Uniform Definition and 1000 Threads

Discuss uniform continuity of the following functions: ##\tan x## in ##[0,\frac{\pi}{2})## ##\frac{1}{x}\sin^2 x## in ##(0,\pi]## ##\frac{1}{x-3}## in ##(0,3),(4,\infty),(3,\infty)## I am completely new to this uniform continuity and couldn't find a lot of examples to learn the solving pattern...

I am trying to follow this link: https://people.math.wisc.edu/~angenent/521.2017s/UniformConvergence.html ...not getting it... of course i know what convergence is, just to mention for e.g given a sequence, ##\dfrac{1}{n}## I know that the sequence tends to ##0## or rather converges to ##0##...

How is there a uniform electric field between two parallel metal plates? Okay, I can understand it through electric field lines. But how can this be quantitatively confirmed? My intuition is that as the charge approaches the metal plate, r becomes very small, so according to Coulomb's law, a...

I was doing a problem with this one detail. It says that the electric potential energy of an uniformly charged hollow sphere and a point charge is (at the surface of the hollow sphere; both positive): $$U = k \frac{q_1 q_2}{r}$$ I guess this assumes that the hollow sphere is a point charge. Now...

Was reading the Reimann integrals chapter of Understanding Analysis by Stephen Abbott and got stuck on exercise 7.2.5. In the solutions they went from having |f-f_n|<epsilon/3(a-b) to having |M_k-N_k|<epsilon/3(a-b), but I’m confused how did they do this. We know that fn uniformly converges to...

Problem : I copy and paste the problem as it appeared in the text. Attempt : ##\texttt{I could make no significant attempt at solving the problem.}## The only thing I realise is that if the two tourists be A and B, A travels on the bicycle for a while, leaves the bicycle and walks. B...

I had an exam ques which was as follows: The first part is clear to me.....it is uniform (or constant) speed. I am in doubt on the account of the second part as the answer key says this: So the overall question concerning the second part is as follows: We know that the direction of...

Consider a circular loop with uniform current flowing around it in a uniform magnetic field. Does it experience no translational force due to its symmetry

Problem: Solution part a) where formula 6.14 is just M x n. We need to do part b without seperation of variables, I'm quite stuck. Will B just be the magnetic field inside a solenoid? How can I find the other fields.

I would like to discuss the nature of the following effect. At whatever angle and with whatever initial speed the particle fly into a uniform potential field, over time the directions of the instantaneous velocity and field strength converge. The kinematics and dynamics here are trivial, but I...

Griffith's E&M problem 4.7 asks to calculate the energy of a dipole in a uniform electric field and I ended up getting a different answer than the one given. I thought that calculating the energy/work done to construct the dipole is the same as dragging two point charges where one is d apart...

I'm able to get the force of the top spring (6.75N upwards) by setting the sum of all forces equal to 0 as everything is stationary. Where I'm stuck is starting the second part of the problem. I initially tried setting the sum of torques equal to 0 using the top spring as a lever arm but unless...

Question : For uniformly accelerated motion ##a(t)=a_0\;\; \forall \text{times}\;t##, we can say that the average velocity for the entire motion ##\bar v = \frac{v_0+v}{2}##, where ##v(t)## is the final velocity at some time ##t## and ##v_0## is the initial velocity. How do we show that? Issue...

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Hi! For this problem, The solution is, However, why did they not include constants of integration in their working shown in red? Many thanks!

So i got some equations but i think i am missing something, my main doubt is what is the relation between dx / dt and v(o) [ here] . Workings in attachment

How and why can charge be evenly or uniformly distributed in a conductor? How can such near perfect configuration of charge be achieved? Is outside influence (or force) or any special scientific tools or instruments required to accomplish that? By definition, electrostatic equilibrium is...

d(ɣmv)/dt = qvB (dɣ/dt)mv + ɣm(dv/dt) = qvB Substituting gamma in and using the chain rule, it ends up simplifying to the following: ɣ^3*m(dv/dt) = qvB Now, I am confused on how to solve for v.

I placed my Oxy coordinate system at the center of the square, the ##x##-axis pointing rightwards and the ##y##-axis pointing upwards. I divided the square into thin vertical strips, each of height ##h=2(\frac{L}{\sqrt{2}}-x)##, base ##dx## and mass ##dm=\sigma h...

Mentors’ note: this thread is forked from https://www.physicsforums.com/threads/free-fall-in-curved-spacetime.1016510/ But what if the gravity field is homogeneous? Like that of an infinite massive plane? The objects in the ship will stay where they are. An infinite massive plane is quite...

Hello everyone, I've been studying centripetal and centrifugal acceleration and derivation of their magnitude. I noticed in one of Walter Lewin's lectures that the velocity is written as both a vector and an arc length which is confusing to me. When velocity is written as a vector, it has a...

I started by making my coordinate system so that the x-axis aligned with the radius of the circle at a certain latitude L and the positive direction was facing away from the center of the circle, and the y-axis was parallel to the vertical axis of the Earth. Then, I wrote the equations for the...

My solution was as follows: $$\frac {d\overrightarrow p} {dt}=q \frac {\overrightarrow v} {c}\times \overrightarrow B_0$$ The movement is in the ##[yz]## plane so ##|\overrightarrow v\times \overrightarrow B_0|=vB_0##, therefore: $$\biggr |\frac {dp} {dt}\biggr |= \frac {qvB_0} {c}.$$ On the...

If I want to calculate the dipole moment of a dielectric cylinder of uniform polarization perpendicular to its axis, I could multiply the polarization by the volume of the cylinder, which is okay. But another method is to consider the cylinder to be a superposition of two cylinders of equal and...

$$rcm = \frac{1}{M}\int_0^\pi(rdm)$$ $$dm = \sigma{dA}$$ $$dA = (\pi*R^2)*\frac{d\theta}{2\pi}$$ $$\sigma = \frac{M}{\frac{\pi*R^2}{2}}$$ $$dm = M*\frac{d\theta}{\pi}$$ $$r = R(cos(\theta)\vec i + sin(\theta)\vec j)$$ $$rcm = \int_0^\pi{\frac{R}{\pi}(cos(\theta)\vec i + sin(\theta)\vec j)} =...

I encountered a problem regarding the appropriate sign needed to be taken for the work done on a dipole when it rotates in a uniform electric field and would appreciate some help. The torque on a dipole can be defined as τ=PEsinθ The work done on a dipole to move it from an angle ##\theta_0##...

https://slidetodoc.com/genetic-algorithms-an-example-genetic-algorithm-procedure-ga/ slide is taken from here. is this done total randomly or is it done pseudorandomly. I mean is there some forumula for randomness used in this case? i learned about single point and double point crossover but...

we haven't learned howw to do parts d-f yest. could you please give me a hand? (a) $$E(X)=E(Y)=\int_0^3\int_0^3\frac{1}{9}dxdy=\frac{3}{2}$$ (b)its typoe suppoesed to be W=Y-2 $$E(Z)=E(X-2)=E(X)-E(2)=-\frac{1}{2}$$ $$E(W)=E(Y-2)=E(Y)-E(2)=-\frac{1}{2}$$ (c) i guess joint pdf from ##E(Z)## and...

This is an offshoot of @Angela G 's thread. I don't want to hijack her thread so I decided to create a new one. Original thread https://www.physicsforums.com/threads/unstable-or-stable-electrostatic-equilibrium.1007881/ @kuruman @PeroK @bob012345 If you have the time I'd appreciate your input...

A random variable is distributed uniformly over a circle of radius R. What does the cdf ##F(x,y)## look like as a function of the Cartesian coordinates? The pdf can be expressed as ##f(x,y)=\frac{\delta(\sqrt{x^2+y^2}-R)}{2\pi R}##, where ##\delta## is Dirac delta function. Integration is...

My understanding is that the uniform electric field ##\vec E## cannot be the net electric field since the dipole creates its own electric field as shown in first diagram below, which must superimpose with the uniform electric field. So, yes, the uniform electric field ##\vec E## around the...

Hi, I found this question online and made an attempt and would be keen to see whether I am thinking about it in the right manner? Question: Find the probability of two line segment intersecting with each other. The end points of lines are informally sampled from an uniform distribution...

For a uniform field like this, I imagine the two plates that creates it are made of multiple atoms with charges, which are points sources that create radial fields. We know that radial fields don't have parallel fields lines, so how are parallel fields lines form when the field is made of...

I have figured out how the force is towards left in the first case. I think it is due to the larger force on - charge. Please help me out with the second and third case.

I was reading Einstein's 1911 paper named "On the Influence of Gravitation on the Propagation of Light" when stated the formula for frequencies measured by observers at different fixed positions (heights) on Earth surface. One observer is at the origin of some coordinate system and measures a...

In NMR, protons that are near each other exchange energy with each other by photon emission and absorption. Proton A is spin down, and proton B is spin up. proton A emits a photon and becomes spin up, and proton B absorbs this photon and becomes spin down. have the two protons been entangled?

Hey guys! I'm having trouble with the solution that I arrived at. Through boundary conditions I'm able to determine ##\vec{D}## as $$\vec{D}=-\frac{4Q}{R_0^2}\hat{e_z}$$ (In CGS units) Trough that I'm able to get the electric field as $$\vec{E}=-\frac{1}{\epsilon(r)}\frac{4Q}{R_0^2}\hat{e_z}$$...

The following is my work: The following is the answer given: While I took FA as the pivot point, the answer seem to have taken FB as the pivot point, and got negative value for FB. As the result, the answer seems to have used the tensile strength for FA and compressive strength for FB...

prove or disprove if the following function is uniformly continuous: sin(sin(x)) using the ε,δ definition

I'm trying to make a comparison between the maximum allowable uniform load of a beam made of steel vs LVL. Here is a table of allowable uniform loads for steel W beams: https://www.engineeringtoolbox.com/w-steel-beam-uniform-load-d_1722.html Here is a table of allowable uniform loads for LVL...

Hi, I was attempting the following question, but got confused on this part: Question: Two radar tracking stations provide independent measurements ##x_1## and ##x_2## of the landing site, ##\mathbf{x} = (x, y) ##, of a returning space probe. Both have Gaussian sensor models, ##p(x_i|X_i ) =...

Hey! :giggle: We define the sequence of functions $f_n:[0,1]\rightarrow \mathbb{R}$ by $$f_{n+1}(x)=\begin{cases}0 & \text{ if } x\in \left[ 0, \frac{1}{2n+3}\right ]\\ |2(n+1)x-1| & \text{ if } x\in \left [\frac{1}{2n+3}, \frac{1}{2n+1}\right ] \\ f_n(x) & \text{ if } x\in \left...

Hello, Apologies if this is in the wrong section, it's related to circles so I figured Geometry was the best place. I found a very good example online that explains how to determine a future position of an object undergoing uniform circular motion: (Note that they made a mistake by writing...

I don't know where to start. This chapter is incredibly confusing for me.

I'd say yes, it is. Suppose ##|f|## is uniformly continuous on ##D##. Then for all ##\epsilon>0## there is ##\delta>0## (call this ##\delta'##) such that if ##x,y\in D##, then ##||f(x)|-|f(y)||<\epsilon##. Define sets: ##D^+=\{x\in D: x>a\}## ##D^-=\{x\in D: x<a\}## Restrict the domain of...

First I figured out the normal force being exerted on the car using the equation above. Cos(40°)*(1050*9.8) = 7883N Next, I tried to find out the horizontal component of the normal force by doing: Cos(50) * 7883 = 5067N I figured out the angle by using certain geometrical properties. Next, I...

I think the answe should be C as shown in the attachment.

Assume a solenoid coil(made up of ##N## windings) placed in the horizontal(##\hat{y}##) direction and in a constant uniform magnetic field. Would an induced current run through the(closed) coil if it spins around its central horizontal ##\hat{y}## axis? My guess is "no", since such a current is...

In the article 'Cellular vacuum' (Int. J. Theor. Phys. 21: 537-551, 1982), Minsky writes: "One can prove that any bounded packet which moves within a regular lattice must have an asymptotically helical trajectory...". But he gives no references whatsoever. I had no success in a search on the...