Divergence Definition and 746 Threads

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...

Hi, i thought a while about the meaning of the following expression \delta t \, \cdot \, \mathrm{div} \, \vec j_m(\vec r,t) \qquad \mathrm{with} \qquad \vec j_m(\vec r,t) = \rho_m(\vec r,t) \cdot \vec v(\vec r, t)Does it indicates the mass, which is produced / annihilated in the Volume...

Homework Statement Let's define the radial vector \vec{v}(r) = \hat{r}/r^{2} where \vec{r} = \vec{OP} (O being the origin of our coordinate system and P being our observation point at point (x, y, z)). Using spherical coordinates, demonstrate that \vec{\nabla } \cdot\vec{v}(r) = 0 everywhere...

Homework Statement Use the divergence theorem to find the outward flux of a vector field F=\sqrt{x^2+y^2+z^2}(x\hat{i}+y\hat{j}+z\hat{k}) across the boundary of the region 1\leq x^2+y^2+z^2 \leq4 Homework Equations The Gauss Divergence Theorem states \int_D dV \nabla \bullet...

Homework Statement find the divergence and curl of the vector field A = (x/(\sqrt{x^2 + y^2 + z^2}))i + (y/(\sqrt{x^2 + y^2 + z^2}))j + (z/(\sqrt{x^2 + y^2 + z^2}))k Homework Statement The Attempt at a Solution Im not going to go through the whole lot but i have done the whole...

Homework Statement I need to prove the identity div (a x b) = b dot (curl a) - a dot (curl b) The Attempt at a Solution I've done the proof about 10 times now, and everytime I get the left hand of the identity equal to this: (all the d's are partial derivatives) d(a3b1)/dx -...

Given two vectors, A and B: A = (x\widehat{x} + 2y\widehat{y} + 3z\widehat{z}) B = (3y\widehat{x} - 2x\widehat{y}) I need to calculate (B [dot] \nabla)A, as part of a problem. The answer should be: \widehat{x}(3y) + \widehat{y}( -4x) I get: (B [dot] \nabla)A = ((3y) \delta /...

I am studying vector calculus, and I saw the following result in a physics text: g = -\frac{m}{r^3}\vec{r} r^2 = x^2 + y^2 + z^2 \vec{r} = ix + jy +kz \nabla \cdot g = 0 I'm not sure how this was done. Is the product rule used somehow? What happened to the extra power of r...

Can the Test for Divergence (limit of an->infinity not equal to zero) be used on an alternating series? For example, if a series has a (-1)^n term. Can we assume that since the limit of that term does not exist, then the series is automatically diverging?

Homework Statement the problem is to calculate \int (\nabla \cdot \vec{F}) d\tau over the region x^2 + y^2 + x^2 \leq 25 where \vec{F} = (x^2 + y^2 + x^2)(x\hat{i} +y\hat{j} + z\hat{k}) in the simplest manner possible.Homework Equations divergence theorem!The Attempt at a Solution...

Hi everyone, I'm trying to work through section 86 of Landau and Lifgarbagez volume 2 (The Classical Theory of Fields). Basically, I am unable to get equation (86.6) from equations (86.4) and (86.5). I've detailed my working/question in the attached jpg file. I would appreciate any inputs...

Homework Statement The electrostatic field of a point charge q is E=\frac{q}{4 \pi \epsilon r^3} r. Calculate the divergence of E. What happens at the origin? Homework Equations The Attempt at a Solution Well the solution is: \nabla.E= \partialEx/\partialx +...

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Homework Statement A tiny laser beam is directed from Earth to moon. If beam's diameter is 2.50 m at the moon, how small much the divergence angle be for the beam? The distance of moon from the Earth is 3.8x10^8 m Homework Equations The Attempt at a Solution

Homework Statement Given that a_{n} > 0 and lim(na_{n}) = l with l\neq0, prove that \sum a_{n} diverges.Homework Equations The Attempt at a Solution lim(na_n)=l (with =/= 0), so I can safely say that: \left|na_{n}-l\right| < \epsilon by the definition of limit. Then isn't it also true that...

Homework Statement Suppose (a_n) is a sequence of non-negative real numbers such that the series {\sum_{n=1}}^\infty a_n diverges. Prove that the series {\sum_{n=1}}^\infty \frac{a_n}{1+a_n} must also diverge. Homework Equations The Attempt at a Solution I was thinking about looking at...

So I am playing around with the differential form of Gauss's Law: \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} Starting off simple with a point charge, the electric field is: \vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \hat{r} And the divergence, in spherical coordinates...

Homework Statement Evaluate the divergence and curl of the following vectors. A(r) is everywhere parallel to the y-axis with a magnitude A = cx + A0 , where c and A0 are constants. Homework Equations The Attempt at a Solution I can evaluate the div and curl, but i don't know...

In h.m. schey, div grad curl and all that, II-25: Use the divergence theorem to show that \int\int_S \hat{\mathbf{n}}\,dS=0, where S is a closed surface and \hat{\mathbf{n}} the unit vector normal to the surface S. How should I understand the l.h.s. ? Coordinatewise? The r.h.s. is not...

To my mind radio waves are different than light because they are described by fields whose energy is not a function of frequency but just the amplitudes. That they always have to surround their source, and are not uniquely associated with individual particles. Light on the other hand is...

Hi guys, trying to solve a problem in MHD, i realized i need to be able to take the divergence of this following integral, but I don't know how to do it. M is a symmetric rank 2 tensor, r is a vector. The integral is as follows \int_{\partial V} (\textbf{r} d \textbf{S} \cdot...

I know that this question was posted before but I just couldn't get it using another way around. So your comments are highly appreciated. In the textbook, \nabla\bullet\left(\widehat{r}/r^{2}\right)=4\pi\delta^{3}\left(r\right). But when I want to calculate the divergence using Catesian...

Im really having troubles understanding the divergence form of gauss's law. I have done research on it and am still not able to understand it. it sates that E=\rho/\epsilon or E=rho/epslom, so does that mean that the upside down triangle has no significance, ie does that mean i can simply solve...

Homework Statement This is a three part problem. My first task is to calculate the divergence of \vec{r}/r^{a}. Next, I am to calculate its curl. Then I'm supposed to find the charge density that would produce the field \vec{E}=\frac{q\vec{r}}{4\pi\epsilon_{0}r^{a}} The Attempt at a...

Why is the divergence operation called the 'divergence?' What is the significance of this operation on a vector-valued function? And what about "the curl?" The curl seems self-explanatory (at least it does in electrodynamics), but I need someone to expound on 'the curl' as well.

what is the effect of placing a slit of say one tenth of an inch infront of a laser beam (3 inch dia )with a divergence of say one degree upon the divergence of the emanating beam?(slit placed symmetrically along the beam axis) ? would the emanating beam also be of the same divergence ?

\sum (n^2-arctan(n)) / (n^3 + sin(n)) n=0 to ∞ I know this series diverges, but how would you use the comparison test to compare it to (n^2 / n^3 meaning the harmonic series 1/n) Thank you very much

Statement: The definition of the Divergence is given by the following, \nabla \cdot \vec{V} \equiv lim_{\Delta v \rightarrow 0}(\frac{\int \int _{surface}\vec{V} \cdot \vec{ds}}{\Delta v}), where v is the unit volume.Relevant questions: The expression \vec{V} \cdot \vec{ds} on the right side...

Question: Can someone remind me why the divergence of the electric flux is equal to the volume charge density, \nabla \bullet \vec{D} = \rho_{v} (where \vec{D} is the electric flux density). Thoughts: The divergence measures the flow of a field out of a region of space. The del operator takes...

Homework Statement sum 1/n*sin (1/n), n=1..infinity I tried the limit comparison test, but I always get 0. The ratio test is impossible Comparison to the harmonic series cannot be used because 1/n*sin (1/n) is smaller than 1/n Can you guys help? Thanks

I think after weeks of study, I'm finally getting a handle on Gauss' Law. A few ? though. The equation does not specifically state that there is not charge inside the surface. One may think that it doesn't matter if there or isn't...it's alway zero. How come that is not state in the...

1. \sum(\sqrt{k^{2}+1}-\sqrt{k^{2}}) from K=0 to K=\infty 2. Hi all. I need some help here. I have to use a test to determine whether the sum series diverges or converges 3. I thought it was the divergence test because I thought that the limit of the sum didn't approach zero...

http://img60.imageshack.us/img60/9696/21249035.jpg I am stuck at the last step. Can anyone give some hints? Thanks in advanced.

Homework Statement Use the divergence theorem to evaluate \int\int_{\sigma}F . n ds Where n is the outer unit normal to \sigma we have F(x,y,z)=2x i + 2y j +2z k and \sigma is the sphere x^2 + y^2 +z^2=9 Homework Equations \int\int_{s}F . dA = \int\int\int_{R}divF dV The...

Homework Statement Water in an irrigation ditch of width w = 3.0 m and depth d = 2.0 m flows with a speed of 0.40 m/s. For each case, sketch the situation, then find the mass flux through the surface: (a) a surface of area wd, entirely in the water, perpendicular to the flow; (b) a surface...

The following isn't actually a homework problem, but this seems to be the natural place to ask questions of this sort. Without further ado, I have a spherically symmetric tensor field, which is written with the help of dyadics as P(r) = P_n(r)\mathbf{e}_r\mathbf{e}_r +...

Homework Statement use the divergence theorem to evaluate the integral F dot dA F = (2x-z)i + x2yj + xz2k s is the surface enclosing the unit cube and oriented outward Homework Equations The Attempt at a Solution is the the region from -1 to 1 for x y and z div F = x2 +...

Use the divergence theorem to compute the surface integral F dot dS , where F=(xy^2, 2y^2, xy^3) over closed cylindrical surface bounded by x^2+z^2=4 and y is from -1 to 1. I've tried doing it and got 32pi/3 (i guess its wrong, so how to do it?) Is it ok to compute Div F in terms of xyz and...

Homework Statement \vec{F}(x,y,z) = x^2y\vec{i} + y^2z^3\vec{j} + xyz\vec{k} Homework Equations The Attempt at a Solution I got: Curl: (xz - 3y^2z^2)\vec{i} + (-yz)\vec{j} + (-x^2)\vec{k} Div: 2xy + 2yz^3 + xy Are these right?

Evaluate the flux integral using the Divergence Theorem if F(x,y,z)=2xi+3yj+4zk and S is the sphere x^2+y^2+z^2=9 answer is 324pi so far i took the partial derivitavs of i j k for x y z and added them to get 9. so i have the triple integral of 9 dzdxdy i think u have to use polar...

Homework Statement F = xi + yj + zk, s = x^2 + y^2 + z^2 Homework Equations The Attempt at a Solution div F = 1+1+1=3 area of sphere = 4pi i can just multiply them to get 12pi as an answer right?

Homework Statement Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem Homework Equations Divergence theorem: ∫∫∫∇.FdV = ∫∫∇.ndS Stokes...

Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem --- Since the divergence theorem involves a dot product rather than a curl,how would it...

Homework Statement Suppose \vec{G} is a vector field with the property that div\vec{G} = 5 for 2 \leq ||\vec{r}|| \leq 14 and that the flux of \vec{G} through the sphere of radius 4 centered at the origin is 20\pi . Find the flux of through the sphere of radius 12 centered at the...

Homework Statement http://img16.imageshack.us/img16/88/fluxm.th.jpg Homework Equations The Attempt at a Solution I've tried to find the divergence of F and I got 3x^2 + 3y^2 + 3z^2 and as this is a variable I need to set up the integral... how do I set the integral

Homework Statement http://img4.imageshack.us/img4/4218/divergenceandcurl.jpg The Attempt at a Solution Totally confused on what the question's asking. Wouldn't the divergence of say x_hat be the partial of x_hat over x which is just 0? So every answer would just be 0 or something? Same...

series; convergence, divergence... Homework Statement 1. sum(infinity,n=1) n!/1.3.5...(2n-1) 2. sum(infinity, n=1) (-1)^n arcsin(-1/n) 3. sum(infinity, n=0) arcsin(1/n^2) / arctan(1/n^2) The Attempt at a Solution 1. i used the ratio test and then i ended up with lim((n+1)(2n-1)/2n+1))...

Homework Statement Test the series for convergence or divergence -1/3+ 2/4 - 3/5 +4/6 - 5/7 + .... Homework Equations I think it's an alternating series The Attempt at a Solution I found that an = (-1) ^n * n/ (n+2) And it approaches 1 as n goes to inifty so the series...

Homework Statement Compute the divergence in cylindrical coordinates by transforming the expression for divergence in cartestian coordinates. Homework Equations F = F_x i + F_y j + F_z k div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z ... (divergence in Cartesian coordinates) I need to...

The question I was given asks to verify the divergence theorem by showing that both sides of the theorem show the same result. With the divergence theorem obviously being \iint_S\mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV . The vector field is...