Divergence Definition and 746 Threads

The series from n=1 to infinity log(n/(n+1)). This was on my quiz, which I got wrong. Here's what I did: lim n-->infinity of log(n/(n+1)) so then that becomes: log(lim n-->infinity n/(n+1)) which becomes the log1, which is 0, so it converges. Whats wrong with my steps?

Homework Statement Use Divergence theorem to determine an alternate formula for \int\int u \nabla^2 u dx dy dz Then use this to prove laplaces equation \nabla^2 u = 0 is unique. u is given on the boundary.Homework Equations u \nabla^2 u = \nabla * (u \nabla u) -(\nabla u)^2 The Attempt at...

Homework Statement The problem statement has been attached with this post. Homework Equations I considered u = ux i + uy j and unit normal n = nx i + ny j. The Attempt at a Solution I used gauss' divergence theorem. Then it came as integral [(dux/dx) d(omega)] + integral...

Hi, I have a question about the divergence of forward Coulomb (Bhabha/Moller) scattering. I guess the classical analog of it is the Rutherford cross-section divergence, but that can be explained by the infinite impact parameter. In the QED version - the Bhabha/Moller scattering, it is...

Hi, I have a question about the divergence of forward Bhabha/Moller scattering. I guess the classical analog of it is the Rutherford cross-section divergence, but that can be explained by the infinite impact parameter. In the QED version - the Bhabha/Moller scattering, it is the matrix...

Homework Statement Calculate div v. v= r sin(θ) r + r sin(2θ) cos(φ) θ + r cos(2θ) φ. Homework Equations The Attempt at a Solution I've never had to do a problem like this using spherical coords, so I am not sure where to start. I have the general formula though.

From Partial Differential Equations: An Introduction, by Walter A. Strauss; Chapter 1.5, no.4 (b). Homework Statement "Consider the Neumann problem (delta) u = f(x,y,z) in D \frac{\partial u}{\partial n}=0 on bdy D." "(b) Use the divergence theorem and the PDE to show that...

Hi everyone, so let me introduce the scalar function \Phi = -(x2+y2+z2)(-1/2) which some of you may recognize as minus one over the radial distance from the origin. When I compute \nabla2\Phi is get 0. Now if I do the following integral on the surface S of the unit sphere x2+y2+z2= 1 ...

Homework Statement Show that the following series diverges \sum_{n=1}^{\infty}\frac{n!}{2^{n}} Homework Equations The Divergence Test: In order for a series to be divergent, the following must be true \lim_{n\rightarrow \infty} a_n \neq 0 , or \lim_{n\rightarrow \infty} a_n \nexists...

Homework Statement Check the divergence theorem using the function: \mathbf{v} = y^2\mathbf{\hat{x}} + (2xy + z^2) \mathbf{\hat{y}} + (2yz)\mathbf{\hat{z}} Homework Equations \int_\script{v} (\mathbf{\nabla . v }) d\tau = \oint_\script{S} \mathbf{v} . d\mathbf{a} The Attempt at a Solution...

The D field due to a point charge(Q) has a divergence zero D = (Q/(4*pi*r2)) ar (ar --- unit vector) if we calculate the divergence we get zero. but guass law says that the divergence is a finite value not zero because a charge is present in there...

\operatorname{div}\,\mathbf{F}(p) = \lim_{V \rightarrow \{p\}} \iint_{S(V)} {\mathbf{F}\cdot\mathbf{n} \over |V| } \; dS This is the definition of divergence from wikipedia... The divergence is property of a point in space. Is that right? If the divergence is zero at a point, that...

Determine Div & Curl from a given vector field V=Kyi-Kxj How do I format this? It's been a while since I've done this and every divergence and curl example I look up has the format V(x,y,z)={V1(x,y,z);V2(x,y,z);V3(x,y,z)} Should I reformat my V to be V{x,y}={V1(x,y);V2(x,y)}={Ky,-Kx}...

Homework Statement Evaluate the integral \int\limits_{V=\infty} e^{-r} \left[ \nabla \cdot \frac {\widehat{r}} {r^2} \right] , d^3 xHomework Equations Divergence theorem: \int\limits_{V} \left ( \nabla \cdot A \right ) \, d^3 x = \oint\limits_{S} A \cdot \, da} The Attempt at a Solution I...

look for some proof for the formula of the divergence operator in cylindrical & spherical coordinate is there any on the net ? TNX ! the formula here: http://hyperphysics.phy-astr.gsu.edu/hbase/diverg.html

Homework Statement Let the surface, G, be the paraboloid z = x^2 + y^2 be capped by the disk x^2 + y^2 \leq 1 in the plane z = 1. Verify the Divergence Theorem for \textbf{F}(x,y,z) = 2x\textbf{i} - yz\textbf{j} + z^2\textbf{k} Homework Equations I have solved the problem using the...

Homework Statement This problem takes a bit of background, so please bear with me. The assignment reads: Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one...

So I have a simple/easy to answer question for any physics buffs out there. I think I'm doing something fundamentally flawed. Can you take the inverse of a divergence? analagous to antiderivative-integral? e.g., I want to find J from the continuity equation with a known \rho(\vec{r},t)...

Here are five separate problems. Show that the series 1/3^(ln(n)) converges and that the series 1/2^(ln(n)) diverges. integral (sqrt(x)*e^-sqrt(x)) dx integral (x/(sqrt(x-1)+2)) dx integral (1/(2+sin(x)+cos(x) integral (1-cos(x))^(5/2)) dx There are no other relevant...

Homework Statement I'm getting through a paper and have a few things I can't wrap my head around. 1. In defining the boundary conditions for a membrane (a function of vector 'r'), the author claims that for a small displacement (u) and a boundary movement (f), the boundary condition can be...

I know how to calculate the divergence and curl of a vector field. What I am lacking is any intuition on what these values mean. example: V= {x, y, z} ∇.V = 3 ∇xV = {0,0,0} F={-y, x, 0} ∇.F = 0 ∇xF = {0,0,2} G={0, 3y, 0} ∇.G = 3 I understand that that the divergence is a measure of how much...

I've been told time and again that any beam of light (think laser) exhibits divergence no matter how perfect. This prompts three questions: 1) Theoretically, if a laser beam is emitted such that each photon is exactly parallel, (obviously more perfect than can be achieved in reality) does the...

The Wikipedia article Divergence, in the section Application to The Wikipedia article Divergence, in the section [i]Application to Cartesian coordinates, says of the del-dot formula for divergence, "Although expressed in terms of coordinates, the result is invariant under orthogonal...

Homework Statement 1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2) A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards. B)Verify Gauss's divergence theorem...

I am stuck on this problem. Use these equations: \textbf{v}(\textbf{r}) = f(r)\textbf{r} \frac{\partial r}{\partial x} = \frac{x}{r} And the chain rule for differentiation, show that: (\nabla\cdot\textbf{v}) = 2f(r) + r\frac{df}{dr} (cylindrical coordinates) Any help greatly...

Homework Statement Verify Gauss Divergence Theorem ∭∇.F dxdydz=∬F. (N)dA Where the closed surface S is the sphere x^2+y^2+z^2=9 and the vector field F = xz^2i+x^2yj+y^2zk The Attempt at a Solution I have tried to solve the left hand side which appear to be (972*pi)/5 However, I...

Homework Statement A smooth vector field F has divF(1,2,3) = 5. Estimate the flux of F out of a small sphere of radius 0.01 centered at the point (1,2,3). Homework Equations Cartesian Coordinate Definition of Divergence: If F= F1i + F2j +F3k, then divF=dF1/dx + dF2/dy + dF3/dz The...

Homework Statement I have to prove the divergence formula for plane polars. The question goes something like: Find the divergence of the vector field F(r,t) = Frer + Ftet where r and t are polar coordinates and er = (cos t, sin t, 0) and et = (- sin t, cos t, 0) (t is theta in the...

Currently, we are covering the topic of convergence and divergence of a series in my calculus 2 class. I was wondering if you could give me in there own words what it means for a series to converge, and what it means for a series to diverge. I know that when a series converges, its limit...

hi, can someone tell me how \nabla\dot(\frac{\widehat{r}}{r^2})=4\pi\delta^3(r) thanks

Another question of a practice test. How do I use the Divergence theorem to find the outward flux of the field F = (x3,x2y,xy) out through the surface of the solid U = (x,y,z): 0 < y < 5-z, 0 < z < 4-x2. The answer is 4608/35.

How can one work out what terms like: (g^{cd}R^{ab}R_{ab})_{;d} are in terms of the divergence of the Ricci curvature or Ricci scalar? One student noted that since: G^{ab} = R^{ab} - \frac12 g^{ab}R {G^{ab}}_{;b} = 0 that we could maybe use the fact that G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12...

Homework Statement Ok well all I am told in the question is that the magnetic field B at a distance r from a straight wire carrying current I has magnitude uoI/2pi r.. The lines of force are circles on the wire and in planes perpendicular to it.. Show that divB = 0 Homework...

Homework Statement Suppose \sum n converges and an is greater than 0 for all n. Show that the sum of 1/an diverges. Homework Equations The Attempt at a Solution

I have been contemplating my confusion about my intuition regarding GR and believe I have tracked down the primary source of confusion. The classical theories I have been taught assumed flat space with independent time and used the divergence theorem to derive inverse squared laws for fields...

Hi, in my book, it says: ----------------------- Beacause of T^{\mu\nu}{}{}_{;\nu} = 0 and the symmetry of T^{\mu\nu}, it holds that \left(T^{\mu\nu}\xi_\mu\right)_{;\nu} = 0 ----------------------- (here, T^{\mu\nu} ist the energy momentum tensor and \xi_\mu a killing vector. The semicolon...

Homework Statement WTS divergence of: \sum_{-\infty}^{\infty} \frac{1}{z-n} Homework Equations Basic algebraic manipulation, standard tests of non-convergence? The Attempt at a Solution I have been playing with algebra, perhaps this equivilant (I hope equivilant, anyway)...

I am having a problem with the definition of divergence in improper integrals. My understanding of the logic behind convergence and divergence is that, for example, as a improper integral approaches infinity the area under the function will be approaching but never reach zero. This implies...

Use the divergence theorem to show that \oint\oints (nXF)dS = \int\int\intR (\nablaXF)dV. The divergence theorem states: \oint\oints (n.F)dS = \int\int\intR (\nabla.F)dV. The difference is switching from dot product to cross product. I have no idea how to start. Can someone please point...

Verify the divergence theorem when F=xi+yj+zk and sigma is the closed surface bounded by the cylindrical surface x^2+y^2=1 and the planes z=0, z=1. I've done the triple integral side of the equation and got 3pi but don't know how to solve the flux side of the equation \oint\ointF.ds. Any...

Alright so I found div F=3x2+3y2+3z2 The integral then becomes the triple integral of the divergence of F times the derivative of the volume. Changing into spherical coordinates, the integral becomes 3\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{1}p^{4}sin{\phi}dpd{\phi}d{\theta} which ends up...

Hi, I'm having some trouble understanding this theorem in Lang's book, (pp. 497) "Fundamentals of Differential Geometry." It goes as follows: \int_{M} \mathcal{L}_X(\Omega)= \int_{\partial M} \langle X, N \rangle \omega where N is the unit outward normal vector to \partial M , X...

Hello. How can I show the Divergence of a vector field is a scalar field(in E^{3}) ? Should I show that Div is invariant under rotation? x^{i'}=a^{ij}x^{j},V^{'}_{i}(\stackrel{\rightarrow}{x})=a_{ij}v_{j}(\stackrel{\rightarrow}{x}) then \frac{\partial...

Homework Statement I want to show that: Grad dot product with r = 2/r where: r is the unit vector r/r r = xi + yj + zk r is magnitude of r The Attempt at a Solution I think the answer should be 3/r since unit vector r = r/r, r = (x/r)i + (y/r)j + (z/r)k then when I do...

Homework Statement A vector field F for which div F = 0, is called incompressible (also called solenoidal). Consider the vector field F(x, y, z) = ⟨y, x + y, −z⟩. (a) (1 point) Show that F is incompressible. (b) (3 points) Find a vector field A such that F=\nabla×A.Homework Equations div F =...

\[ \nabla \cdot \frac{{\vec e_r }}{{r^2 }} = 4\pi \delta (\vec r) \] This can be seen from\[ \nabla \cdot \frac{{\vec e_r }}{{r^2 }} = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}(r^2 \cdot \frac{1}{{r^2 }}) = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}(1) = 0(r \ne 0) \] And...

What do Divergence and Curl of a vector function actually mean? They are nice to understand as mathematical operators and then we can work on with them, but what do they mean physically and why are they so important in our study of electromagnetism?

Why is the concept of divergence defined to be the sum of the partial derivatives of the x, y and z components of a vector field E with respect to x, y and z, instead of being defined as the sum of the partial derivatives of E itself with respect to x, y and z? What would this operation I just...

Homework Statement Use Comparison Theorem to determine whether the integral is convergent or divergent: integral from 0 to infinity of: arctan(x) / (2 + e^x) Should look like this: http://bit.ly/cAhytV Homework Equations -- The Attempt at a Solution I tried to compare...

Compute area using divergence and flux?? Consider the curve given by g(t) =acos^3(t),asin^3(t), where t is [0; 2pi] and a > 0 is a constant. (a) Find the unit tangent and outward normal vectors. (b) Compute the area enclosed by this curve. I have done part a), and I know that flux of F...