After doing my homework on testing for convergence/divergence in infinite series, I noticed that if you are testing for divergence of a rational function, if the difference in the degree of the functions (bottom - top) \leq 1, then it is divergent, and if the difference in the degree of the...
So I've been practicing several series that can be solved using the alternating series test, but I've came to a question that's been bothering me for sometime now.
If a series fails the alternating series test, will the test for divergence always prove it to be divergent?
Typically, in...
Homework Statement
Let S be a smooth surface enclosing the volume V, and let \vec{n} to be the unit outward normal. Using the Divergence Theorm show that:
∫∫ x \vec{r} ° \vec{n} dS = 4 * ∫∫∫ x dV,
where \vec{r}=(x,y,z)
Homework Equations
Divergence theorm...
Homework Statement
Hi. I think it will be easiest to understand if I upload a word document as it has some complicated characters. The bold letters stand for vectors.
http://www.sendspace.pl/file/1f830d4ff025d966f71b62c
Homework Equations
Divergence theorm...
Reading through Spivak's Calculus on Manifolds and some basic books in Analysis I notice that the divergence theorem is derived for surfaces or manifolds with boundary. I am trying to understand the case where I can apply the divergence theorem on a surface without boundary.
Recently I was reading some paper on design surface plasmon polaritons(so called "SPPs") on corrugated surfaces, mainly these two papers listed below:
1.Pendry, J. B., et al. (2004). "Mimicking surface plasmons with structured surfaces." Science 305(5685): 847-848.
2.Garcia-Vidal, F. J...
I'm trying to understand divergence of a sequence (not series). What methods can I use to prove divergence? I know that convergence can be proven using various methods, such as squeeze theorem and sum, difference, product and quotient rule etc.
Could I use the following to prove divergence...
Hi,
on page 63 of David J. Griffiths' "Introduction to Electrodynamics" he calculates the electric field at a point z above a line charge (with a finite length L) using the electric field in integral form.
E_z = \frac{1}{4 \pi \epsilon_0} \int_{0}^{L} \frac{2 \lambda z}{\sqrt{(z^2 + x^2)^3}}...
Homework Statement
NOTE: don't know see the phi symbol so I used theta. this is cylindrical coordinates not spherical.
Given the field D = 6ρsin(θ/2)ap + 1.5ρcos(θ/2)aθ C/m^2 , evaluate both sides of the divergence theorem for the region bounded by ρ=2, θ=0 to ∏, and z = 0 to 5...
Homework Statement
Homework Equations
Definitely related to the divergence theorem (we're working on it):
The Attempt at a Solution
I'm a bit confused about multiplying a scalar field f into those integrals on the RHS, and I'm not sure if they can be taken out or not. If they can be, I...
Homework Statement
∫Bdot[∇×A]dV=∫Adot[∇×B]dV
Prove this by integration by parts. A(r) and B(r) vanish at infinity.
Homework Equations
I'm getting stuck while trying to integrate by parts - I end up with partial derivatives and dV, which is dxdydz?
The Attempt at a Solution
I...
Dear All,
I am working on electrical modeling which I cannot change divergence to matrix form to solve it with finite difference or finite element, anybody can help me?
J Current density
E Electrical field
I Current...
Homework Statement
Given \nabla\frac{1}{r}, show \nabla\bullet\nabla\frac{1}{r} = -4πδ(r), where δ(r) is the delta dirac function.The Attempt at a Solution
I've used divergence theorem and also solved the equation itself, so I know that outright solving is zero and the divergence theorem gives...
We expect the stress-energy tensor to have zero divergence, because this is required for local conservation of energy-momentum, which has been verified to high precision in laboratory and solar system experiments. The standard review article is Will, "The Confrontation between General Relativity...
We were given an electric field defined by Kr^3 , and asked to calculate what the total flux would be given a sphere of a radius R. I had already calculated the divergence of E to be equal to 5kr^2 . So the first integral is calculating what the divergence over the area of the sphere is...
say you have a function f(x,y)
\nablaf= \partialf/\partialx + \partialf/\partialy
however when y is a function of x the situation is more complicated
first off \partialf/\partialx = \partialf/\partialx +(\partialf/\partialy) (\partialy/\partialx)
( i wrote partial of y to x in case y was...
I work with a grid-based code, this means that all of my quantities are defined on a mesh. I need to compute, for every point of the mesh the divergence of the velocity field.
All I have is, for every cell of my mesh, the values of the 3-d velocity in his 26 neighbors.
I call neighbors the...
Hey guys, was wondering if anyone could help walk me through this problem. I am fairly sure it converges by the ratio test. Thank you.
http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427e1sgqilv5iv
why do most modern books claim the divergence of the E field is ρ/ε_{0} but in more classical books, and when you actually derive it mathematically you arrive at 4πρ
Homework Statement
Verify that the infinite series diverges.
I have the series from n=1 to infinity of (2^(n)+1/2^(n+1)
Homework Equations
Nth term test(This is the way the book did it but I did it used the geometric series test
and I just want to verify if my Algebra was correct)...
I have been studying gradience; divergence and curl. I think i understand them in cartesian coordinate system; But i don't understand how do they get such complex stuffs out of nowhere in calculating divergence in spherical and cylindrical coordinate system. Any helps; links or suggestion...
One can easily prove that \nabla \cdot f is invariant under a rotation of the reference frame, however I would like to prove that the divergence operator itself is invariant (same principle, different approach). In other words I want to prove that \mathbf \nabla = \mathbf e_x...
Homework Statement
use divergence theorem to evaluate ∫s∫F dot n dA if
F=[sinh yz, 0, y4] , S: r=[u,cosv,sinv], -4≤u≤4 , 0≤v≤pi
The Attempt at a Solution
Instructor surprised us with this one, I have no idea how to attempt. I know that ∫vdiv v dV=∫sn dot v dA, which is the...
Hi all
I basically have two questions that are very closely related to each other about divergence, specifically the divergence of a vector function 1/r2\widehat{r}
First, I will be referencing pages 17, 18, and 45 from the 3rd edition of Intro to Electrodynamics.
The first question...
So I am wondering about one thing. The charged propagators in weak theory are W+- bosons. The mathematical expression for them, while drawing the Feynman diagrams is:
-i\frac{g_{\mu\nu}-\frac{q_\mu q_\nu}{m_W^2}}{q^2-m_w^2}.
The problems that are usually given to me are simple and involve...
Hello everyone,
I am revising for my final examination. I came across this simple problem which I can not solve. the problem is from a textbook (Introduction to Structural Dynamics and Aeroelasticity
By Dewey H. Hodges, G. Alvin Pierce). It is the 7th question in the problem sets of the 3rd...
When a vector field representing a physical quantity (e.g. B) has ∇\cdotB = 0 what is then the physical interpretation of this? Some people have said that the field doesn't diverge away from anything, but as far as I can tell magnetic field can easily get weaker and weaker the further you go...
Homework Statement
Compute the flux of F=xi+yj+zk through the curved surface of the cylinder x2+y2=1 bounded below by the plane x+y+z=1, above by the plane x+y+z=7, and oriented away from the z-axis.
Homework Equations
div(F) = (dF/dx) + (dF/dy) + (dF/dz)
The Attempt at a Solution...
The divergence of electric field at a point is proportional to the charge density at the point. Divergence is the rate of change with distance, the rate of change of electric field due to a distant charge is not zero, so how can it be said that the divergence at a point depends only on the...
A ZERO Divergence Vector Field
There is theorem that is widely used in physics--e.g., electricity and magnetism for which I have no proof, yet we use this theorem at the drop of a hat. The theorem is this:
Given sufficient continuity and differentiability, every vector function A such that...
I had a bit of trouble in testing series like this for convergence
$$\sum_{ n=1 }^{ \infty } \frac { 1 }{ 2n+1 } $$
If by the comparison test, ##\frac{ 1 }{ 2n+1 } < \frac{ 1 }{ 2n }## for all of n>0,
and ##\lim_{ n \rightarrow \infty} \frac{ 1 }{ 2n }## =0, then the series should be...
Homework Statement
Prove the divergence theorem for the vector field A = p = (x,y) and taking the volume V to be the cylinder of radius a with its base centred at the origin, its axis
parallel to the z-direction and having height h.
I can find the dV side of the equation fine (I think)...
Homework Statement
Evaluate the surface integral F * dr, where F=<0, y, -z> and the S is y=x^2+y^2 where y is between 0 and 1.
Homework Equations
Divergence theorem
The Attempt at a Solution
I just got out of my calculus final, and that was a problem on it. I used the divergence theorem...
Homework Statement
Find the Divergence or Convergence of the series
\sum^{∞}_{n=1}\frac{2n^2+3n}{\sqrt{5+n^5}}
Homework Equations
Ratio Test, Comparison Test, Limit Comparison Test, Integral test etc.
The Attempt at a Solution
This question was on my final exam and the only question of...
I'm trying to make a little manipulate/interactive box that shows the vector divergence of the E-field coming from a sphere. I have no idea how to start as I'm really new to Mathematica. Does anyone have any pointers? I can't find anything particularly helpful on the Wolfram reference or...
Homework Statement
Homework Equations
So I have that v \otimes n = \left( \begin{array}{ccc}
v_{1}n_{1} & v_{1}n_{2} & v_{1}n_{3} \\
v_{2}n_{1} & v_{2}n_{2} & v_{2}n_{3} \\
v_{3}n_{1} & v_{3}n_{2} & v_{3}n_{3} \end{array} \right)
The Attempt at a Solution
I've tried applying the...
Homework Statement
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
##\sum _{n=1}\left( -1\right) ^{n}\dfrac {n} {n^{2}+1}## the sum goes to infinity.
Homework Equations
Theorem for absolute convergence.
Test for divergence
The Attempt...
Homework Statement
##\sum _{n=1}^{\infty }\left[ \left( -1\right) ^{n}\right] \dfrac {\sqrt {n}} {1+2\sqrt {n}}##Homework Equations
Alternating Series test, Absolute convergence theorem, p-series, and test for divergence.
The Attempt at a Solution
The alternating series test tells us that the...
Homework Statement
Folks,
Verify the divergence theorem for
F(x,y,z)=zi+yj+xk and G the solid sphere x^2+y^2+z^2<=16
Homework Equations
##\int\int\int div(F)dV##
The Attempt at a Solution
My attempt
The radius of the sphere is 4 and div F= 1, therefore the integral...
Homework Statement
There are 3 parts to this problem:
(a) \; \sum^{\infty}_{n=1} \frac{n^4}{4^n}
(b) \; \sum^{\infty}_{n=1} \left( \frac{n+8}{n} \right)^n
(c) \; \sum^{\infty}_{n=1} \frac{5^n-8}{4^n+11}
The attempt at a solution
(a) I've used the Ratio test.
So, u_n=\frac{n^4}{4^n} and...
Homework Statement
##\sum _{n=1}ne^{-n}##
Homework Equations
Ratio Test
Integral Test
The Attempt at a Solution
I know that by the ratio test, it converges absolutely. But, I am unable to determine its convergence through the integral test . Could someone help? I thought that the...
Hi, can you tell me which theorem they have used here: http://everything2.com/title/proof+that+the+sum+of+the+reciprocals+of+the+primes+diverges
i'm thinking on part: Well, there's an elementary theorem of calculus that a product (1-a1)...(1-ak)... with ak->0 converges to a nonzero value iff...
When the exercise tells me to calculate the flux, how do I know when I need to use each of these theorems (Green's, Stokes or Divergence)?
Can anyone tell me the difference between them? I'm a LOT confused about this. If anyone knows any good material about this on internet, it'll help me a...
Homework Statement
##\sum \dfrac {1+2^{n}} {3^{n}}##
According to Wolfram Alpha the sum is 5/2. But, I think that my method is fine and shows another result.
The Attempt at a Solution
##\sum \dfrac {1+2^{n}} {3^{n}}=\sum \left[ \left( \dfrac {1} {3}\right) ^{n}+\left( \dfrac {2} {3}\right)...
Ʃ n=1 to infinity of cos(n∏)
letting an=cos(n∏), I rewrote this as (-1)^n=an.
Using the nth term test i let the limit as n->∞ go to infinity. This value bounces back and forth between positive and negative, but I know clearly the value =/= 0, therefore it diverges by the nth term test.
Is...
Now for the proof of convergence/divergence of the geometric series we first deduce the Nth partial sum which is given by:
\frac{r(1-r^n)}{1-r}
Now for 0<r<1 this become \frac{1}{1-r} which clearly converges by AOL
At r>1 it's similarly obvious why it diverges.
But at r=1, I'm a bit...
Homework Statement
F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k
Find divergence
Homework Equations
The Attempt at a Solution
The gradient is
-i + j + -k
Dotting that with F, I get
x - y + y + z + z - x
=
2z
My book lists the answer as -1. What the heck are they talking...