Divergence Definition and 775 Threads

Homework Statement Let V = (sin(theta)cos(phi))/r Determine: (a) ∇V (b) ∇ x ∇V (c) ∇∇V Homework Equations The Attempt at a Solution Uploaded

Homework Statement Evaluate the divergence of the following vector fields (a) A= XYUx+Y^2Uy-XZUz (b) B= ρZ^2Up+ρsin^2(phi)Uphi+2ρZsin^2(phi)Uz (c) C= rUr+rcos^2(theta)Uphi Homework Equations The Attempt at a Solution Uploaded

Hey guys! So I've been trying to get my head around Divergence of a vector field. I do get the general idea, however I thought of a hypothetical situation I can't get my head around. Look at the second vector field on this page, http://mathinsight.org/divergence_idea it has a negative...

Homework Statement Find the divergence of \vec v = \frac{\hat{v}}{r} Then use the divergence theorem to look for a delta function at the origin. Homework Equations \int ∇\cdot \vec v d\tau = \oint \vec v \cdot da The Attempt at a Solution I got the divergence easy enough...

Let ##\vec {F}(\vec {r}')## be a vector function of position vector ##\vec {r}'=\hat x x'+\hat y y'+\hat z z'##. I want to find ##\nabla\cdot\frac {\vec {F}(\vec {r}')}{|\vec {r}-\vec{r}'|}##. My attempt: Let ##\vec {r}=\hat x x+\hat y y+\hat z z##. Since ##\nabla## work on ##x,y,z##, not...

Let ##\vec {F}(\vec {r}')## be a vector function of position vector ##\vec {r}'=\hat x x'+\hat y y'+\hat z z'##. Question is why: \nabla\cdot\vec {F}(\vec{r}')=\nabla\times\vec {F}(\vec{r}')=0 I understand ##\nabla## work on ##x,y,z##, not ##x',y',z'##. But what if \vec {F}(\vec {r}')=\frac...

I want to verify: \vec A=\hat R \frac{k}{R^2}\;\hbox{ where k is a constant.} \nabla\cdot\vec A=\frac{1}{R^2}\frac{\partial (R^2A_R)}{\partial R}+\frac{1}{R\sin\theta}\frac{\partial (A_{\theta}\sin\theta)}{\partial \theta}+\frac{1}{R\sin\theta}\frac{\partial A_{\phi}}{\partial \phi}...

Hello, I am approaching the end of my multivariable/ vector analysis "Calc III" class and have a question about flux. My book states that flux, ∫∫ F \bullet N dS measures the fluid flow "across" a surface S per unit time. Now, the divergence theorem ∫∫∫ divF dV measures the "same...

Homework Statement Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges find its limit. an = (1*3*5*...*(2n-1))/(2n)n Homework Equations lim n->infinity an = L The Attempt at a Solution The answer in the book shows: 1/2n *...

I would like to take divergence of the following expression ∇.((xi × xj × xk)/r^3), which is a triadic. here × denote a dyadic product and r=mod(r vector) and xi, xj and xk are the components of r vector. So, the above eq. can also be written as ∇.((xixjxk ei×ej×ek)/r^3), where ei...

Homework Statement This is the problem: http://i.imgur.com/AtISqng.jpg its the first series, not the second one with the cos Homework Equations The Attempt at a Solution so anyways i did this question but i just had one doubt. in every other question like this that I've done, the series...

What is the theoretical connection intuitively justifying that the trace of the jacobian=the divergence of a vector field? I know that this also equals the volume flow rate/original volume in the vector field but leaving that aside, what is the mathematical background behind establishing this...

In peskin at page 319 right above equation (10.6) he writes "If the constant term in a taylor expansion of the self energy were proportional to the cutoff ##\Lambda##, the electron mass shift would also have a term proportional to ##\Lambda##. But the electron mass shift must actually be...

Homework Statement Compute the flux of \vec{F} through z=e^{1-r^2} where \vec{F} = [x,y,2-2z]^T and r=\sqrt{x^2+y^2} . EDIT: the curve must satisfy z\geq 0 .Homework Equations Divergence theorem: \iint\limits_{\partial X} \Phi_{\vec{F}} = \iiint\limits_X \nabla\cdot\vec{F}\,dx\,dy\,dz...

Homework Statement A surface S in three dimensional space may be specified by the equation f(x, y, z) = 0, where f(x, y, z) is a real function. Show that a unit vector nˆ normal to the surface at point (x0, y0, z0) is given by Homework Equations The Attempt at a Solution r...

Is it safe to say when an integral has an infinite boundary \int_n^∞ a_{n} and the limit yields a finite number, then the integral is said to converge. And when a series has an upper limit of infinity \sum_n^{∞}a_{n} and the limit yields a finite number, then the series is said to diverge.

Alright, I have my final for Calc 2 on Monday. I am only stuck on two sections of problems because I am terrible at convergence and divergence/alternating series. I have questions for each one really. They are as follows: 4) a. Can I simply factor out the alternating series and apply a...

Homework Statement Verify the divergence theorem by computing both integrals for the vector field F = <x^3, y^3, z^2> over a cylindrical region define by x^2+y^2 ≤ 9. Homework Equations Divergence Theorem, and Flux Integrals. The Attempt at a Solution I did the divergence...

Homework Statement Find The Divergence Of The Vector Field: < ex2 -2xy, sin(y^2), 3yz-2x> Homework Equations I know that divergence is ∇ dot F. The Attempt at a Solution When I did it by hand I got 2xex2 + 2ycos(y2) + 3y However wolfram alpha says it should be 2xex2 +...

I understand how to derive the divergence in polar. Ok, so I have the formula. But what I am confused about is this: say u=(0,0,w(r)), so as you can see the third component of this vector u is in fact a function of r. If I plug this into the polar divergence formula, I get zero, fine. But...

Homework Statement We have R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \equiv 0 Show that by multiplying above with g^{im}g^{kn} we'll get \left( R^{ik}-\frac{1}{2} g^{ik} R \right)_{;k} 2. The attempt at a solution g^{im}g^{kn} \left( R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \right) \equiv 0...

Hey. I want to use integrals-math to get from Gauss law in divergence form to the one in integral form. I know you can do it by simply accepting ∇*E dV = ρ/ε => ∫ ∇*E dV= ∫ρ/εdV = Q/ε = ∫E*dA, but I want to do it another way. I want to begin with ∫∇*E*dV and end up with Q/ε. So: E =...

Homework Statement Not really a problem, more of a general question. When exactly can you use the Divergence test. Does it only work on both series and sequences?Homework Equations The series Diverges if lim ƩAn ≠ 0 The Attempt at a Solution If you take the lim of the series n^3/2n^3 ≠ 0 there...

Homework Statement Determine either absolute convergence, conditional convergence or divergence for the series.Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5}The Attempt at a Solution It converges conditionally i know, but i can't figure out how. 1. I applied the...

(My question is simpler than it looks at first glance.) Here is Reynolds Transport Theorem: $$\frac{D}{Dt}\int \limits_{V(t)} \mathbf{F}(\vec{x}, t)\ dV = \int \limits_{V(t)} \left[ \frac{\partial \mathbf{F}}{\partial t} + \vec{\nabla} \cdot (\mathbf{F} \vec{u}) \right] \ dV$$ where boldface...

Homework Statement Let ##\mathit{F}(x,y,z) = (e^y\cos z, \sqrt{x^3 + 1}\sin z, x^2 + y^2 + 3)## and let ##S## be the graph of ##z = (1-x^2-y^2)e^{(1-x^2-3y^2)}## for ##z \ge 0##, oriented by the upward unit normal. Evaluate ##\int_{S} \mathit{F} \ dS##. (Hint: Close up this surface and use the...

Prove that the series \displaystyle\sum_{k=1}^{\infty}\sqrt[k]{k+1}-1 diverges. I thought that I could show the n^{th} term was greater than \frac{1}{n} but this is turning out to be more difficult than I imagined. Is there a neat proof that n^n>(n+1)^{n-1}?

Homework Statement The divergence theorem states that ∫∫∫V div F dV = ∫∫S F(dot)Ndσ Suppose that div F = 1, then ∫∫∫V div F dV = ∫∫S F(dot)Ndσ If divF = 2, does the following hold true?∫∫∫V div F dV = 2∫∫S F(dot)Ndσ Homework Equations Since the divergence theorem computes the volume, if...

Is it possible for a spherically symmetric field, on all of R^3, to have a divergence of 0? (assuming the field is nonzero) Relevant equation: F=f(ρ)a (a is a unit vector of <x,y,z>) and f(ρ) is scalar fxn, and ρ = lal

I'm trying to understand when a vector field is equal to the curl of a vector potential. Why is it possible that there is always a vector potential with zero divergence? Relevent Equation: B=∇χA I'm trying to understand the proof that the above vector potential A can be one with zero divergence.

Hi to all Homework Statement ∫∫∫∇ψdv = ∫∫ψ ds over R over S R is the region closed by a surface S here dv and ψ are given as scalars and ds is given as a vector quantitiy. and questions asks for establishing the gradient theorem by appliying the divergence theorem to each component...

\nabla \cdot \frac{\mathbf{r}}{|r|^3}=4 \pi \delta ^3(\mathbf{r}) What's the proof for this, and what's wrong with the following analysis? The vector field \frac{\mathbf{r}}{|r|^3}=\frac{1}{r^2}\hat{r} can also be written \mathbf{F}=\frac{x}{\sqrt{x^2+y^2+z^2}^3}\hat{x}+...

I need to prove the identity: \nabla(\vec{A} \times \vec{B})=\vec{B} \bullet(\nabla \times \vec{A}) - \vec{A} \bullet( \nabla \times \vec{B}) I need to prove for an arbitrary coordinate system, meaning I have scaling factors. The proof should be quite straight forward if you use the levi...

I am new to tensor algebra. I have an expression involving a 2nd rank tensor (actually a dyadic) and a vector. I want to take divergence of the product i.e. ∇. (T.V) However, I am not sure if the simple product rule would work here. If I use that ∇. (T.V)= (∇.T).V + T. (∇.V)...

Simple question: Are there any series which we don't know whether or not they converge?

Hi! The velocity field as a function of poisition of an incompressible fluid in a uniform acceleration field, such as a waterfall accelerated by gravity can be found as follows: The position is \vec{x}. The velocity field is \vec{v} = \frac{d\vec{x}}{dt}. The constant acceleration field...

Hi can anyone explain what a quadratic divergence is? and if so how it effects the mass of the scalar field i.e why m^2 = m^2_{0} + \delta m^2, why are these things squared? Also how would this divergence affect the standard model as a natural concept, because from reading books it would...

Hi guys, this is in regards to a problem from Wald from the section on linearized gravity. We have a quantity t_{ab} very, very similar to the L&L pseudo tensor and have the quantity (a sort of total energy) E = \int_{\Sigma }t_{00}d^{3}x where \Sigma is a space - like hypersurface of a...

I see identity in one mathematical book div \vec{A}(r)=\frac{\partial \vec{A}}{\partial r} \cdot grad r How? From which equation?

Homework Statement I'm trying to understand where the Cartesian components of the rotor and the divergence of a vector field derived. I read that the divergence of a vector field is defined by: \vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V...

Hi guys, I've run across a problem. In finding the potential energy between two electrical quadrupoles, I've come across the expression for the energy as follows: U_{Q}=\frac{3Q_{0}}{4r^{4}}\left[(\hat{k}\cdot \nabla)(5(\hat{k}\cdot \hat{r})^3-2(\hat{k}\cdot \hat{r})^2-(\hat{k}\cdot...

I have a larger problem involving divergences and curls, but the correct answer requires ∇°B (divergence of B) = 0. I understand the proof of this in Griffiths, but the definition of divergence in cylindrical coordinates is: After using the product rule to split the first term, we get the...

Homework Statement \sum_{n=1}^{\infty}{(-1)^n*n} The Attempt at a Solution Well, this is a series I came upon when analyzing the endpoints of a particular power series, the thing is, my book says it's divergent by the Test for divergence, however, I can't find this result, what I tried to do...

Homework Statement A cube of side 5cm is placed in a fluid. For the following velocity values, calculate the average divergence of the flow. Is there a source or sink within this box? The velocities are given in cm/s. The cube is oriented so that the +z axis exits the front face, the +x axis...

Hello, I am new to calculus, and am having problems with divergence, I was reading something to explain the physical interpretation of divergence and i got stuck in the very first part. it says that if we have a small volume dxdydz at the origin, and that a fluid flowing into this volume...

Homework Statement if a vector can be written as the curl of another vector, its divergence vanishes. Can you justify the statement: "any vector field whose divergence vanishes identically can be written as the curl of some other vector"? Homework Equations Prove this by construction. Let...

Hey guys, I have a general question about example 4 in section 16.8 of the book "Thomas' Calculus, Early Transcedentials". So far I understand the material given in the book without any problems but this particular example is a little bit unclear to me. Homework Statement Given a vector field...

Homework Statement I'm trying to find the divergence of a vector field (a fluid flow vector), but the vector takes the form u = u(x,y,z,t) The Attempt at a Solution I only really know how to take the divergence of a time-independent vector, so I'm guessing I just take the partial...

What is the Physical significance of Divergence and Curl?

Homework Statement Please evaluate the integral \oint d\vec{A}\cdot\vec{v}, where \vec{v} = 3\vec{r} and S is a hemisphere defined by |\vec{r}| \leqa and z ≥ 0, a) directly by surface integration. b) using the divergence theorem. Homework Equations -Divergence theorem in...