Divergence Definition and 746 Threads

Use the Ratio Test to determine whether the series is convergent or divergent $$\sum_{n=1}^{\infty}\dfrac{(-2)^n}{n^2}$$ If $\displaystyle\lim_{n \to \infty} \left|\dfrac{a_{n+1}}{a_n}\right|=L>1 \textit{ or } \left|\dfrac{a_{n+1}}{a_n}\right|=\infty...

Hello all, I was working on some homework regarding testing for convergence and divergence of series and I was having trouble with a particular series (doesn't really matter which one) and tried almost all the methods; then tried the Integral Test, my series met the conditions of the...

If the comma means ordinary derivative, then ##(A_\mu A_\nu^{,\nu} - A_\mu^{,\nu} A_\nu)^\mu = A_\mu^{,\mu}A_\nu^{,\nu} + A_\mu A_\nu^{,\nu,\mu} - A_\mu^{,\nu,\mu} A_\nu - A_\mu^{,\nu}A_\nu^{,\mu} = A_\mu^{,\mu}A_\nu^{,\nu} - A_\mu^{,\nu}A_\nu^{,\mu} ##, where ##A## is some vector field...

What is the physical significance of fundamental law del.E=0 in free space ?

If within a volume v ,there exists 10 velocity fields at different points then can anyone please suggest how to compute ##\int_v(\nabla•v)## within the volume?? using matlab For exm if the velocity vector field be ##v=x\hat x+y\hat y+z\hat z## and for x=1 to 10,y=1 to 10 and z= 1 to 10 the 10...

Assume that ##\partial M_{ab}/\partial \hat{n}_c## is completely symmetric in ##a, b## and ##c##. Then, it is stated in the book I read that the divergence of the traceless part of ##M## is proportional to the gradient of the trace of ##M##. More precisely, $$ \partial /\partial \hat{n}_a...

In section 1-5 of the third edition of Foundations of Electromagnetic Theory by Reitz, Milford and Christy, the authors give a coordinate-system-independent definition of the divergence of a vector field: $$\nabla\cdot\mathbf{F} = \lim_{V\rightarrow 0}\frac{1}{V}\int_S\mathbf{F\cdot n}da$$...

Given $$\vec E = -\nabla \phi$$ there $$\vec d \rightarrow 0, \phi(\vec r) = \frac {\vec p \cdot \vec r} {r^3}$$ and ##\vec p## is the dipole moment defined as $$\vec p = q\vec d$$ It's quite trivial to show that ##\nabla \times \vec E = \nabla \times (-\nabla \phi) = 0##. However, I want to...

As far as I can tell the divergence theorem might be one of the most used theorems in physics. I have found it in electrodynamics, fluid mechanics, reactor theory, just to name a few fields... it's literally everywhere. Usually the divergence theorem is used to change a law from integral form to...

I am checking the divergence theorem for the vector field: $$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$ The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2## This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...

From gauss divergence theorem it is known that ##\int_v(\nabla • u)dv=\int_s(u•ds)## but what will be then ##\int_v(\nabla ×u)dv## Any hint?? The result is given as ##\int_s (ds×u)##

Homework Statement I was working on a problem from Maxwell Equations. Why is the below zero? Homework EquationsThe Attempt at a Solution

according to continuity equation (partial ρ)/(partial t) +divergence J = 0 . there is such a situation that there is continuous water spreads out from the center of a sphere with unchanged density ρ, and at the center dm/dt = C(a constant), divergence of J = ρv should be 0 anywhere except the...

Homework Statement Test the following series for convergence or divergence. ##\sum_{n = 1}^{\infty} \frac {\sqrt n} {e^\sqrt n}## Homework Equations None that I'm aware of. The Attempt at a Solution I know I can use the Integral Test for this, but I was hoping for a simpler way.

How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)

One way to get the gradient of polar coordinates is to start from the Cartesian form: ##\nabla = \hat x \frac{\partial}{\partial x} + \hat y \frac{\partial}{\partial y}## And then to use the following four identies: ##\hat x = \hat r\cos\theta - \hat{\theta}\sin\theta## ##\hat y = \hat...

I want to visualize the concept of divergence of a vector field.I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a...

I am looking for some resources describing the following content: A light with wavelength ##\lambda## is propagating in flat spacetime. The light redshifts as its wavelength gets larger and larger. In quantum field theory, this causes an infrared divergence of the field. What I want to know...

1. The problem statement In calculating the amplitude for the diagram[1], view 1.jpg. [1] Voja Radovanovic, Problem Book Quantum Field Theory 2. Homework Equations View 2.jpg. The Attempt at a Solution View 3.jpg.[/B] Why the integrals is divergent? Why the other terms are finite?

Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...

I am dealing with an expression in a large amount of literature usually presented as: \frac{\partial}{\partial \phi_i}\left(\nabla \phi_i \cdot \nabla \phi_j \right) I'm looking at tables of vector calculus identities and cannot seem to find one for the exact expression given, even if I...

If I have a vector field say ## v = e^{z}(y\hat{i}+x\hat{j}) ##, and I want to calculate the divergence. Do I only take partial derivatives with respect to x and y (like so, ## \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} ##) or should I take partial derivatives with respect...

Homework Statement prove (5-n^2)/(3n+1) diverges to negative infinity as n approaches infinity Homework Equations For all M>0 there exists an N in the natural numbers such that for all n >= N, x_n <= -M The Attempt at a Solution Let M be an element of the field of the real numbers. Let N in...

Hello, I was given f(-4x)= 1/(1+4x), and I used the geometric series to find the power series representation of this function. I then took the limit of (-4x)^k by using ratio test. The answer is abs. value of x. So -1/4<x<1/4 I then plugged in those end points to the series going from k=0 to...

Homework Statement Prove sequence ((-1)^m)m diverges. Homework Equations for all epsilon greater than zero, there exists a natural number M such that for all natural numbers m greater than or equal to M, Ix_m-xI is less than or equal to epsilon.[/B]The Attempt at a Solution Assume it...

Hello I have a question if it possible, Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M. Is it true that X(exp-f)=-exp(-f).X(f) And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)? Thank you

I've been thinking about this problem and would like some clarification regarding the value of the divergence at a theoretical point charge. My logic so far: Because the integral over all space(in spherical coordinates) around the point charge is finite(4pi), then the divergence at r=0 must be...

The general definition for a sequence to diverge is the negation of what it means for a sequence to converge: ##\forall L\in\mathbb{R}~\exists\epsilon>0~\forall N\in\mathbb{N}~\exists n\ge N##, ##|a_n - L| \ge \epsilon##. How does this general definition of divergence relate to the definition of...

For the vector valud function F in the image, the three components of the output vector at a point are functions of (x,y,z)the three coordinates of the point.But while calculating divergence, why is the rate of change of x component of the output along x direction alone is accounted(similarly...

Homework Statement [/B] ∇ * B = 0 and ∇ X B = Mu * J. This is proved to hold not only for infinite wires but for magnetostatics in general. Magnetostatics = steady current Closed wire loop with constant current is certainly a magnetostatics example. Magnetic field on z axis above loop around...

This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam: There are two cases but the excercise is pretty much the same: Compute $$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...

If I am given a function f( x , y , z , ...) = C then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?

Homework Statement I have in the picture attached a screenshot from Peskin's textbook. I don't understand how did they get that for the two last diagrams that ##D=-2##. The question is from pages 316-317 of Peskin's textbook. Homework Equations $$D=4-N_{\gamma}-3/2N_e$$ where ##N_e##=number of...

I need to prove that in a vacuum, the energy-momentum tensor is divergenceless, i.e. $$ \partial_{\mu} T^{\mu \nu} = 0$$ where $$ T^{\mu \nu} = \frac{1}{\mu_{0}}\Big[F^{\alpha \mu} F^{\nu}_{\alpha} - \frac{1}{4}\eta^{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}\Big]$$ Here ##F_{\alpha...

Hi, maybe as you know ##\nabla. J = -\frac {\partial p} {\partial t}## where J is current density p is charge density. But also we know current density flux outward the circuit is 0 because current density does not flow out of circuit an this actually volume integral of ##\nabla. J## is zero (...

Homework Statement Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent. Homework Equations We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent. The Attempt at a Solution Applying the ratio test, we find that...

Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...

Homework Statement I have to calculate the partial derivative of an arctan function. I have started to calculate it but I wonder if there is any simpler form, because if the simplest solution is this complex then it would make my further calculation pretty painful... Homework Equations $$\beta...

Hello PF, I was reading through “A First Course in General Relativity” by Schutz and I got to the part where he derives the divergence formula for a vector:$$V^α { } _{;α} = \frac {1} {\sqrt{-g}} ( \sqrt{-g} V^α )_{,α}$$I’m having trouble with a couple of the steps he made. So we start with the...

Homework Statement Griffiths Introduction to Electrodynamics 4th Edition Example 1.10 Check the divergence theorem using the function: v = y^2 (i) + (2xy + z^2) (j) + (2yz) (k) and a unit cube at the origin. Homework Equations (closed)∫v⋅da = ∫∇⋅vdV The flux of vector v at the boundary of the...

Basically a case where a positive charge q is placed in space which for convenience is taken as the origin. This electric field must have a large positive divergence but yet when evaluated mathematically we get 0. Also when we find divergence, we find it for a point right ? or is it possible to...

Homework Statement Show that $$div ( \frac{x}{|x|^2} ) = 2 \pi \delta_{(0,0)}$$ with ## x \in R^2 \ \{ 0 \} ## and ## \delta_{(0,0)} ## beeing the dirac delta distribution with pole in ## (0,0) ##. Homework Equations ## div (f(x)) = \nabla \cdot (f(x)) = f_{x_1} + f_{x_2} ## The distribution...

whyT^[ab][;b]≠T_[ab][;b] for spatially flat FLWR cosmology ((ds)^2=(c^2)* (dt)^2-a(t)^2[(dx)^2+(dy)^2+(dz)^2])? τ[ab][/;b] gives the right answer, but not τ[ab][/;b]. (T^(ab) or T_(ab)) contra-variant and co-variant energy momentum tensor of perfect fluid (;) covariant derivative, (c) spped of...

Homework Statement Show that the sequence with two distinct accumulation points must diverge. (Hint: look at the proof of divergence for {##(-1)^k##}. Homework Equations Some definitions and propositions I'm trying to use: 2.2.3: A sequence cannot converge to two different numbers. If...

I just learned that an incompressible fluid must have zero divergence within a given control volume. Given that the divergence of a fluid at a point(x,y,z) can be found by taking the scalar sum of the of the x, y, z acceleration vectors at the given point, wouldn't this mean that water flowing...

Hi Folks, Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the...

Homework Statement If the sum of a sub n to infinity (n=1) converges then the limit of n as n tends to infinity of an = 0 Homework EquationsThe Attempt at a Solution an =(a1+a2+...an)-(a1+...+an-1) = limit of an (n tends to infinity) = sn -s(n-1) =0 The area I'm confused is why do we assume...

1. Problem: Consider vector field A##\left( \vec r \right) = \frac {\vec n} {(r^2+a^2)}## representing the electric field of a point charge, however, regularized by adding a in the denominator. Here ##\vec n = \frac {\vec r} r##. Calculate the divergence of this vector field. Show that in the...

Prove that the sequence :$$(-1)^n$$ diverges by using the ε-definition of the limit of a sequence

Consider the following integral $$\int \frac{d^4k}{k^2}$$ It is UV divergent but is it IR finite or IR divergent? The integrand is singular as ##k \rightarrow 0## so this suggest an IR divergence but this is no longer the case if I make a shift of the loop momenta by say ##p_1## and write the...