Smooth Definition and 229 Threads

Can unquantized fields be considered smooth curved abstract manifolds? Say free particle solutions of the Dirac equation or the Klein Gordon equation? Can quantized fields also be considered curved abstract manifolds? Thanks for any help!

I wish it wasn't out of desperation that I'm making this first post! I have a neural network that is making predictions, the next 5 time points per training. Back testing consists of appending these 5 point sets together to produce a data set that spans time over a much longer period...

Homework Statement A uniform circular disc has mass M and diameter AB of length 4a. The disc rotates in a vertical plane about a fixed smooth axis perpendicular to the disc through the point D of AB where AD=a. The disc is released from rest with AB horizontal. (See attached diagram) (a)...

Homework Statement A mass of 30kg on a smooth horizontal table is tied to a cord running along the table over a frictionless pulley mounted at the edge of the table. A 10kg mass is attached to the other end of the cord. When the two masses are allowed to move freely the tension in the cord...

I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p\inM Proposition: TpF : TpM → TF(p) is...

Homework Statement an block of mass 6kg is sliding down an inclined plane inclined at 45° to the horizontal. Find the acceleration of the mass. Homework Equations F=ma The Attempt at a Solution Is it correct that the weight times sin45 will equal ma, because that's how i first did...

Say you have an object on a smooth friction-less slope, and a force 10N (acting parallel to the slope) is applied to it to move it at a distance 5m up the slope. Work done by the force is: force*distance. In this case, 10*5 = 50 joules of energy Does all that energy get transferred to...

Author: John Lee Title: Introduction to Smooth Manifolds Amazon link https://www.amazon.com/dp/0387954481/?tag=pfamazon01-20 Prerequisities: Topology, Linear algebra, Calculus 3. Some analysis wouldn't hurt either. Level: Grad Table of Contents: Smooth Manifolds Topological Manifolds...

Homework Statement Construct a C∞ natural differential structure on the ellipsoid \left\{(x_{1}, x_{2}, x_{3})\in E | \frac{x_{1}^{2}}{a^{2}}+\frac{x_{2}^{2}}{b^{2}}+ \frac{x_{3}^{2}}{c^{2}}=1\right\} Is this diffeomorphic to S2? Explain. Homework Equations Do I need to prove...

Hello, Homework Statement I am asked to show that the time it would take a bead to slide down a smooth cord, positioned at an angle beta wrt the vertical axis, is independent on that angle (between the cord and the axis). The bead starts its slide from rest. Homework Equations The...

How might I prove the following? 1) If f ∈ C(Rn) and f has compact support, then f ∈ Lp(Rn) for every 1 ≤ p ≤ ∞. 2) If f ∈ C(Rn), then f ∈ Lp_{loc}(Rn) for every 1 ≤ p < ∞. (Where C(Rn) is the space of continuous functions on Rn)

Hello, I have a question regular values and smooth homotopies. Usually in giving the definition of regular value, they disregard the regular values whose inverse image is empty set (although they should be called regular values if we want to be able to say that set of regular values is dense for...

I tried to solve a time independent schroedinger equation with a finite potential well today by solving it in 3 pieces, one for in the box and 2 for the outsides. By setting the equations equal to each other where they met at the edges of the box, by setting the integral of everything squared =...

Recently I have been working through a text on Differential Topology and have come across the notion of smooth homotopy. Now the textbook (along with every other source I can find on the matter) defines a smooth homotopy of maps f,g:M \rightarrow N as a smooth map h:M \times [0,1] \rightarrow N...

Okay, here is a question that I have not been able to get answered. My Grandmother, when she was alive made crafts out of broken glass. The goal was to have broken glass with no sharp edges. These are the steps. First she would heat the bottle in the oven. Then she would remove the bottle using...

I downloaded this a couple days ago: https://chrome.google.com/webstore/detail/lfkgmnnajiljnolcgolmmgnecgldgeld. It is amazing - right click mouse gestures. Tons of different gestures, you can disable the built in ones, add new ones, modify the gesture movements, etc. It speeds up browsing...

Hey, I'm struggling with this question, any help would be great. 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 My attempt...

Here is the situation I am concerned with - Consider a smooth curve g:[0,1] \to M where M is a topological manifold (I'd be happy to assume M smooth/finite dimensional if that helps). Let Im(g) be the image of [0,1] under the map g . Give Im(g) the subspace topology induced by...

I recently read an article that stated something to the effect that it is thought that space is divided up into quantized little indivisible chunks, the size of which is called the Planck length. is this a new theory and where can i find more information on it? ANY information Specifically...

I'm trying to prove that if a function is continuous on [a,b] and smooth on (a,b) then there's a point x in (a,b) where f'(x) exists. The definition of smoothness is: f is smooth at x iff \lim_{h \rightarrow 0} \frac{ f(x+h) + f(x-h) - 2f(x) }{ h } = 0 . I'm starting with the simpler case...

Homework Statement Prove that f(x) is a smooth function (i.e. infinitely differentiable) Homework Equations ln(x) = \int^{x}_{1} 1/t dt f(x) = ln(x) The Attempt at a Solution I was thinking about using taylor series to prove ln(x) is smooth but I'm strictly told to NOT assume f(x) = ln(x)...

Hi, I've started to write a simulator for SPH, and I'm fallowing a paper on SPH simulation, The problem is that in that paper they show the formulas for computing force but they don't tell you on what vector do you need to apply this force. So here are the formulas for pressure, viscosity...

Confused with the use of the word "smooth". [Multi-Variable Calc course] A couple weeks ago we went over the Fundamental Theorem of Line Integrals, which requires "smooth" simple connected curves. My professor's definition of smooth was a curve having "no corners". Now, with Green's Theorem...

Hi, I want to ask a problem from Lee ' s book Introduction to Smooth Manifolds: Let F_p denote the subspace of C^\inf(M) consisting of smooth functions that vanish at p and let F^2_p be the subspace of F_p spanned by functions fg for some f,g \in F_p. Define a map \phi: F_p----->...

Give me a function that is piecewise continuous but not piecewise smooth

Hi, I'm having trouble understanding why is tangent space at point p on a smooth manifold, not embedded in any ambient euclidean sapce, has to be defined as, for example, set of all directional derivatives at that point. To my understanding, the goal of defining tangent space is to provide...

In my textbook, piecewise continous/piecewise smooth is always defined on interval[a,b]. Can they be defined on open interval (a.b)?

I have difficulty understanding the following Theorem If U is open in ℝ^2, F: U \rightarrow ℝ is a differentiable function with Lipschitz derivative, and X_c=\{x\in U|F(x)=c\}, then X_c is a smooth curve if [\operatorname{D}F(\textbf{a})] is onto for \textbf{a}\in X_c; i.e., if \big[...

Constructing a "smooth" characteristic function Suppose I'd like to construct a C^\infty generalization of a characteristic function, f(x): \mathbb R \to \mathbb R, as follows: I want f to be 1 for, say, x\in (a,b), zero for x < a-\delta and b > x + \delta, and I want it to be C^\infty on...

Hello everyone, I just had a quick question I was hoping somebody could answer. If f: M \times N \rightarrow P is a smooth map, where M,N and P are smooth manifolds, then is it true for fixed m that f_m : N \rightarrow P is smooth, where f_m (n) = f(m,n)? Any help would be appreciated.

how to extend a locally defined function to a smooth function on the whole manifold ,by using a bump function?

Hi, I want to charectize the function whose cube is smooth from R to R. For example x^1/3 is smooth and olsa any polynomial but how can i charectrize it? Thanks

Hi! I want to know if any smooth manifold in n-dimensional euclidean space can be compact or not. If it is possible, then could you give me an example about that? I also want to comfirm whether a cylinder having finite volume in 3-dimensional euclidean space can be a smooth manifold. I...

Homework Statement A uniform ladder of length L and mass M has one end on a smooth horizontal floor and teh other end against a smooth vertical wall. The ladder is initially at rest in a vertical plane perpendicular to the wall and makes an angle \theta0 with the horizontal. (a) Write down the...

This is really just a general question of interest: can every smooth n-manifold be embedded (in R2n+1 say) so that it coincides with an affine variety over R? Does anyone know of any results on this?

Homework Statement Consider the ellipsoid L \subsetE3 specified by (x/a)^2 + (y/b)^2 + (z/c)^2=1 (a, b, c \neq 0). Define f: L-S^{2} by f(x, y, z) = (x/a, y/b. z/c). (a) Verify that f is invertible (by finding its inverse). (b) Use the map f, together with a smooth atlas of S^{2}, to...

Right now, I'm reading through Lee's Intro to Smooth Manifolds and I was wondering if there is a website somewhere that has problems different from the book. Or if there is another book out there that covers about the same material as Lee's that would be good to. Thanks in advance.

Hello, I am interested in making plots similar to those attached (I have my own date :P). Any recommendations on what software to use? I suppose there are two things I would like it to do (although either one alone would be nice): 1. Create nice radial/polar diagrams with the angles and...

"smooth" curves with cusps in 3d While reviewing basic calculus, I noticed that the curve (1+t^2,t^2,t^3), which clearly has a cusp at (1,0,0), has a derivative curve (2t,2t,3t^2) which is clearly smooth. This struck me as odd since differentiation usually seems to turn cusps into...

obviously in one coordinate neighborhood it can't.. I'm thinking of a line which smoothly develops into a surface : -----<< what particular properties would this object have.. Thanks :)

compute the integral of ...i can't find the integral symbol, but is the standard integral sign with a circle in the middle of it and a c to the bottom right of it. F*dr where C is an arbitrary closed smooth contour that does not enclose the origin.

Hello, I am learning about smooth manifolds through Lee's text. One thing that I have been pondering is describing manifolds such as |x| which are extremely well behaved but not smooth in an ordinary setting. It is simple to put a smooth structure on this manifold, however that is...

Homework Statement Hello everyone. I'm trying to finish the following problem: Show that \intz^n dz = 0 for any closed smooth path and any integer not equal to -1. [If n is negative, assume that γ does not pass through the origin, since otherwise the integral is not defined.] Homework...

Hi, I have been told that in R^2 the unit circle {(x,y) | x^2 + y^2 = 1} is smoothly mappable to the curve {(x,y) | x^4 + y^2 = 1}. Can someone please tell me what this smooth map is between them? I can only think of using the map (x,y) --> (sqrt(x), y) if x is non-negative and (sqrt(-x), y)...

One form of the third law of thermodynamics states that ; in absolute zero temperature,entropy is absolute that means it does not depend on any of the properties of the system.the question is ; is this a radical feature for absolue zero or absolute zero is only a limit i.e. the dependence of...

Homework Statement A particle is free to slide along a smooth cycloidal trough whose surface is given by the parametric equations: x = \frac{a}{4}(2 \theta + \sin{2 \theta}) y = \frac{a}{4}(1 - \cos{\theta}) where 0 <= \theta <= \pi and a is a constant. (sorry, TeX is not working for...

We can show that any vector field V:M->TM(tangent bundle of M) is smooth embedding of M, but how do we show that these smooth embeddings are all smoothly homotopic? How to construct such a homotopy?

does friction in any way increase when the surfaces get very smooth ...? a teacher told us that it can increase due to electromagnetism...is it true...i searched the net but couldn't get any useful info...

Homework Statement Block A has a mass of 4.90 kg and rests on a smooth table. It is connected to Block B which has a mass of 3.50 kg. Block B is released from rest. How long does it take Block B to travel 0.840 m? Description of figure; Mass A is on the frictionless table. Mass B is...

This should hopefully be a quick and easy answer. I'm running through Lee's Introduction to Smooth Manifolds to brush up my differential geometry. I love this book, but I've come to something I'm not sure about. He states a result for which the proof is an exercise: I'm not quite clear on...