Divergence Definition and 746 Threads

Homework Statement Verify the divergence theorem for the function V = xy i − y^2 j + z k and the surface enclosed by the three parts (i) z = 0, s < 1, s^2 = x^2 + y^2, (ii) s = 1, 0 ≤ z ≤ 1 and (iii) z^2 = a^2 + (1 − a^2)s^2, 1 ≤ z ≤ a, a > 1. Homework Equations [/B]...

Dear All, I'm doing some tensor calculation on the divergence of gradient (of a vector) inverse. Am I allowed to first use the nabla operator on gradient and then inverse the whole product? In other words, I'm searching for the divergence of a 2nd order tensor which is itself inverse of...

Suppose I have electric field of the form ##\mathbf{E} = 3x\mathbf{i} + 3y\mathbf{j}##. Calculating the charge density gives me ##\rho = \epsilon_0 \nabla\cdot\mathbf{E} = 6\epsilon_0##. But now if I turn one of the components of the field in the opposite direction, for example ##\mathbf{E} =...

What is the intuition behind divergences for the sunset diagram? I know that there is quadratic divergence by why no quartic divergence or higher?

Homework Statement For a volume charge, ##\textbf{E}(\textbf{r}) = \frac{1}{4\pi\epsilon_0}\int_{all space}\frac{\hat{\gamma}}{\gamma^2}\rho(r')d\tau'## and I am trying to get the divergence of it. Homework Equations The book says ##\nabla\cdot\textbf{E} = \frac{1}{4\pi\epsilon_0}\int_{all...

eq.1 eq2. eq.3 eq.4Hello, I have a question about eq.4 If we find the closed surface flux integral of J, would it be current?

Homework Statement If ##\vec{E}=k\frac{x\hat i +y\hat j}{x^2+y^2}##, find flux through a sphere of radius R centered at origin. Homework Equations ##\int E.da=\int(\nabla\cdot E)\cdot da## The Attempt at a Solution I was able to solve this problem without finding divergence of electric field...

Homework Statement Using the fact that \nabla \cdot r^3 \vec{r} = 6 r^2 (where \vec{F(\vec{r})} = r^3 \vec{r}) where S is the surface of a sphere of radius R centred at the origin. Homework Equations \int \int \int_V \nabla \cdot \vec{F} dV =\int \int_S \vec{F} \cdot d \vec{S} That is meant...

Hi everyone My professor just asked us a question that I can't get my head around. So we have the original vector in Cartesian format, <y^2,z^2,x^2> Then I am asked to convert to cylindrical coordinates: z= z; θ==arctan(z^2/y^2); r = \sqrt(y^4+z^4) However , I am then asked to take the...

Hi everyone, I am wondering if anybody could help me out. For my study I got the following question but I got stuck in part C (see image below). I Found at A that due to symmetry all components which are not in Ar direction will get canceled out I found at B that there is only charge density at...

Homework Statement The problem is given in the following photo: Actually I did the first proof but I couldn't get the second relation. (Divergence of E cross H). Homework Equations They are all given in the photo. (a) (b) and (c). The Attempt at a Solution What I tried is to interchange...

Homework Statement I need to show that $$\del*\vec{A(\vec{r})}=\frac{\mu}{4\pi}\int{\frac{\vec{J{vec\r'}}}{\vec{R}}}d\tau=0$$ where A is the vector potential and R refers to "script r" or (r-r') where r is source point of charge and r' is the measurement point. tau refers to a volume integral...

Homework Statement The problem puts forth and identity for me to prove: or . It says that I can use "straight-forward" calculation to solve this using the definition of nabla or I can use Gauss's and Stoke's Theorum on an example in which I have a solid 3D shape nearly cut in two by a curve...

Hi, The Hamiltonian for the free scalar field, expressed in terms of the creation/annihilation operators, is H = \int d^{3}p [\omega_p a^{\dagger}_p a_p + \frac{1}{2}\omega_p \delta^{3}(0)] \hspace{3mm} I thought: \omega_p is a function of p as \omega^{2}_p = |p|^{2} + m^2 and so the...

Hi guys! I am using many different sources to self teach myself about divergence. I understand it, however there is one thing that is confusing me. For example, a divergence of 0 could mean that i, j, and k don't change at all, or it could mean that i changes by 1, j by -1, and k by zero (or...

Hello, I've been struggling with this question: Let q be a constant, and let f(X) = f(x,y,z) = q/(4pi*r) where r = ||X||. Compute the integral of E = - grad f over a sphere centered at the origin to find q. I parametrized the sphere using phi and theta, crossed the partials, and got q, but I...

Hi! I have recently been independently studying vector calculus. I understand that divergence measures change in magnitude and curl is the change in direction, however, I don't understand what certain divergences and curls represent. For example, how would you describe a field with a divergence...

So we all know the divergence/Gauss's theorem as ∫ (\vec∇ ⋅ \vec v) dV = ∫\vec v \cdot d\vec S Now I've come across something labeled as Gauss's theorem: \int (\vec\nabla p)dV = \oint p d\vec S where p is a scalar function. I was wondering if I could go about proving it in the following way...

Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it. Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result: \vec...

Why is the divergence of electric field not zero for a material with non-constant conductivity?

So my question here is: the divergence theorem literally states that Let \Omega be a compact subset of \mathbb{R}^3 with a piecewise smooth boundary surface S. Let \vec{F}: D \mapsto \mathbb{R}^3 a continously differentiable vector field defined on a neighborhood D of \Omega. Then...

What is the difference between \int_{-\infty}^{\infty} \frac{x}{1+x^2}dx and \lim_{R\rightarrow \infty}\int_{-R}^{R} \frac{x}{1+x^2}dx ? And why does the first expression diverge, whilst the second converges and is equal to zero?

Homework Statement [/B] Here is an nth term test for determining divergence, I think I have it, but wanted another opinion -- 1/34 + 1/35 + 1/36+ … + 1/1,000,034 -- IHomework Equations ∑(upper limit ∞)(lower limit n=0) 1/(n+34) The Attempt at a Solution 1/34 + 1/35 + 1/36+ … + 1/1,000,034...

Homework Statement I am suppose to calculate the relative entropy between two sets of data: Base set Set 1: A C G T 0 0 0 10 0 0 0 10 0 0 10 0 0 10 0 0 10 0 0 0 * * * * //Randomized 0 0 0 10 0 10 0...

I have the integral 1/(x^0.25 - 2) dx between 500 to 16, and am trying to find whether it converges or diverges. I have sketched the graph and noticed that their is an asymptote at x=16 (hence why the integral is improper for these boundaries). I am now trying to evaluate the limits to see if...

If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0 I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...

How can I find out if 1/n! is divergent or convergent? I cannot solve it using integral test because the expression contains a factorial. I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using...

Homework Statement ∞ ∑ tan(1/k) k=5 show that it is convergent or divergent Homework EquationsThe Attempt at a Solution i used ratio test, but it's equal to 1, it means no works... i used divergence test, it equals to 0, no work too... so what should i do? i don't know how to use...

Homework Statement Homework EquationsThe Attempt at a Solution I thought of using the divergence theorem where I find that ∇.F = 3z thus integral is ∫ ∫ ∫ 3z r dz dr dθ where r dz dr dθ is the cylindrical coordinates with limits 0<=z<=4 0<=r<=3 0<=θ<=2π and solving gives me 216π Can I...

Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...

Homework Statement If lim(k>inf) 1/k, goes to 0, why does it diverge? Homework Equations Divergent series test The Attempt at a Solution i don't understand why 1/k (harmonic series) diverges, when according to the divergent series test, it should converge to 0. [/B]

Homework Statement (3 i) Using \nabla . \mathbf{F} = \frac{\partial \mathbf{F_{\rho}}}{\partial \rho} + \frac{\mathbf{F_{\rho}}}{\rho} + \frac{1}{\rho} \frac{\partial \mathbf{F_{\phi}}}{\partial \phi} + \frac{\partial \mathbf{F_{z}}}{\partial z} calculate the divergence of the vector field...

Homework Statement Let ν(x,y,z) = (xi + yj + zk)rk where v, i, j, k are vectors The k in rk∈ℝ and r=√(x2+y2+z2). Show that ∇.v=λrk except at r=0 and find λ in terms of k. Homework Equations As far as I understand it, ∇.v=∂/∂x i + ∂/∂y j + ∂/∂z k, but this may very well be wrong. The Attempt...

I am having a problem with this concept, when looking at the fields from a point source. My problem is that the field gets weaker the further it gets from the source, so at any point away from the source should there not be more entering that point than leaving it, and so have a negative...

Hello All, If I apply the Divergence Operator on the incompressible Navier-Stokes equation, I get this equation: $$\nabla ^2P = -\rho \nabla \cdot \left [ V \cdot \nabla V \right ]$$ In 2D cartesian coordinates (x and y), I am supposed to get: $$\nabla ^2P = -\rho \left[ \left( \frac...

Hi, I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them. This works great. I don't know if what I am...

Homework Statement determine series convergence of divergence summation (n=1 to infinity) n/n^2 +1 Homework EquationsThe Attempt at a Solution I take the limit comparison limit (1/n)/ (n/(n^2 +1) =1 for 1/n if i use p series the series diverge if i use the method to take limit of sequence...

Homework Statement Does sum from n=1 to n=infinity of 1/[n^(1+1/n)] converge or diverge. Homework Equations ^^^^^^^^^^^^^^^ The Attempt at a Solution The general term goes to 0 and its a p-series with p>1, but for large n the series becomes 1/n pretty much so, even tho p>1 is it divergent?

Homework Statement Prove Green's theorem \int_{\tau} (\varphi \nabla^{2} \psi -\psi\nabla^{2}\varphi)d\tau = \int_{\sigma}(\varphi\nabla\psi -\psi\nabla\varphi)\cdot d\vec{\sigma} Homework Equations div (\vec{V})=\lim_{\Delta\tau\rightarrow 0} \frac{1}{\Delta\tau} \int_{\sigma} \vec{V} \cdot...

Homework Statement Does \frac{2^{n}}{n!} converge or diverge? The Attempt at a Solution Is there more than one way to prove this? I would appreciate a few directions. I've been trying the Squeeze theorem for a long time. I said 1/n! was smaller, but I have no damn idea how to say what's...

In Jackson's 'classical electrodynamics' he re-expresses a volume integral of a vector in terms of a moment like divergence: \begin{align}\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot \mathbf{J} ) d^3 x\end{align} He calls this change "integration by parts". If this...

Homework Statement So my question was Sum- (n=2) ln(n)/n Homework Equations I noticed that you can only limit comparison, because so far, I have tried doing all the other test such as the nth term test, p-series, integral(i have no idea how to integrate that). The Attempt at a Solution

Homework Statement Homework Equations The Attempt at a Solution (a)[/B] Divergence of a gradient is a Laplacian. (b) I suppose to do it in Cartesian coordinates. Let \nabla=\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z} and...

Homework Statement Find the flux of the field F(x) = <x,y,z> across the hemisphere x^2 + y^2 + z^2 = 4 above the plane z = 1, using both the Divergence Theorem and with flux integrals. (The plane is closing the surface) Homework Equations The Attempt at a Solution Obviously, the divergence...

I'm trying to figure out what the physical meaning of divergence is for a vector field. My textbook offered the following example: if v = <u, v, w> represents the velocity field of a fluid flow, then div(v) evaluated at P = (x, y, z) represents the net rate of the change of mass of the fluid...

Homework Statement Find the outward flux of the radial vector field F(x,y,z) = x i^ + y j^ + z k^ through the boundary of domain in R^3 given by two inequalities x^2 + y^2 + z^2 ≤ 2 and z ≥ x^2 + y^2. Homework Equations Divergence theorem: ∫∫_S F ⋅ n^ = ∫∫∫_D div F dV The Attempt at a...

I'm learning time-dependent Maxwell's Equations and having difficulty understanding the following derivative: Given f(\textbf{r}, \textbf{r}', t) = \frac{[\rho(\textbf{r}, t)]}{|\textbf{r} - \textbf{r}'|} where \textbf{r} = x \cdot \textbf{i} + y \cdot \textbf{j} + z \cdot \textbf{k}, in...

Homework Statement This problem came when I was learning the Poisson's equation (refer to http://farside.ph.utexas.edu/teaching/em/lectures/node31.html). when it came to the step to find the Green's function G which satisfies \nabla^2 \cdot G(\textbf{r}, \textbf{r}') =...

Homework Statement Hi, I'm trying to show the four divergence of the stress energy tensor of the sourceless klein gordon equation is zero. I got to the part in the solution where I am left with the equations of motion which is identically zero and 3 other terms. I managed to find a solution...

I'm exploring the divergence theorem and Green's theorem, but I seem to be lacking some understanding. I have tried this problem several times, and I am wondering where my mistake is in this method. The problem: For one example, I am trying to find the divergence of some vector field from a...