Smooth Definition and 229 Threads

Now F:S^2->R^4 is a map of the following form: F(x,y)=(x^2-y^2,xy,xz,yz) now using the smooth covering map p:S^2->RP^2, p is the composition of inclusion map i:S^2->R^3 and the quotient map q:R^3\{0}->RP^2. show that F descends to a smooth embedding of RP^2 into R^4. Is the problem asked to...

Suppose M and N are smooth manifold with M connected, and F:M->N is a smooth map and its pushforward is zero map for each p in M. Show that F is a constant map. I just remember from topology, the only continuous functions from connected space to {0,1} are constant functions. With this be...

Is there a "standard" probability measure one would use for the set of smooth real-valued functions on [a, b]? My intuition is picturing a setup where you cut out shapes in the x-y plane, and then the set of functions whose graphs are contained in that shape have a measure proportional to the...

How does one prove that the smooth structure on 2-manifolds is unique? Source? Thx!

Show from Eq. 1.1 that the below function is smooth at t = 1 and at t = 2. Is it smooth at any 1 < t < 2? x(t) = 1.0 + 2.0 t 0 ≤ t ≤ 1 3 + 4(t − 1) 1 ≤ t ≤ 2 7 + 3(t − 2) 2 ≤ t for equation 1.1 my book gives me: lim dt→0 [x(t + dt) − x(t) ]/dt= 0 This problem...

"standard" Square is not a Smooth Submanifold of R^2 Hi, everyone: I am trying to show the standard square in R^2, i.e., the figure made of the line segments joining the vertices {(0,0),(0,1),(1,0),(1,1)} is not a submanifold of R^2. Only idea I think would work here is using the...

Homework Statement A smooth ring with a mass m is threaded through a light inextensible string .The ends of the string are tied to two fixed points A and B on a horizontal ceiling so that the ring is suspended and can slide freely on the string.A hotizontal force acts on the ring in a...

I can't find anything on the internet about this... How is it done? I've got a smooth curve given by these parametric equations: y = 5cos( theta )+5; x = 5sin( theta ) taking g = 9.81 how can I model the position of the ball as a function of time? Or how can I model it so that i can...

Ball sliding down smooth curve Homework Statement A mass of 2 kg is dropped vertically into a frictionless slide located in the x-y plane. The mass enters with zero velocity at (-5,5) and exits traveling horizontally at (0,0). Assuming the slide to be perfectly circular in shape construct a...

Hey there! I'm working on a bit of a game modification, and I'm looking for a little help with some physics-related math. I'm only in my second year of high-school physics, so I'm not completely sure where to go with this question. And sorry if this is in the wrong forum. Essentially, I'm...

Homework Statement I have been asked to explain what Endoplasmic Reticulum Smooth and Rough is and what the functions are of it. Homework Equations N/A The Attempt at a Solution There are two recognized parts of the ER - Rough and Smooth - with the rough part of the ER...

I haven't been studying quantum mechanics for very long, and I've only just started reading about the path integral formulation, so I don't know many of the details yet, but I noticed something peculiar. The path integral formulation is possible because of the similarities of the Schroedinger...

I've been trying to find the answer to this question for a*long* time. I think I've finally come to the right place for the answer. Can a smooth ball be thrown so it curves? A lacrosse ball is is basically smooth. There's a small seem where the halves are glued -or melted or whatever and...

Sorry I am new to the forum, I don't know how to type in the integrals and stuff. Homework Statement Let S be the portion of the surface y = x2 where 0 <= z <= X <= 2. Compute the surface area of S. Homework Equations r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k A(S) = integral ru X rv dAThe Attempt...

Homework Statement I'm currently working on this question: A particle moves on the smooth inside surface of the hemisphere z = -(a^2 - r^2)^(1/2), r <= a, where (r, theta, z) denote cylindrical polar coordinates, with the z-axis vertically upward. Initially the particle is at z =...

1. A mass of 5kg moves horizontally across a smooth surface at 1ms-1. Another mass of 5kg is lowered gently onto the first mass. a) What is the change in velocity of the mass? b) What impulse has been applied to the mass? c) Is this an elastic 'collision'? 2. force x time = change in...

What is a "smooth" function? If someone tells you to consider a "smooth" function on the complex plane, what does that mean, exactly? Does it mean that the partial derivatives of the real and imaginary parts exist and are continuous? Does it mean something else?

smooth and L^2 on R^n. Will it be bounded?? Hello, If a function, say u, is smooth and L^2 on R^n. Will it be bounded?? In the case of n=1 I would say that it obviously is so. Because if it were unbounded then it wouldn't be L^2. But in the case of n=2 (or higher). I can imagine a...

Hi everyone, this is a problem I posted here a month ago but it wasn't given any attention from the helpers here. I still cannot solve it. So I will greatly appreciate any help. 1. Homework Statement Three small balls of masses m, 2m and 3m are placed on a smooth horizontal surface so that...

Smooth transition map (easy!?) Homework Statement Check the transition map http://img132.imageshack.us/img132/4341/18142532.png is smooth in the set for which their images intersect The Attempt at a Solution I have thought of two ways to show this. (1) Show that Φ is a composition...

Hi! I have the following problem: Let P be a principal bundle over a manifold M, p: P -> M. Let G be the lie group acting on P from the right. Now let U be an open set on M and s : U -> P a smooth section. Now it has been said that we can define a local trivialization of P by t ...

TL;dr: "I am not sure how to calculate a smooth transition of thrust between one vector and another." Hi, this looked like the right place to post this being "The mathematics of change and motion". I am programming a simple game where an enemy chases after the player in an open space (no...

Dear friends, I am a new member of physics forums, so this is may new message. Already thanks to you for your helps to my question. I research some applied mathematician problem's numerical solutions. There are initial-boundary value problems. I need an initial condition function which must...

Homework Statement α(t) = (sint, cost + ln tan t/2) for α: (0:π) -> R2 Show that α is a smooth, parametrized curve, which is regular except for t = π/2 The Attempt at a Solution I am familiar with the definitions of smooth and regular, which I have provided below, however I am...

Is Spacetime Smooth? Smooth: infinitely differentiable If there were a limit to the differentiability of matter's motion through time, I'd assume it would be at the quantum level (where particles are not actually point particles). Example: When I accelerate in my car, the value of my...

Homework Statement f, \hat{f} \in C_c^\infty(\mathbb{R}^n) Homework Equations \hat{f} = \int_{\mathbb{R}^n} f(x) e^{-2\pi i \xi \cdot x} \,dx \check{f} = \int_{\mathbb{R}^n} \hat{f}(x) e^{2\pi i \xi \cdot x} \,d\xi The Attempt at a Solution As C_c^\infty \subset...

(I already have self-taught calculus) Are there functions f:\mathbb{R} \rightarrow \mathbb{R} that are EVERYWHERE smooth (infinitely many times differentiable \forall x \in \mathbb{R}) but NOWHERE analytic (Taylor series does not equal f(x) for any real x. don't gimme a bump function.)? example...

I'm trying to find a Fourier series for exp(-ax) where a is a positive constant. How is exp(-ax) piecewise smooth?

Homework Statement I've got a rigid rod, length 2a, held at an angle \alpha above a smooth horizontal surface. Its end is in contact with the ground. My two generalized coordinates are x, the horizontal position of the center of the rod and \theta the angle between the rod and the vertical...

If a body is in a state of acceleration, thus a changing velocity and it collides with another body. When I am calculating the force of the body do I take into consideration the changing rate of acceleration or just the instantaneous velocity (velocity reached at this point) upon the point of...

https://www.getsmoothaway.com/ver3/index.asp?refcode=smooth3

Hi, I'm confused about what differentiation on smooth manifolds means. I know that a vector field v on a manifold M is a function from C^{\infty}(M) to C^{\infty}(M) which is linear over R and satisfies the Leibniz law. This should be thought of, I'm told, as a 'derivation' on smooth...

My lecturer gave me a question that included giving a proof that a particular function is smooth. I have taken a course on analysis and have no problems when it comes to proof of continuity; i was just wondering what the usual steps are in proving that a function is smooth. I would guess that...

Hi, I'm looking to find a function y=f(x), invertible to x=f(y) and written in terms of elementary functions and operations, that can represent a straight line Ax+B where x<<T and another straight line Cx+D where x>>T, where T is the x position where the two lines would cross. In the region...

Homework Statement When a function is piecewise smooth, then f and f' (the derivative of f) are piecewise continuous. In my book they mention "a function f, which is continuous and piecewise smooth". How can f be both continuous and piecewise continuous?

Homework Statement Assume that f:[a,b] \rightarrow \mathbb{R} is continuously differentiable. A critical point of f is an x such that f'(x) = 0 . A critical value is a number y such that for at least one critical point x, y = f(x). Prove that the set of critical values is a zero set...

I have a question that i am struggling to come to terms with. I know how to get the answer but i am not 100% sure why what i am doing works. The question : A smooth plane is inclined at X degrees to the horizontal. A block of mass 5KG is held at rest on the plane by a horizontal force...

My question is simple : Suppose that f is in C^\infty(U, [0,1]) where U is an open of R^n . Is there g in C^\infty(R^n,[0,1]) such that f=g on U ? I would say yes, but I don't know how to prove it. Thanks

Hello, all. I know what I want, but I just don't know what it's called. This has to do with regression (polynomial fits). Given a set of N (x,y) points, we can compute a regression of degree K. For example, we could have a hundred (x,y) points and compute a linear regression (degree 1). Of...

An object rolls down a smooth plane (whose angle and length are given) starting from rest. At the end of the plane it reaches horizontal rough surface, it rolls again on this surface and comes to rest. Given the friction coefficient how do we calculate the distance traveled by the object on...

Is rotating with whatever angular velocity we want in a vacuum, it and you being the only objects around (assume no intrinsic properties like charge/mass). How do you tell that it's doing so? And if it has perfect rotational symmetry, does it even make sense to say that it's rotating? This has...

Let G\subset\mathbb{R}^n be some open set, and x_1, x_2:[0,1]\to G be differentiable paths with the same starting and ending points. Assume that there exists a homotopy f:[0,1]^2\to G between the two paths. That means that the f is continuous, and the following conditions hold...

What is the direction of beating off some rubber ball, throwed on some smooth surface? It depends of the angle that I will throw the ball and why?

1. A sphere of mass m impinges obliquely on a sphere of mass M, which is at rest. The coefficient of restitution between the spheres is e. Show that if m=eM, the directions of motion after impact are at right angles 2. Coefficient of restitution - 3. I really don't think I have a...

Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous? How can I prove it? Thanks.

Hi, hope this is the right area. Also please excuse me completley ignoring the template, I don't think it's applicable for the problem. I'm an honours year student in a comp sci course and I've decided to do an implimentation of Smooth Particle Hydrodynamics in a 3d application as my topic...

A={ {{cos x, -sin x},{sin x, cos x}}|x \inR}, show that set A is smooth manifold in space of 2x2 real matrix. What is tangent space in unity matrix? My questions about problem: 1. What is topology here? (Because I need topology to show that this is manifold) 2. In solution they say that...

Hello So, my advanced calculus book (Folland) has this theorem... If f is of class C^k on an open set S, then... \partial i_1 \partial i_2 ... \partial i_k f = \partial j_1 \partial j_2 \partial j_k f on an open set S whenever the sequence {j_1 ,..., j_k} is a reordering of the sequence {i_1...

Homework Statement In calculus II, vector valued function in space. The vector function r(t)=f(t)i+g(t)j+h(t)k. The curve traced by r is smooth if dr/dt is continuous and NEVER 0. Homework Equations I don't understand why there is "NEVER 0" in the above statement, in order for the...

Hi, everyone: I have been reading Boothby's intro to diff. manifolds, and in def. 4.3, talking about 1-1 immersions F:N->M , n and m-mfld. and M an m-mfld. That: "...F establishes a 1-1 correspondence between N and the image N'=F(N) of M. If we use this correspondence to give...