Smooth Definition and 229 Threads

I have been thinking about rotation number of regular smooth curves in different surfaces. Here is how I’ve been defining these things: a regular smooth curve is a map from ##S^1 \rightarrow \mathbb{R}^2## whose derivative is non-vanishing. If we have a regular smooth curve ##\gamma## as well...

Hi, in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space ##\mathbb R^3##. On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts. I believe the same view might also be...

As discussed in a recent thread, I'd ask whether any smooth section over a Mobius strip must necessarily take value zero on some point over the base space ##\mathbb S^1##. Edit: my doubt is that any closed curve going in circle two times around the strip is not actually a section at all. Thanks.

Hello! There is a proof that Grassmannian is indeed a smooth manifold provided in Nicolaescu textbook on differential geometry. Screenshots are below There are some troubles with signs in the formulas please ignore them they are not relevant. My questions are the following: 1. After (1.2.5)...

Hi, I'm puzzled about the definition of smooth coordinate chart for a manifold (e.g. spacetime). From my point of view there is no invariant way to define a smooth coordinate chart since a coordinate system is smooth only w.r.t. another coordinate system already defined on the given manifold...

Definitions: 1. A map ##p : X → Y## of smooth manifolds is called a trivial fibration with fiber ##Z## which is also a smooth manifold, if there is a diffeomorphism ##θ : X → Y ×Z## such that ##p## is the composition of ##θ## with the natural projection ##pr_1:Y × Z → Y##. 2. A map ##p: X →Y##...

Given the definition of a smooth map as follows: A continuous map ##f : X → Y## is smooth if for any pair of charts ##\phi : U →R^m, \psi:V →R^n## with ##U ⊂ X, V ⊂Y##, the map ##\phi(U ∩f^{-1}(V)) → R^n## given by the composition $$\psi ◦ f ◦ \phi^{-1}$$ is smooth. Claim: A map ##f : X → Y##...

Let ##f : \mathbb{R}^n \to \mathbb{R}^n## be a smooth proper map that is not surjective. If ##\omega## is a generator of ##H^n_c(\mathbb{R}^n)## (the ##n##th de Rham cohomology of ##\mathbb{R}^n## with compact supports), show that $$\int_{\mathbb{R}^n} f^*\omega = 0$$

Roughly, how much force does it take to make 1 mm surface bump of diameter 16 mm circle area out of 1 mm thick steel plate of far larger area (e.g. a muscled hand pounding it laid over the base with 16 mm dia. hole by M16 bolt medium is viable enough) ?

Hello. I read about smooth infinitesimal analysis and I have several questions: 1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6) 2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2)...

Hello. How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.) Thanks.

Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$. I need to prove that the map is differentiable. But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable? MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math

Statement : Here is the statement from the text that I paste to the right. Diagram : Does anyone have a diagram (image) as to how does the situation look? Normal Reaction : When a rod rests against a smooth wall, we know that the direction of the reaction is normal to the wall. I understand...

Attempting to draw the FBD for this problem but was wondering what to do about the cable, especially when I determine the internal loadings at E. At C, do I only draw one force arrow (CB), or draw both on the left and right side?

Hi, I've a doubt about the definition of curve. A smooth curve in ##\mathbb R^2## is defined by the application ##\gamma : I \rightarrow \mathbb R^2##. Consider two maps ##\gamma## and ##\gamma'## that happen to have the same image (or trace) in ##\mathbb R^2##. At a given point on the...

Hello everyone, I'm looking for help for the problem 3 of the chapter III. Schrödinger's equation, §25 The transmission coefficient of the Volume 3 of the Landau-Lifshitz book (non-relativistic QM). In this exercise Landau considers a smooth potential wall $$\frac{U_0}{1 + \exp{\left(-\alpha x...

Hi, I have the following function, which is computed by: (x+n)/(x+y+n+m), where x, y are real numbers n, m are natural numbers What techniques I can use to smooth the function preventing it to jump up or down at an early stage. I would appreciate your suggestion. Thanks

Is there a function that takes positive values only in the unit ball not including the boundary points defined by the set ##\{x^2+y^2+z^2<1\}##, and ##0## everywhere else?

Hi all, My question is about the attenuation and delay terms in part (1). what are attenuation and delay terms describing in physical phenomenon? thank you. What do "attenuation" and "delay" mean in terms of real-life physical phenomena? Consider the wave equation for spherical waves in...

I am studying on Zorich, Mathematical Analysis II, 1st ed. pag. 174-175. After having properly explained how orientations (equivalence classes) are defined for smooth k-dimensional surfaces in ##\mathbb {R} ^ n## that can be described with a single map, move on to the more general case by...

From Zorich, Mathematical Analysis II, 1st ed., pag.163: where the referred mapping (12.1) is a map ##\varphi:I_k\to U_S(x_0)##, in which: 1. ##I_k\subset\mathbb{R}^n## is the k-dimensional unit cube, 2. ##x_0## is a generic point on the surface ##S## and ##U_S(x_0)## is a neighborhood of...

From Zorich, Mathematical Analysis II, sec. 11.5.2: where as one can read from the statement, the sets could also be unbounded. I do not report here the proof of the fact a), beacuse I have no doubt about it and one can, without the presence of dark steps in the reasoning, assume a) as...

I'm studying differential geometry basics for general relativity (no specific source, just googling around). I know that spacetime is modeled as a ##4##-dimensional smooth manifold. Smooth manifold means that we consider a restriction of the maximal atlas such that all charts in it are...

Edit, the vector that rotates below might not rotate at all. Please forgive any mistaken statements or sloppiness on my part below. I think that by some measure a helicoid can be considered a smooth curved 2 dimensional surface except for a line of points? Consider not the helicoid above...

Figure of the problem. There are many exercises like these, and I've read the whole chapter and I got no clue where to start here. Kinetic energy from gravity is: E = mgh = 39.3m I could try change v to ωr in the K equation but it will leave me nowhere because I don't have the mass nor the...

Answers- 1,3,4 My attempt, the wedge being massless, there shoul not be any force acting on as it will then have infinite acceleration, so by that i really can't think of how force is applied on pully.

Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that...

Hi I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This...

I do not know to start. Here is the problem.Determine if the given function is piecewise continuous, piecewise smooth, or neither. Here $x\neq0$ is in the interval $[-1,1]$ and $f(0)=0$ in all cases. 1. $f(x)=sin(\frac{1}{x})$ 2. $f(x)=xsin(\frac{1}{x})$ 3. $f(x)={x}^{2}sin(\frac{1}{x})$ 4...

1) Are there any periodic alternating series functions other than sine and cosine (and series derived from them, like the series for cos(a) * cos(b))? 2) What is the following series called when x is (0,1) and (1,2]? Quasiperiodic? Semi? \sum_{n=0}^\infty \, (-1)^n \...

I am reading Bjorn Ian Dundas' book: "A Short Course in Differential Topology" ... I am focused on Chapter 2: Smooth Manifolds ... ... I need help in order to fully understand Example 2.2.6 ... ... Example 2.2.6 reads as follows: My questions are as follows: Question 1 In the above text...

Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8 ##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)## As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...

Let ##X## be the set of all functions infinitely differentiable at ##0##. Let's define an equivalence relation on $X$ by saying that ##f\sim g## if there exists a sufficiently small open interval ##I## containing ##0## such that ##f(x)=g(x)## for all ##x## in ##I##. Then the set of germs of...

Hi there. It is obvious that if you have two differentiable functions ##f(x)## and ##g(x)##, then the product ##h(x)=f(x)g(x)## is also smooth, from the chain rule. But if now these functions are multivariate, and I have that ##h(x,y)=f(x)g(y)##, that is ##f(x,y)=f(x)## for all y, and similarly...

Hi, Given a smooth distribution of lines in R2, could we assert that there is a unique distribution of curves such that: - the family of curves "fill in" R2 completely - every curve is tangent at every point to one of the smooth distribution of lines

Homework Statement Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve? Homework Equations $\phi(\theta) =...

I have been stuck several days with the following problem. Suppose M and N are smooth manifolds, U an open subset of M , and F: U → N is smooth. Show that there exists a neighborhood V of any p in U, V contained in U, such that F can be extended to a smooth mapping F*: M → N with...

Hello everyone, my question is: what are the criteria that must be satisfied for the pushforward of a smooth vector field to be a smooth vector field on its own right? Consider a smooth map \phi : M \longrightarrow N between the smooth manifolds M and N. The pushforward associated with this map...

Hi. All technical implementations of the Magnus effect I can find on Google (such as ships) seem to use fairly smooth cylinders. Why? Shouldn't the efficiency increase with increasing friction between cylinder and fluid, for example with a rough surface or even attaching blades?

Two threads share a common buffer (an image, say). One thread (visualization which has an almost fixed delay). The other, image generation can have a delay either much greater or much lesser than the fixed one. What is the best paradigm to synchronize these threads using pthreads in order to...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 4: Complex Integration, Section 1.2 Smooth and Piecewise Smooth Paths ... I need help with some aspects of Example 1.3, Section 1.2, Chapter 4 ... Example 1.3, Section 1.2, Chapter 4...

I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...

Ok, so this is an undergraduate level question probably but consider this R. C. Hibbeler's Textbook on Mechanics of Materials says this about thrust bearings and smooth journal Bearings : In other words, it says nothing about reaction moments. Now this chart has other 2D connection types...

For example, the curvature due to a mass; does that curvature continue passing from within to outside the mass's light cone? If so, is the mass subject to the external curvature? If not, does the curvature have a discontinuity at the light cone surface?

I want to represent data with 2 variables in 2D format. The value is represented by color and the 2 variables as the 2 axis. I am using the contourf function to plot my data: load('data.mat') cMap=jet(256); F2=figure(1); [c,h]=contourf(xrow,ycol,BDmatrix); set(h, 'edgecolor','none'); Both xrow...

Hi there! I chanced upon this problem whilst trying to brush up my Classical Mech knowledge and found it confusing. Hope someone out there can provide an insight! Homework Statement A bowling ball of mass m and radius R sits on the smooth floor of a subway car. If the car has a horizontal...

Why can't spacetime or manifold be crumpled like a piece of rug or paper.. why should it be smooth? Does it have to do with time or the geodesics discontinuous or time sporatic if it is not smooth?

Homework Statement For a1 < a2 < b2 < b1 and c > 0 real numbers show that there exists a smooth (bump) function f : R → R with the following property: f(x) = ... - 0 if x ≤ a and x ≥ b - c if a2 ≤ x ≤ b2 - monotonic...

Homework Statement The problem statement is in the attached picture file and this thread will focus on question 7 Homework Equations The length of a curve formula given in the problem statement Take a polygon in R^n as an n-tuple of vectors (a0,...,ak) where we imagine the vectors, ai, as the...

I am reading Larson's Calculus textbook and come across this paragraph about vector-valued function: The parametrization of the curve represented by the vector-valued function $$\textbf{r}(t) = f(t)\textbf{i} + g(t)\textbf{j} + h(t)k$$ is smooth on an open interval $I$ when $f'$, $g'$ and...