What is Chain: Definition and 976 Discussions

A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A chain may consist of two or more links. Chains can be classified by their design, which can be dictated by their use:

Those designed for lifting, such as when used with a hoist; for pulling; or for securing, such as with a bicycle lock, have links that are torus shaped, which make the chain flexible in two dimensions (the fixed third dimension being a chain's length). Small chains serving as jewellery are a mostly decorative analogue of such types.
Those designed for transferring power in machines have links designed to mesh with the teeth of the sprockets of the machine, and are flexible in only one dimension. They are known as roller chains, though there are also non-roller chains such as block chain.Two distinct chains can be connected using a quick link, carabiner, shackle, or clevis.
Load can be transferred from a chain to another object by a chain stopper

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  1. M

    Solving Differential Equations

    Solve the following: d/dt cos(theta) d/dt t sin(theta) d/dt r cos (theta) d/dt r^2 (theta) d/dt e^ (-3x) d/dt (x^2 + y^2) I would assume all by the second one are 0 since your solving for terms dt and not theta, x, y, or r... I don't think its right at all. I know it goes something...
  2. A

    Derivative Chain Rule Question

    Hi, Say x=position, v=velocity, a=acceleration, t=time. Thanks! EDIT: I just realized that 2/x is not a constant and thus I shouldn't have treated it as a constant (taking the derivative of it as 0). However, I don't understand how to take the derivative with respect to t of it.
  3. Telemachus

    Second order mixed derivative and chain rule

    I want to find the second order derivative for f(x,y),x(u,v),y(u,v), f depends on x and y, and x and y depends on u and v. I'm trying to find \frac{{\partial^2 f}}{{\partial v \partial u}}This is what I did: \frac{{\partial f}}{{\partial u}}=\frac{{\partial f}}{{\partial x}}\frac{{\partial...
  4. S

    Confusion on chain rule substitution

    I'm confident in my math ability, but how is it that by using the chain rule... W_{x_1 \rightarrow x_2} = \int^{x_2}_{x_1} m \frac{dv}{dt} dx can be turned into W_{x_1 \rightarrow x_2} = \int^{x_2}_{x_1} m \frac{dv}{dx} \frac{dx}{dt} dx = \int^{v_2}_{v_1}mv dv ? I understand the...
  5. M

    How is Epsilon Defined in the Proof of the Chain Rule in James Stewart Calculus?

    From james stewart calculus Early Transcendentals.Before he states the proof he intoduced a property of differentiable funcion My problem is how we defined \epsilon to be 0 when \Delta x=0 where this is not in the Domain.
  6. D

    Proof of the Chain Rule: An Elegant and Simple Approach

    Is there an elegant and simple proof of the Chain Rule? Every proof I've found is complex and mind-boggling
  7. S

    Any gentle introduction of Fibonacci chain?

    Hi, I have been searching the internet for some gentle introductions of Fibonacci chain, but so far I haven't found anything. I wonder if anyone can recommend some good introductions to me, e.g. a book, an article... Thank you.
  8. J

    Uncountable union of a chain of countable sets can be uncountable?

    Let X be a non-empty set, and let S contain all countable subsets of X. Partially order S by inclusion. Let C be a totally ordered subset ("chain") of S, and let U = \cup_{E \in C} E It appears that U is not always countable: if it were, U would be an upper bound of the chain C, and U would...
  9. S

    Chain rule problem: proper method?

    Homework Statement Use the chain rule, the derivative formula Dxsinu=cosuDxu, together with the identities cosx=sin(\pi/2 -x) and sinx=cos(\pi/2 -x) to obtain the fomula for Dxcosx. Homework Equations Chain rule: dy/dx=dy/du\cdotdu/dx The Attempt at a Solution For my second...
  10. L

    How to Generate a 100 State Sequence Using a Markov Chain Model?

    I have a state transition probability matrix and a state probability vector [0.9 0.1; 0.1 0.9] & [0.4 0.6] respectively. Now, I want to generate the states of 1 and 0 according to this. say 100 state sequence. Any sort of help would be appreciated. Thanks.
  11. X

    Persistence of State i in a Markov Chain

    In a Markov chain, show that a state i is persistent if and only if the mean number of visits to the state i is infinite given the chain started in state i. I thought about looking at the mean recurrence time, but that's all I have so far.
  12. X

    Show all other states are transient in Markov chain

    Let X be a Markov chain with a state s that is absorbing, i.e. pss(1) = 1. All other states communicate with s i.e. i → s for all states i ∈ S. Show that all states in S except s are transient. I understand this intuitively, but I'm not really sure how to start the proof.
  13. V

    Polymerase Chain Reaction Primers

    I have learned that the primers in PCR are added to the DNA in the annealing phase, but what exactly do these primers do to the DNA? Thank you
  14. K

    What is the equation for the length of a falling chain through a hole?

    Homework Statement A flexible chain of mass M and length L lies on a frictionless table, with a very short portion of its length L0 hanging through a hole. Initially the chain is at rest. Find a general equation for y(t), the length of chain through the hole, as a function of time. (Hint...
  15. G

    How Does a Chain Hoist Work Mathematically?

    Can anyone explain the mechanics of a chain hoist and its mechanical advantage, in mathematical terms,? thank you Bashyam
  16. QuarkCharmer

    Implicit Differentiation, and the Chain Rule

    Homework Statement Use implicit differentiation to find dy/dx 2x^3+x^2y-xy^3 = 2 Homework Equations Chain Rule et al. The Attempt at a Solution My questions is this. When deriving something like xy^3, apply the product rule to get 1y^3 + x\frac{d}{dx}y^3 I am confused on...
  17. S

    Any fast way to compute the fixed vector of a Markov chain transistion matrix?

    * I have already posted this in the General Math, but I guess the problem is more like a linear algebra problem. Currently I am using a rather simple way, to solve vector w from (M-I)w=0 (replace one equation by w1+w2+...wn=1). Is there any faster way to do this? Thank you.
  18. H

    Population Estimates Over 2000 Years: My Questions

    From this day , if we take in account . How much population will be my following generation at the end of 2000 years(from this day). My question is , will the population be thousands or millions? Will it likely be fade away in the middle and no trace remains in the end? Any...
  19. P

    Partial derivitives chain rule proof

    Homework Statement If u=f(x,y) where x=escost and y=essint show that d2u/dx2+d2u/dy2 = e-2s[d2u/ds2+d2u/dt2 The Attempt at a Solution i have no idea! question though, do the partial derivitives have to be solved and expanded then just show that one side equals the other or can...
  20. QuarkCharmer

    Solving the Chain Rule: A Visual Guide

    Homework Statement \frac{d}{dx}(x+(x+sin^2(x))^3)^4 Homework Equations Calc up to Chain Rule. The Attempt at a Solution Using product and chain rule I got: \frac{dy}{dx}=4(x+(x+sin^2(x))^3)^3(1+3(x+sin^2(x))^2)(1+\frac{d}{dx}sin^2(x)) Then I calculated the derivative of sin^2(x)...
  21. K

    Small problem understanding application of chain rule

    Homework Statement I have proven in two ways (correctly) that the derivative of ln|x| = 1/x (note absolute value does vanish) Now I open my textbook and see a general rule that \frac{d}{dx} ln (u) = \frac{u'}{u} And the not so general derivative of |x| is \frac{d}{dx} |x| =...
  22. J

    Why Is My Markov Chain Simulation Producing Unexpected Output?

    Homework Statement Homework Equations Seems pretty straightforward ... The Attempt at a Solution Here's what I put: P = zeros(10,10); P(1,2) = 1; P(2,1) = 1/2; P(2,3) = 1/2; P(3,1) = 1/2; P(3,4) = 1/2; P(4,1) = 1/3; P(4,2) = 1/3; P(4,5) = 1/3...
  23. J

    Do Markov Chain Transition Matrices Sum by Row or Column?

    Homework Statement Homework Equations The Attempt at a Solution A few things. First of all, the homework problem notes that "all the columns should sum to 1," whereas Wikipedia says ∑Pij = 1 when we sum all along the the row i. Second of all, I don't know where to...
  24. M

    Most Common Pu-239 Fission Chain Reaction?

    Anyone on this forum know what the highest-probability fission chain reaction of plutonium-239 is? Mike Fontenot
  25. V

    Understanding the Von Neumann Chain Concept in Modern Quantum Theory

    Is anyone thoroughly familiar what von Neumann was trying to say in his von Neumann Chain concept where one can locate anywhere the observer and the observed? I've been reading the original and analyzing it for hours and can't seem to completely get the context in light of present day concept...
  26. M

    Motor torque calculation for lift having chain drive

    i want to know how to calcuate Starting and running for motor in 2 post lift for lifting 3.0T. Actually i have 2 columns having trapezoidal screw rod and nut in each side. One column is getting drive by motor through pulley belt mechanism. Another shaft is getting driven by chain at the...
  27. E

    Fatty Acid chain length and its effect on binding to serum albumin

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  28. M

    The Chain Rule: Mth 251 Homework

    Hello! I'm currently taking Mth 251 and have been working on this chain rule for a bit. It's kind of straight forward, there's just a lot of chains inside of chains and so forth. I think I have a solution, but I'm not entirely sure its correct. Homework Statement Find the derivative of...
  29. F

    Chain drive design, long length

    So I have a chain driven system on a tricycle that I am kind of stuck with. the center distance of the two sprockets is 65 inches, and the chain is a 530 series, which has a pitch of 5/8. Now I have seen some sources reference or mention a 60 to 80 maximum of center distance to pitch ratio...
  30. G

    Deriving arcsin(1-2e^-t): What Chain Rule to Use?

    1. Derive arcsin(1 - 2 e ^-t) 2. The derivative of arcsin is 1/√(1-x^2) 3. I tried using the chain rule for 1 - 2 e ^-t, but that didn't work out. What should I take the chain rule of?
  31. Q

    Chain rule for 2nd derivatives

    hi does anyone know why the 2nd derivative chain rule is as such? i roughly know that if u = f(x,y) and x=rcos(T) , y = rsin(T) then du/dr = df/dx * dx/dr + df/dy * dy/dr but if i am going to have a second d/dr, then how does it work out?
  32. P

    A Flexible chain between two hooks (statics)

    Homework Statement A flexible chain weighing 41.0 N hangs between two hooks located at the same height. At each hook, the tangent to the chain makes an angle θ = 41.5° with the horizontal. (a) Find the magnitude of the force each hook exerts on the chain. (b) Find the tension in the chain at...
  33. S

    Proof of Multivariable Chain Rule in higher dimensions

    Homework Statement Let \textbf{F}: \textbf{R}^m \rightarrow \textbf{R}^n and \textbf{G}: \textbf{R}^p \rightarrow \textbf{R}^m Prove that ({\textbf{F} \circ \textbf{G}})'(x) = {\textbf{F}}'(\textbf{G}(\textbf{x})) {\textbf{G}}'(\textbf{x}) Homework Equations Assume the single...
  34. M

    Find F(x) when v(x) is given. Chain rule?

    Homework Statement The movement of an object with a mass of 1500kg is given by v(x)=(4.0 [1/ms]) * x^2 Determine the net force acting on the object as a function of x. Homework Equations F=ma The Attempt at a Solution I know I'm supposed to use the chain rule to solve this but...
  35. C

    Second order partial derivatives and the chain rule

    Homework Statement http://www.math.wvu.edu/~hjlai/Teaching/Tip-Pdf/Tip3-27.pdf Example 7. Not this question in particular, but it shows what I'm talking about. I understand how they get the first partial derivative, but I'm completely lost as how to take a second one. I have tried...
  36. O

    Debye Frequency for 1D atomic chain

    Homework Statement Consider phonons propagating on a one-dimensional chain of N identical atoms of mass M interacting by nearest-neighbour spring constants of magnitude C. Show that the Debye frequency can be written as w_{D}=\pi \left(\frac{C}{M}\right)^{1/2}. Homework Equations The...
  37. H

    Using chain rule to derive the path equation

    r' = 4r - rf f' = -3f + rf In this question, there was three parts: a) find all the critical points of this system. b) Derive the linearised system about each critcal point... c) Use the chain rule to derive the path equation of the trajectories in the phase plane. I managed to get a...
  38. C

    What is the derivative of the square root of 2x using the chain rule?

    Homework Statement Find the derivative of: f(x)=\sqrt{2x} Homework Equations So using the chain rule: \frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx} The Attempt at a Solution Isn't it just a simple matter of setting u=2x, therefore du/dx=2, and y=\sqrt{u}=u^1/2, therefore dy/du=1/2 *...
  39. A

    Why Does the Chain Rule Work in Calculus?

    I just learned about chain rule in calculus, but I was wondering why exactly chain rule works. I understand how to use it, just not exactly why it works. Thanks in advance
  40. P

    Second Order Partial Derivatives + Chain Rule

    Homework Statement Let z = z (x,y) be a function with x = x(s), y = y(t) satisfying the partial differential equation (Ill write ddz/ddt for the partial derivative of z wrt t and dz/dt for the total derivative of z wrt t, as I have no idea how to use Latex.) ddz/ddt +...
  41. S

    Partial Derivative Chain Rule for u(x, t) in terms of f and its Derivatives

    Homework Statement Say that f(x) is some function whose second derivative exists and say u(x, t)=f(x + ct) for c > 0. Determine \frac{\partial u}{\partial x} In terms of f and its derivatives. Homework Equations PD Chain rule. The Attempt at a Solution Say that x and y are...
  42. A

    Difference between martingale and markov chain

    What is the difference between martingale and markov chain. As it seems apparently, if a process is a martingale, then the future expected value is dependent on the current value of the process while in markov chain the probability of future value (not the expected value) is dependent on the...
  43. D

    Calculating Torque Required for Bicycle Chain Drive on an Incline

    Homework Statement I am having some trouble with a bicycle chain drive. I need to find the torque required to have a bicycle travel up an incline. The bicycle is already moving Homework Equations \tau = r*F Torque is equal to the radius of the wheel, multiplied by the force...
  44. B

    Understanding Variable Changes in the Chain Rule for Mathematicians

    I have a question about the variable changes in the proofs of Proposition 1.3.4 and Proposition 1.3.6. In the first one, it seems like the author does the variable change but once he applies the chain rule he doesn't do it completely. While in the second it seems like it does the variable...
  45. G

    Finding Centre of Mass of a Chain

    Homework Statement A chain hangs b'n two points in the vertical plane, find its centre of mass. I have no idea where to start. Homework Equations The Attempt at a Solution
  46. R

    What Is the Tangential Acceleration of a Chain Sliding Down a Sphere?

    A chain of length 'l' and mass 'm' lies on the surface of a smooth sphere of radius 'R' > 'l', with one end tied to the top of the sphere. (a) Find the gravitational potential energy of the chain with reference level at the center of the sphere. (b) Find the tangential acceleration dv/dt...
  47. M

    Speed of half a chain falling off of a table

    Homework Statement A uniform chain of total mass m is laid out straight on a frictionless table and held stationary so that one-quarter of its length, L = 2.87 m, is hanging vertically over the edge of the table. The chain is then released. Determine the speed of the chain at the instant when...
  48. P

    Steady-state vector of infinite markov chain

    Today we encountered a problem with markov chains and are wondering if this can be solved analytically. Suppose we have a banded transition probability matrix M of the following form: M= [ P P 0 0 0 ... Q 0 P 0 0 ... 0 Q 0 P 0 ... 0 0 Q 0 P ... 0 0 0 Q 0 ... . . . . . . . . . . ]...
  49. R

    Calculating df/dg with Chain Rule: Romeo's Guide

    If I have two functions, f(x,y,z) and g(x,y,z), do I use the chain rule to calculate df/dg? e.g. df/dg=df/dx df/dy df/dz Cheers! Romeo
  50. Simfish

    How do you knock off a halogen from a hydrocarbon chain?

    Without introducing any other atoms or double bonds into the chain? (just replace the halogen with a hydrogen atom). Assume for now that the halogen was introduced in the middle of the carbon chain.
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