Smooth Definition and 229 Threads

Firstly I must say if this is in a wrong sub-forum I apologise. This is my first post and I'm new to the website so please bare with me. Also I was unsure of the Prefix so I again apologise if that to, was incorrect. But my Question is that is the acceleration (or deceleration) of an object...

https://www.inverse.com/article/24863-dark-matter-might-be-smoother-than-we-thought Scientists have yet to actually observe dark matter in the flesh, but most research up to now posits it’s the kind of stuff that clumps up and aggregates into unwieldy masses around the universe. New research...

I am trying to figure out how to get the Hamiltonian for a mass on a fixed smooth hemisphere. Using Thorton from example 7.10 page 252 My main question is about the Potential energy= mgrcosineθ is the generalized momenta Pdotθ supposed to be equal to zero because θ is cyclic? Or is Pdotθ=...

Let $f:\mathbf R\to \mathbf R$ be a smooth map and $g:\mathbf R\to \mathbf R$ be defined as $g(x)=f(x^{1/3})$ for all $x\in \mathbf R$. Problem. Then $g$ is smooth if and only if $f^{(n)}(0)$ is $0$ whenever $n$ is not an integral multiple of $3$. One direction is easy. Assume $g$ is smooth...

Homework Statement [/B]Mass m lies on a Weighing scale which is on Wagon M. the inclined surface is smooth, between m and M there is enough friction to prevent m from moving. 1) What does the weigh show? 2) What is the minimum coefficient of friction between m and M to prevent slipping? 3)...

A 10 Kg block lies on a smooth plane inclined at 51 degrees. What force parallel to the incline would prevent the block from slipping?

Consider a ladder slanted like \ of length ##l## slipping against a smooth wall and on a smooth floor. I come to the contradiction that there must be a deceleration in the x direction but there is no force opposing the velocity of the ladder. Its free-body diagram contains a rightward normal...

This isn't about a specific physics problem, but rather a question: Given I have a ball or cylinder rolling smoothly along some path, is it generally true that mechanical energy is conserved? I.e. if ##E_mech = K+U = K_{trans} + K_{rot} + U##, then ##\Delta E_mech = 0##? I have been able to...

Hello, Given a Lie group G and a smooth path γ:[-ε,ε]→G centered at g∈G (i.e., γ(0)=g), and assuming I have a chart Φ:G→U⊂ℝn, how do I define the derivative \frac{d\gamma}{dt}\mid_{t=0} ? I already know that many books define the derivative of matrix Lie groups in terms of an "infinitesimal...

Hello Here is my question So I solved Euler DE and find and when we apply the boundary condition we obtain y=0 . My teacher said that we should write it as two different function as where H is (1/2).He solved this equation with this way So Here are my questions: a) why don't we accept...

I was doing more reading in John Lee's "Introduction to smooth manifolds" and he mentioned that for every n \in \mathbb{N} such that n \neq 4 , the smooth structure that can be imposed on \mathbb{R}^n is unique up to diffeomorphism, but for \mathbb{R}^4 , there are uncountably many smooth...

Because a triangle comes out to 180 degrees, and yet it can only have three sides. A circle has 360 degrees, but its number of "sides" are uncountable. Can someone explain this?

Homework Statement [/B] Mass m starts sliding down on a rough surface with coefficient of friction μ. it reaches point B and starts sliding frictionlessly till it reaches point D without velocity, i.e. without escaping the arc. What is the maximum length AB=x0 not to escape the arc. What is...

Homework Statement A ring of mass m slides on a smooth circular hoop with radius r in the vertical plane. The ring is connected to the top of the hoop by a spring with natural length r and spring constant k. By resolving in one direction only show that in static equilibrium the angle the...

Hi, I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is...

Came across this in a discussion of essential self-adjointedness: Let P be the densely defined operator with Dom(P) = C^{\infty}_c (\mathbb{R}) \subset L^2 ( \mathbb{R} ) and given by Pf = -i df/dx. Then P is essentially self-adjoint. It is the C^{\infty}_c (\mathbb{R}) \subset L^2 (...

Homework Statement Mass m lays on the smooth triangle of mass M. what is the acceleration of M so that m will stay in place. Homework Equations Newton's second law: ##F=ma## The Attempt at a Solution $$\tan\alpha=\frac{ma}{mg}\;\rightarrow\; a=g\tan\alpha$$

Homework Statement Given a 2.0 kg mass at rest on a horizontal surface at point zero. For 30.0 m, a constant horizontal force of 6 N is applied to the mass. For the first 15 m, the surface is frictionless. For the second 15 m, there is friction between the surface and the mass. The 6 N force...

How do I exit? Time to write this down. :-)

Homework Statement For a smooth (“low jerk”) ride, an elevator is programmed to start from rest and accelerate according to $$a(t) = \frac{a_m}{2}[1 − \cos{\frac{2\pi t}{T}}] \:\:\:\:0 ≤ t ≤ T$$ $$a(t) = -\frac{a_m}{2}[1 − \cos{\frac{2\pi t}{T}}] \:\:\:\:T ≤ t ≤ 2T$$ Where ##a_m## is the...

Let $f:\mathbf R^n\times \mathbf R^k\to \mathbf R^n\times \mathbf R^k$ be a diffeomorphism such that: 1) $f(\{\mathbf p\}\times \mathbf R^k)=\{\mathbf p\}\times \mathbf R^k$, for all $\mathbf p\in \mathbf R^n$. 2) $f|\{\mathbf p\}\times\mathbf R^k:\{\mathbf p\}\times \mathbf R\to \{\mathbf...

Hello! My question is quite a quick one- I was wondering whether it is ever possible to have a smooth wall exerting a force parallel to it (and not just perpendicular to it). For example, if you were to place a see-saw by a smooth wall so that the wall is holding one of the see-saw ends below...

Homework Statement [/B] A circular table of radius rotates about its center with an angular velocity 'w'. The surface of the table is smooth. A groove is dug along the surface of the table at a distance 'd' from the centre of the table till the circumference. A particle is kept at the starting...

Homework Statement Suppose a can, after an initial kick, moves up along a smooth hill of ice. Make a statement concerning its acceleration. A) It will travel at constant velocity with zero acceleration. B) It will have a constant acceleration up the hill, but a different constant acceleration...

Let ##V## be a real vector space and assume that ##V## (together with a topology and smooth structure) is also a smooth manifold of dimension ##n## with ##0 < n < \infty##, not necessarily diffeomorphic or even homeomorphic to ##\mathbb R^n##. Here's my question: Does this imply that addition...

I am currently reading the paper given here by Acharya+Gukov titled "M Theory and Singularities of Exceptional Holonomy Manifolds", and in particular right now am following section 4 where the field content of the effective 4-dimensional theory is derived by harmonic decomposition of the 11D...

I feel I understand what happens, and how to solve the equation of motion x(t) for a mass attached to a spring and released from rest horizontally on a smooth surface. We typically end up with x(t) = x_0 cos(ωt) as the solution, with x_0 as the amplitude of the oscillation. But I've...

Why do things like grease, cheese, butter, jam, etc. stick to smooth surfaces like a butter knife or teflon? What are the ways in which they would not stick and be allowed to release without being heated?

I'm looking to prove the Global Frobenius theorem, however in order to do so I need to prove the following lemma: If ##D## is an involutive distribution and and ##\left\{N_\alpha\right\}## is collection of integral manifolds of ##D## with a point in common, then ##N = \cup_\alpha...

Suppose I have a smooth curve \gamma:[0,1] \to M, where M is a smooth m-dimensional manifold such that \gamma(0) = \gamma(1), and \hat{\gamma}:=\gamma|_{[0,1)} is an injection. Suppose further that \gamma is an immersion; i.e., the pushforward \gamma_* is injective at every t\in [0,1]. Claim...

Homework Statement Determine where r(t) is a smooth curve for -pi <t<pi R(t)= (x(t),y(t))=(4sin^3(t), 4cos^3(t)) Homework Equations The Attempt at a Solution To be honest I have no idea where to start. I know what a smooth function is but my understanding is that the sin(t) and...

Consider a smooth map ##F: M \to N## between two smooth manifolds ##M## and ##N##. If the pushforward ##F_*: T_pM \to T_{F(p)} N## is injective and ##F## is a homeomorphism onto ##F(M)## we say that ##F## is a smooth embedding. In analogy with a topological embedding being defined as a map...

I have 50 data sets. Each set has three related time series: fast, medium, slow. My end purpose is simple, I want to generate a number that indicates a relative degree of change of the time series at each point. That relative degree of change should range between 0-1 for all the time series and...

Let ##M## and ##N## be smooth manifolds and let ##F:M \to N## be a smooth map. Iff ##(U,\phi)## is a chart on ##M## and ##(V,\psi)## is a chart on ##N## then the coordinate representation of ##F## is given by ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \to \psi(V)##. My question is...

In the definition of smooth manifolds we require that the transition functions between different charts be infinitely differentiable (a circle is an example of such a manifold). Topological manifolds, however, does not require transitions functions to be smooth (or rather no transition functions...

A function f:\mathbb{R} \to \mathbb{R} is called "smooth" if its k-th derivative exists for all k. A function is called analytic at a if its Taylor series \sum_{n\geq 0} \frac{f^{(n)}(a)}{n!} (x-a)^n converges and is equal to f(x) in a small neighborhood around a. The challenge...

Here's my work: http://i.imgur.com/UMj72Ub.png I used the surface area differential for a parametrized surface to solve for the area of that paraboloid surface. My friend tried solving this by parametrizing with x and y instead of r and theta which gave him the same answer. I would greatly...

Homework Statement A small lump of ice is sliding down a large, smooth sphere with a radius R. The lump is initially at rest. To get it started, it starts from a position slightly right to the sphere's top, but you can count it to start from the top. The lump is fallowing the sphere for a...

Sorry guys, I have some differential topology homework, and I may be asking a lot of questions in the next few days. Problem Statement Suppose M_1,...,M_k are smooth manifolds and N is a smooth manifold with boundary. Then M_1×..×M_k×N is a smooth manifold with a boundary. Attempt Since...

Hi guys, i would need some help with movement on ellipse. I am using basicequation for figuring out position on ellipse : basically getting degree, converting that to radians and using radians to figure out posX and Y. Basic stuff. degree += speed * Time; radian = (degree/180.0f) *...

Help slove pls!:( A uniform smooth sphere P,of mass 3m, is moving in a straight line with speed u on a smooth horizontal table.Another uniform smooth sphere Q, of mass and m and having the same radius as P, moving with speed 2u in the same straight line as P but in the opposite direction to P...

A sphere of mass 'm' collides with a fixed plane with initial speed 'u' at an angle 'α'(alpha). The sphere rebounds with speed 'v' at an angle 'β' with the normal. The plane being fixed remains at rest. We applied Newton's Experimental law( along the common normal(CN) The equation after...

I'd like someone to confirm whether I am on the right track here. Most formulations of the twin paradox involve a sharp turn-around with infinite acceleration. I suppose that there is an SR-only description of a non-infinite acceleration - a kind of 'smooth' version of the twin paradox. But my...

Just came across this article, which details findings from the Fermi telescope that have an interesting consequence to quantum gravity theories: www.space.com/19202-einstein-space-time-smooth.html First, what do you guys think of this finding? Is it legitimate, or flawed? It's obviously...

As you should know, a function can be smooth in some neighborhood and yet fail to be analytic. A canonical example is ##\exp (-1/x^2)## near ##x = 0##. My question is this: suppose I want to express a given function as a double series, f(x) = \sum_{m = 0}^\infty \sum_{n = 0}^\infty a_m x^m...

Note: If this is the wrong sub-forum for this question please move it. I was not sure if this question should go in the General section or not. Question: I want to create a smooth non-piecewise curve in ℝ^{3} (3-space) such that it's intersection with the xy-plane consists of the integer...

can a photon have a perfectly smooth orbit? say for e.g. you have a photon orbiting a point, if its wavelength were to become twice the diameter of its orbit then would the wave not become a replica of the orbit offset by the amplitude? similarly say the amplitude is the radius of the...

Homework Statement If B is Hermitian, show that BN and the real, smooth function f(B) is as well. Homework Equations The operator B is Hermitian if \int { { f }^{ * }(x)Bg(x)dx= } { \left[ \int { { g }^{ * }(x)Bf(x) } \right] }^{ * } The Attempt at a Solution Below is my...

This is a prediction that is made every day. If I do a back test assemble a curve composed of each days' prediction, I get fair results. However, if I smooth this backtest curve, I get fantastic results. So what I need to do is take today's prediction and the prediction time history...

Homework Statement Consider the map \Phi : ℝ4 \rightarrow ℝ2 defined by \Phi (x,y,s,t)=(x2+y, yx2+y2+s2+t2+y) show that (0,1) is a regular value of \Phi and that the level set \Phi^{-1} is diffeomorphic to S2 (unit sphere) Homework Equations The Attempt at a Solution So I...