries for the definition of boundaries the question is whether every closed n-1 dim. closed submanifold C of an arbitrary n-dim. manifold defines a volume V; i.e. whether \partial V = C can be turned around such that V is defined as the...

Homework Statement I am trying to calculate the transmission and reflection coefficients for rectangular finite potential barrier between (-a, a) for a particle of mass m with energy equal to the height of the barrier (E = V0 > 0). Homework Equations...

Dear experts, I´m trying to model in ANSYS Mechanical (v14.5) the linear buckling behavior of a cylinder made of BEAM4 elements under combined loading (axial compression and bending moment) applied at the ends. How should I set up the boundary conditions of a cylinder to keep rigid the ends...

Hello I'm getting confused when I want to use magnetic boundary equation could you tell me how we define the unit vector(an) in this equation? for example you assume that we have two different region (A in red and B in yellow) which vector (1,2,3,4) is right for equation and which is right...

Hello! When considering the boundary conditions for the electromagnetic field \mathbf{E}, \mathbf{H} on the surface of a Perfect Eletric Conductor we have: \mathbf{E} \times \mathbf{\hat{n}} = 0 \mathbf{J}_S = \mathbf{\hat{n}} \times \mathbf{H} the tangential electric field should vahish...

Hello! In http://my.ece.ucsb.edu/York/Bobsclass/201C/Handouts/Chap1.pdf, pages 19-20, the Love's Theorem in Electromagnetism is declared. In presence of some electric sources \mathbf{J} and magnetic sources \mathbf{M} enclosed by an arbitrary geometrical surface S, which produce outside S a...

All, I recently completed a project where transient thermal boundary conditions are rotated around a cylinder for a general number of revolutions. In reality, the cylinder rotated but it was much easier to rotate the thermal conditions around the model in the ANSYS environment. I used 360...

Homework Statement An #a*b*c box is given in x,y,z (so it's length #a along the x axis, etc.). Every face is kept at #V=0 except for the face at #x=a , which is kept at #V(a,y,z)=V_o*sin(pi*y/b)*sin(pi*z/c). We are to, "solve for all possible configurations of the box's potential" Homework...

Let f(x,y) be defined by f(x,y) = [x2y2]/[x2y2 + (x-y)2] a) Find the domain of the function f. b) show that (0,0) is a boundary point of the domain of f c) Compute the following limit if it exists: lim (x,y) ---> (0,0) f(x,y) The Attempt at a Solution a) I first change the value (x-y)2 to...

Is anyone aware of how to numerically solve the (1D) SE with periodic boundary conditions?

Homework Statement I have a hollow, grounded, conducting sphere of radius R, inside of which is a point charge q lying distance a from the center, such that a<R. The problem claims, "There are no other charges besides q and what is needed on the sphere to satisfy the boundary condition". I...

Does anybody know papers in which the asymptotic safety approach has been applied to the "no boundary" proposal?

Hi, I am a bit confused about the terminology used for the boundary conditions describing open and closed strings. For the open string, Ramond case: \psi^+(\sigma = \pi, t) = \psi^-(\sigma = \pi, t) Neveu-Schwarz case: \psi^+(\sigma = \pi, t) = -\psi^-(\sigma = \pi, t) Question 1: Is it...

Suppose we have this rectangle that is stretched equally on both sides with some force, F. Neglect shear force or moments and assuming transverse waves, is the solution still ε = Ae^(i(wt-kx))+Be^(i(wt+kx)) With boundary conditions: X = +L/2, ∂ε/∂x = 0 and X = -L/2, ∂ε/∂x =...

Homework Statement Determine the boundary of the following set. As usual, z=(x,y). 0<\left| z-z_0 \right|<2 2. The attempt at a solution The book's solution says "The circle \left| z-z_0 \right|=2 together with the point (0,0)" Why should the answer not be "... together with the...

Homework Statement I have an infinite plate of which two electrodes are attached at a distance ##2a## and the electric potential between them is ##U##. Now I have to find a function ##\phi (x,y)## that satisfies Laplace's equation ##\nabla ^2 \phi =0## and is equal to ##0## at all possible...

Hello, I am at present analyzing the electromagnetic interaction of a layer of paritcles in air when illuminated by an electromagnetic wave. This can be done by considering the layer of particles as an interface with surface current (as opposed to a 'normal' interface with Fresnel...

Hi, I am wondering as to how to define the boundary condition for a shape in a unit cell. Just imagine that the shape is the hole for the unit cell. Hence, for a constant thickness on the untextured boundary, thickness is, let's say C, then for the circle, it's C+depth. For example for...

Homework Statement I'm doing question 23 in Chapter 4 of Spivak's Calculus on Manifolds. The question asks, For R > O, and n an integer, define the singular l-cube, c_{R,n} :[0,1] \rightarrow \mathbb {R}^2 - 0 by c_{R,n} (t) = (Rcos2\pi nt, Rsin2\pi nt). Show that there is a singular...

Hello, I was just after an explanation of how people get to this conclusion: Say you are looking at the Helmholtz equation in spherical co-ordinates. You use separation of variables, you solve for the polar and azimuthal components. Now you solve for the radial, you will find that...

To prevent grain boundary sliding so that creep is less likely to occur, usually engineers would design components of larger grains or have columnar grain structure to prevent grain-boundary sliding. Why this two method can prevent grain-boundary sliding? For columnar grains, would they be more...

Homework Statement . Let ##X## be a nonempty set and let ##x_0 \in X##. (a) ##\{U \in \mathcal P(X) : x_0 \in U\} \cup \{\emptyset\}## is a topology on ##X##. (b) ##\{U \in \mathcal P(X) : x_0 \not \in U\} \cup \{X\}## is a topology on ##X##. Describe the interior, the closure and the...

Hi there Can anyone tell what is the meaning of boundary conditions for temperature distribution in a flowing viscous fluid in a pipe ? for example I need some one explane for me this: T = T1 at r = R, x<0 T = T0 at x = 0, r<R where T1 is a temperature of well and T0 is a temperature...

Homework Statement ##\mathscr{C}: x=1,x=3,y=2,y=3## ##\int_\mathscr{C} (xy^2-y^3)dx+(-5x^2+y^3)dy## Homework Equations The Attempt at a Solution ##\frac{\partial Q}{\partial x} = -10x^2 \,\,; \frac{\partial P}{\partial y} = 2xy-3y^2## ##\int\int_\mathscr{C} \frac{\partial...

Calculate the thickness of the boundary Layer δ at a location x= 0.3m along the chord length of an aircraft wing at each of the following velocities. (u = 20, 40,60,80,100 knots) Assume ISA P=101325 R=287 T=288.5 μ =18 x 10-6 (1)Re transition=5 x 10^5 (2)δ Laminar = x 4.91 Rex^-0.5 (3)δ...

4m3 rigid tank contains saturated H2O vapour at 3.5 bar. When this tank is left for a long time in a laboratory at 25.4oC, its temperatures reduces to this temperature. The thermodynamic properties of H2O is attached. Questions What would the boundary of the system be and what would the...

Hi, Can anybody help me withg the following problem: A rectangular plate with points starting from top left corner and going clockwise:: A B C D. Sides CD and DA are simply supported, and a point force F is applied anywhere on the surface. I am looking for the bending stress distribution...

I have a PDE of the following form: f_t(t,x,y) = k f + g(x,y) f_x(t,x,y) + h(x,y) f_y(t,x,y) + c f_{yy}(t,x,y) \\ \lim_{t\to s^+} f(t,x,y) = \delta (x-y) Here k and c are real numbers and g, h are (infinitely) smooth real-valued functions. I have been trying to learn how to do this...

Hi, I am studying fluid mechanics and I am trying to get to grips with slip and no-slip boundaries. I know that: Slip ---> Occurs when fluid is inviscid so no viscous stress at boundary. No-slip ---> Viscous effects mean the the tangential velocity must be zero, relative to the boundary...

Hey! :o I have to solve the following initial and boundary value problem: $$u_t=u_{xx}, 0<x<L, t>0 (1)$$ $$u_x(0,t)=u_x(L,t)=0, t>0$$ $$u(x,0)=H(x - \frac{L}{2} ), 0<x<L, \text{ where } H(x)=1 \text{ for } x>0 \text{ and } H(x)=0 \text{ for } x<0$$ I have done the following: Using the method...

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential $V(r,z)$ with the help of changing the Poisson equation into cylindrical polar co ordinates? Where V is...

hi pf! i was reading a sample problem in a text on ode's and came across a boundary condition that didnt really make sense to me. the physical scenario is: a liquid ##L## measured in moles/cubic meter (##mol / m^3##) is injected into a stream of water. ##L## is being injected at a rate...

If I have a grounded conducting material, then I know that $\phi=0$ inside this material, no matter what the electric configuration in the surrounding will be. Now I have a conducting material that is not grounded, then there will be (as long as I am dealing with static problems) no electric...

Homework Statement Part (a): List the boundary conditions Part (b): Show the relation for potential is: Part (c): Find Potential everywhere. Part (d): With a surface charge, where does the Electric field disappear? Homework Equations The Attempt at a Solution Part (a) Boundary conditions...

Hey! :o I have a question.. (Wasntme) When we have the following boundary value problem: $$u_{xx}+u_{yy}=0, 0<x<a, 0<y<b (1)$$ $$u_x(0,y)=u_x(a,y)=0, 0<y<b$$ $$u(x,0)=x, u_y(x,b)=0, 0<x<a$$using the method of separation of variables, the solution would be of the form $u(x,y)=X(x) \cdot Y(y)$...

Hey! :o I have to solve the following boundary value problem: $$u_{xx}+=u_{yy}=0, 0<x<a, 0<y<b$$ $$u_x(0,y)=u_x(a,y)=0, 0<y<b$$ $$u(x,0)=x, u_y(x,b)=0, 0<x<a$$ By using the method of separation of variables, the solution is of the form $u(x,y)=X(x)Y(y)$. By substituting this it the problem we...

Hi, Given a holomorphic function u(x,y) defined in the half plane ( x\in (-\infty,\infty), y\in (-\infty,0)), with boundary value u(x,0) = f(x) , the solution to this equation (known as the Poisson integral formula) is u(x,y) = \int_{-\infty}^{\infty} \frac{y\ f(t) }{(t-x)^2 +y^2} \...

For an imperfect conductor, when there is current, an electric field is set up inside the wire along the direction of the current flow, and is parallel to the wire. If this is true, then what I don't understand is boundary condition tells me the tangential E-field is always continuous, if...

I don't really understand boundary conditions and I've been trying to research it for ages now but to no real avail. I understand what boundary conditions are, I think. You need them along with the initial conditions of a wire/string in order to describe the shape of motion of the string. I...

Hi all, I was reviewing some old material on the representation of orientable surfaces in terms of disks and bands , in page 2 of: http://www.maths.ed.ac.uk/~jcollins/Knot_Theory.pdf Please tell me if I am correct here. Assume there is a horizontal line dividing the surface into an...

the boundary conditions on p are homogenous dirichlet on this equation where q(0,τ)=0 and q(l,t)=0 for all τ>0. the initial condition p(x,0)=p_o(x) also translates to an initial condition on p. how do i show what the new initial condition is on q

Homework Statement A tight string lies along the positive x-axis when unperturbed. Its displacement from the x-axis is denoted by y(x, t). It is attached to a boundary at x = 0. The condition at the boundary is y+\alpha \frac{\partial y}{\partial x} =0 where \alpha is a constant. Write the...

Homework Statement I have been given a complex function I have been given a complex function \widetilde{U}(x,t)=X(x)e(i\omega t) Where X(x) may be complex I have also been told that it obeys the heat equation...

Homework Statement A unit sphere at the origin contains no free charge or conductors in its interior or on its boundary. It is, however, embedded in a dielectric medium. The dielectric is linear, but the permitivity varies by angle about the origin. It is constant along any radial direction...

Homework Statement Hello, I have to demonstrate that multiplying a differential equation: -d/dx[a(x)*d/dx{u(x)}]=f(x), 0<x<1 subject to u(0)=0 and u(1)=0. by some function v(x) and integrating over an interval [0,1], I get a new equation that can be used in an optimisation problem, that...

Hi all, Generally boundary condition (Dirichlet and Neumann) are applied on the Load Vector, in FEM formulation. The equation i solved, is Generalized eigenvalue equation for Scalar Helmholtz equation in homogeneous wave guide with perfectly conducting wall ( Kψ =λMψ ), and found, doesn't...

Homework Statement Stuck on two similar problems: "State the normal stress boundary condition at an interface x_3-h(x_1,x_2,t)=0between an invisicid incompressible fluid and a vacuum. You may assume that the interface has a constant tension." The second question in the same but the fluid is...

Homework Statement I am trying to extract the constant of integration after integrating the following stress equation (I got this by solving a system of ODEs for the upper and lower tablets (attached), quite tedious to paste it all here): σ = β * sinh(λx/2) * cosh( λ(L-x)/2 ) Where σ...

This is my first post so I hope this in the right place. I am fairly sure this is quite a straight forward question but I having trouble working out the details of it. "State the boundary conditions for an inviscid fluid at an impermeable fixed boundary x_3-h(x_1,x_3)=0 where we do...

I've been trying to prove that the closed unit ball is a manifold with boudnary, using the stereographic projection but I cannot seem to be able to get any progress. Can anyone give me a hint on how to prove it? Thanks in advance :)