What is Smooth: Definition and 225 Discussions

Smooth muscle is an involuntary non-striated muscle, so-called because it has no sarcomeres and therefore no striations. It is divided into two subgroups, single-unit and multiunit smooth muscle. Within single-unit muscle, the whole bundle or sheet of smooth muscle cells contracts as a syncytium.
Smooth muscle is found in the walls of hollow organs, including the stomach, intestines, bladder and uterus; in the walls of passageways, such as blood, and lymph vessels, and in the tracts of the respiratory, urinary, and reproductive systems. In the eyes, the ciliary muscle, a type of smooth muscle, dilate and contract the iris and alter the shape of the lens. In the skin, smooth muscle cells such as those of the arrector pili cause hair to stand erect in response to cold temperature or fear.

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  1. K

    Masses sliding on a smooth wedge

    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)...
  2. J

    MHB Force on 10 Kg Block on 51° Inclined Plane

    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?
  3. H

    I Ladder slipping against a smooth wall

    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...
  4. S

    Smooth rolling motion - conservation of energy?

    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...
  5. mnb96

    A Derivative of smooth paths in Lie groups

    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...
  6. baby_1

    Piece wise smooth function

    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...
  7. J

    Unique smooth structure on Euclidean space

    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...
  8. C

    Why are circles infinitely smooth if they have degrees?

    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?
  9. K

    Mass sliding on rough and smooth surfaces

    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...
  10. O

    Finding Tension in a Ring on a Smooth Hoop

    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...
  11. T

    Approximation of second derivative of a smooth function

    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...
  12. pellman

    Are all smooth functions square-integrable?

    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 (...
  13. K

    Mass not sliding on a smooth, accelerating base

    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 $$\tan\alpha=\frac{ma}{mg}\;\rightarrow\; a=g\tan\alpha$$
  14. R

    Kinetic friction on smooth then rough surface

    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...
  15. S

    Exiting 101: Tips & Tricks for a Smooth Transition

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

    Low Jerk Elevator Ride: max speed & acceleration

    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...
  17. caffeinemachine

    MHB To Show that a Certain Function is Smooth

    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...
  18. 2

    Force parallel to a smooth wall?

    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...
  19. Adithyan

    Particle inside a smooth groove performing circular motion

    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...
  20. A

    Kinematics Acceleration question

    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...
  21. G

    Real Vector Space: Is Addition & Scalar Multiplication Smooth?

    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...
  22. D

    Compactification of M Theory on Smooth G2 Manifolds

    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...
  23. E

    Simple Harmonic Oscillator on a smooth surface

    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...
  24. A

    Why do grease, cheese, butter, jam, etc., stick to smooth surfaces?

    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?
  25. C

    Uniqueness of smooth structure

    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...
  26. O

    Is every smooth simple closed curve a smooth embedding of the circle?

    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...
  27. M

    Analyzing a Smooth Curve for -π < t < π

    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...
  28. C

    Is a Smooth Embedding the Same as a Diffeomorphism?

    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...
  29. E

    MHB Calculating a smooth 90% limit for differences in a time series

    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...
  30. C

    Smooth maps between manifolds domain restriction

    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...
  31. C

    Example of a topological manifold without smooth transition functions.

    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...
  32. O

    Challenge 14: Smooth is not enough

    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...
  33. D

    Area of a Parametrized Surface

    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...
  34. M

    A small lump of ice sliding down a large, smooth sphere.

    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...
  35. A

    Product of Smooth Manifolds and Boundaries

    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...
  36. Z

    Constant Speed Motion on Elliptical Path

    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) *...
  37. T

    Collision of Uniform Smooth Spheres: Finding Coefficients of Restitution

    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...
  38. andyrk

    Oblique Impact of a smooth sphere against a fixed plane

    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...
  39. E

    Confirm: Smooth Twin Paradox Intuition

    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...
  40. soothsayer

    Spacetime is smooth after all?

    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...
  41. B

    Question about smooth functions

    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...
  42. S

    Help creating a specific smooth curve

    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...
  43. L

    Can a Photon Have a Perfectly Smooth Orbit?

    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...
  44. Ikaros

    Show a real, smooth function of Hermitian operator is Hermitian

    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...
  45. C

    Must On-Line Smooth Forecasted Point relative to previous points.

    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...
  46. B

    Regular Values (Introduction to Smooth Manifolds)

    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...
  47. Spinnor

    Can unquantized fields be considered smooth curved abstract manifolds?

    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!
  48. C

    First Post: How to Smooth End point of Finite Data Series time series

    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...
  49. Z

    Rotation of a uniform rigid disc about a fixed smooth axis

    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)...
  50. C

    A mass of 30kg on a smooth horizontal table is tied to a cord running

    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...
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