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. S

    Solving Calculus Chain Rule: Step by Step Guide

    Hi, I have been doing research in my spare time this summer on calculus proofs. I am working on a mathematics degree and I am working to understand calculus inside and out. It has been going really well but I have sort of hit a bump with the calc 1 chain rule. Here is my attempt: lim h -> 0...
  2. M

    Understanding the Straight Chain Rule: Help Needed!

    Can someone help me out with what I believe who.d be a straight chain rule application? D/dx (dy/dx)^2 I would think, applying the chain rule, you get 2 (dy/dx) d/dx (dy/dx) = 2 (dy/dx) (d2y/dx2) But, from the papers I checked, the (dy/dx) in the answer goes away, leaving just 2...
  3. W

    Learning Markov Chain: Clarifying Persistent vs Regular

    Hi all, I am trying to understand the concept of Markov Chain (a scan copy of 2 pages is attached), before this text I already studied the notes on Markov Chains at: http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf I am lil' confused at the...
  4. V

    Chain Rule Paradox or Am I Doing Something Wrong?

    If h(x) = ax, g(x) = bx and f(x) = g(h(x)). Wouldn't h'(x) = a? And g'(x) = b? And f'(x) = ab? But the chain rule says f'(x) must equal h'(x)g'(h(x)), so that means f'(x) = ab(ax) = (a^2)bx. Am I missing something obvious?
  5. B

    Making Sense of the Chain Rule: Can I Multiply to Find dy/dx?

    I have read a few sources regarding the chain rule, and a pervasive explanation that most of the sources share is this, which is way to sort of make sense of it: "Regard du/dx as the rate of change of u with respect to x, dy/du as the rate of change of y with respect to u, and dy/dx as the...
  6. A

    A uniform chain problem (Mechanics)

    Homework Statement A uniform chain with a mass of M and a length of L is put on a horizantal table in a way that half of it is hanging from the air. At the moment t=0 the chain is released from rest. 1. What is the speed of the chain as its tip will leave the table? 2. Answer question 1...
  7. F

    Find the derivative of the function using the chain rule

    1. Find the derivative of the function 2. \left(y= x sin\sqrt{x}\right) 3. I started using the product rule and then proceeded to use the chain rule, but I am wondering if I should have used the chain rule twice rather than starting with the product rule. Since I know that x is the...
  8. G

    Calculating Young's Modulus for a Ring Chain of Springs

    I know that Young's modulus for a spring is Y= K*L/A where K: is the stiffness of the spring L: the original length of the spring A: the cross sectional area How does this formula change in the case of continuously distributed springs over a ring chain of radius R and a...
  9. G

    Proton beta plus decay -proton proton chain

    proton beta plus decay --proton proton chain Im a biologist so forgive the ignorance. In beta-plus decay, a proton decays into a neutron and emmits a β+ and an electron neutrino. If the neutron is more massive than the proton where did the extra mass come from? Im asking in the context...
  10. U

    Chain rule with multiple variables

    I was reading over a textbook that stated the following, where y(s,t) = v(x(s,t),t) \frac{\partial y}{\partial t} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t} and \frac{\partial^2y}{\partial t^2} = \frac{\partial^2 v}{\partial x^2}\left (...
  11. A

    Markov Chain aggregation method

    I am using a Markov Chain to get the 10 best search results from the union of 3 different search engines. The top 10 results are taken from each engine to form a set of 30 results. The chain starts at State x, a uniform distribution of set S = {1,2,3,...30}. If the current state is page i...
  12. R

    Mass Hanging in Center of Chain Problem

    Homework Statement A mass of 200 kg is hanging directly in the center of a chain; the chain makes a 20o angle from its horizontal. The chain will break if more than 2000 N of force are applied at any point on the chain. Will the chain break?Homework Equations F=ma Tfy=mg(sin\Theta) g=10m/s The...
  13. M

    How is Ising Model a Markov Chain?

    The title says it all. It looks like the configuration probability only depends on where you want to go, not what state you are in now. Yet when I watch simulations, there is clearly a dependence on the previous state. Is there something pretty basic I'm misunderstanding about configuration...
  14. B

    Simplify derivative after using product and chain rule

    Homework Statement (x^{2}-x^{-1}+1)(x^{3}+2x-6)^{7} Homework Equations Chain Rule & Power RuleThe Attempt at a Solution (x^{3}+2x-6)^{6}[(x^{3}+2x-6)(2x+x^{-2})+7(3x^{2}+2)(x^{2}-x^{-1}+1)] This is the farthest I've gotten but when I do additional computation I do not arrive at the correct...
  15. R

    Need help with chain rule for relating ds/dt to dx/dt and dy/dt

    Homework Statement s=\sqrt{(3x^2)+(6y^2)} Homework Equations None The Attempt at a Solution \stackrel{ds}{dt}=\stackrel{d}{dt}\sqrt{(3x^2)+(6y^2)} \stackrel{3x}{\sqrt{(3x^2)+(6y^2)}} The problem with that is its only d/dx if y is a set number. I don't know how to...
  16. Y

    Any proof for the CHAIN RULE ?

    Any proof for the CHAIN RULE ?? Can somebody please show me the proof of the chain rule?? even though i have been applying that concept since i touch differentiation but i still have doubt and question on this concept!
  17. A

    Calculus 3: Chain Rule for Finding dx/dy with x=yz and y=2sin(y+z)

    Homework Statement If x=yz and y=2sin(y+z), find dx/dy Homework Equations Chain rule The Attempt at a Solution From y = 2sin(y+z) we get dz/dy= (1-2cos(y+z))/(2cos(y+z)) dz/dy=((1/2)sec(y+z) - 1) dx/dy = ∂x/∂y + ∂x/∂z dz/dy = z + y ((1/2)sec(y+z) - 1) = z...
  18. Kushwoho44

    Chain Rule for Functions of Two Variables Partial Differentiation Question

    Homework Statement Let x=ts^2 -1 and y=ln(s)-t Use the chain rule for functions of two variables to determine ∂f/∂t at (s,t)=(1,1) The Attempt at a Solution y=ln(s)-t ∂f/∂t= ∂f/∂s X ∂s/∂t -1 t=x+1/s^2 ∂t/∂s= -2(x+1)/s^3 ∂s/∂t=s^3/-2(x+1) ∴ ∂f/∂t= s^2/-2(x+1)...
  19. M

    Total pressure load on a chain

    Scenario- I'm riding a bike up a mountain road, at a constant speed of 5 mph, at a constant grade of 11%. The bike plus me (and all gear) weighs in at 200 pounds. Based on this (and excluding frictional losses, wind, etc.) how much TOTAL pressure, in pounds is on the chain? Would it be...
  20. M

    Trying to determine actual load (PSI) on a drive chain

    I think I have a rough idea how to get there...but I'm not sure. Let's assume I'm riding a bicycle, and I'm peddling at a constant pressure (torque). This is a peddling pressure which would be the exact amount which would just lift a 50 pound weight off the ground and hold it there...
  21. T

    Chain Rule and Partial Derivatives

    Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/MATH%20253/help.PNG The Attempt at a Solution http://dl.dropbox.com/u/64325990/Photobook/Photo%202012-05-24%209%2037%2028%20PM.jpg This seems to be wrong... Since I have fx and fy which I cannot cancel out. Why...
  22. T

    Multivariable Calculus: Chain Rule and Second Derivatives

    Homework Statement Here is the problem with the solution: http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because...
  23. G

    Stationary distribution of a Markov chain

    Homework Statement find the stationary distribtion of ##\left( \begin{array}{ccccc} \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\ \frac{1}{3} & \frac{2}{3} & 0 & 0 & 0 \\ 0 & \frac{1}{3} & \frac{2}{3} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right)##Homework Equations...
  24. H

    Help in evaluating the chain rule?

    $${x = r \cos \theta}$$, $${y = r \sin \theta}$$, $${r^2 = x^2 + y^2}$$ and $${\theta = \tmop{ \arctan} (y / x)}$$ (with some caveats for the last formula). Suppose $${u = u (x, y)}$$. Show that $${\frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \cos \theta +...
  25. W

    Solving for y' as a Function of y: Exploring the Equation

    Hi, If I have the equation y' = ax - by where y = y(t) , x= x(t) and y' = \frac{dy}{dt} then what is \frac {d}{dy} y' = \frac {d}{dy}(ax - by) ? I think it would come out to \frac {dy'}{dy} = a \frac {dx}{dt}\frac {dt}{dy} - b Is that right? In general...
  26. K

    Is this the same organic side chain?

    Attached is the image of the desired structure, my question is based on what is pictured as (CH3)2HC- I represented this group as C3H7 which does seem like the logical equivalent, but just being safe here. Am I right? Thank you!
  27. M

    How Do You Calculate Torque in Bicycle Chain Drives?

    Its not an actual homework question just principals I am unclear on. Its in relation to a standard bicycle being driven by a rider How would I calculate the minimum torque on the back wheel? How would I calculate the maximum torque on the back wheel? How would I calculate toque on the...
  28. C

    Multivariable Calculus: Directional Derivatives, Differentiablity, Chain Rule

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  29. C

    Markov Birth Death Chain Show Stationary Distribution

    Homework Statement Logistics model: Consider the birth and death chain with birth rates π(n)=a + Bn and death rates μ(n) = (S + yn)n where all four constants (a, B, S, y) are positive. In words, there is immigration at rate a, each particle gives birth at rate B and dies at rate S+yn, i.e...
  30. M

    Applying the Product Rule to Vector Dot Products

    Homework Statement I have vector R. I need to show the R dot dR/dt = 0 => 1/2 d/dt[R dot R] Homework Equations The Attempt at a Solution I guess I've never really applied the chain rule to dot products and its throwing me off. How does one go from R.dR/dt=0 to 1/2 d/dt[R.R] = 0. I...
  31. L

    Markov Chain Statistics

    Homework Statement A taxicab moves between the airport, Hotel A, and Hotel B according to a Markov chain with transition probabilities: P(airport → A) = 0.7, P(airport → B) = 0.3, P(A → airport) = 0.9, P(A → B) = 0.1, P(B → airport) = 0.8, P(B → A) = 0.2. A-If the taxicab starts...
  32. G

    When was the Chain rule first used?

    Since Calculus has supposedly been around for a long time, when is there actual evidence of the chain rule first being used?
  33. W

    Help with Chain Rule: Step by Step Explanation

    I'm looking at one step in my thermodynamics book and they go from pV = \nu*R*T to p*dV + V*dp = \nu * R * dT I think there's an application of the chain rule in here but I don't see exactly how it's working. Could someone show me the steps in between? Thanks!
  34. A

    A complicated derivative using chain rule

    Homework Statement I have a function z, and I need to find the derivative dz/dt "using the chain rule without substitution" Homework Equations z = x^{2}y^{3} + e^{y}\cos x x = \log(t^{2}) y = \sin(4t) The Attempt at a Solution \frac{\mathrm{d} z}{\mathrm{d} t} =...
  35. R

    Find all Invariant Probability Measures for P (Markov Chain)

    Homework Statement Find all Invariant Probability Measures for P (Markov Chain) E = {1,2,3,4,5} The screenshot below has P and my attempted solution. I am wondering if it acceptable to have infinitely many answers ("all" seems to indicate that is acceptable). Basically, I had too many unknowns...
  36. R

    Find Lambda for Simple Walk on Z (Markov Chain)

    Homework Statement Find \lambda= \{\lambda_i\} with \lambda_0 = 1 if P(X_n=i+1|X_n=i) = p, P(X_n = i-1|X_n = i) = 1 - p, i \in Z, 0<p<1 Homework Equations \lambda P = \lambda The Attempt at a Solution I used my relevant equation to write out: (1-p)\lambda_0 + P\lambda_{-2} =...
  37. B

    Decay chain of radioactive isotopes

    How can I efficiently calculate the amount of material decayed after a specific time in a two-step decay chain? In my specific example, I have 56Ni -> 56Co -> 56Fe. The half life of the first process is 6.1 days, the second - 77.7 days. How can I accurately calculate the amount of 56Fe that...
  38. M

    Noise in Engines: Investigating Chain & Sprocket Design

    hi.. can someone help me out with noise in engines.. i am specifically interested in noise due to running of chain on sprockets.. it is a sort of whistling noise... which are those parameters in design of sprocket and chain which contribute in noise.. we can also talk in terms of different...
  39. Z

    Changing A,B,C and D Beams to Chain for Perfect System Performance

    Homework Statement http://data.imagup.com/10/1146211599.jpg A,B,C and D are 4 beams ..what beam we can change it to chain and this system till stay work perfectly?
  40. B

    Conductivity in a one-dimensional chain of flourine atoms

    If we prepare a chain of flourine atoms: F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F we can construct the band structure shown. I'm using flourine as an example but my question can be generalized: What do we know about the conductivity of a material (such as this 1D chain) when the fermi level lies in...
  41. N

    Help Understanding Stewart Chain Rule Proof [Picture Provided]

    Hi I've spent a long time trying to understand this chain rule proof but I just can't get it... I have attached 2 pictures: the second one is an intuitive chain rule proof that turns out to be bogus and the first is the correct proof. So I am trying to understand first of all what does the...
  42. T

    A Mechanics tries to remove an engine from a car by attaching a chain

    Homework Statement A Mechanics tries to remove an engine from a car by attaching a chain to it from a point directly overhead and then pulling sideways with a horizontal force F. If the engine has mass 180kg, what is the tension in the chain when it makes an angle of 15 degrees with vertical...
  43. 1

    Calc III Chain rule - which vars to put in?

    Homework Statement z = cos(x^2 + 3y^2) x = ucosv y=usinv find dz/dv Homework Equations The Attempt at a Solution I think I can do these fairly well, but I'm a little unsure of the "protocol" for which variables to put back in. Sometimes (in this case) I can't really put...
  44. H

    Concept question on irreducibility (markov chain)

    Homework Statement This is not really a homework question but just something that I'm confused about. I'm having trouble with understanding the movement of Markov chain. Say, if S = {0,1,2...} and 0<q<1. Let p(i,i+1)=q and p(i,i-1)=1-q. I can see how this Markov chain moves (back and forth...
  45. M

    About the chain rule what's wrong with me?

    What I know from the chain rule is that if y and u are differentiable with respect to x then dy/dx = (dy/du)*(du/dx) Now, why is this example doesn't work: y = x^2 u = c then we have dy/dx = (dy/du) * (du/dx) = (dy/du) * 0 = 0 doesn't equal 2x I want an answer irrelated to the chain...
  46. A

    How Does Diffusion Affect the Decay Chain of Uranium?

    Ok guys, I've got a problem which I'd like to solve in an elegant way, but I don't how to solve it (if it can be solved analytically). I'm considering the decay chain of Uranium. So Uranium decays to Thorium, which decays to Protactinium, and so on. I know how to solve such a linear system...
  47. K

    Chain Rule & PDES: Solving ∂z/∂u

    Im new on the forum, so I hope you guys will have some patience with me :-) I have a question about the chain rule and partial differential equations that I can't solve, it's: Write the appropriate version of the chain rule for the derivative: ∂z/∂u if z=g(x,y), where y=f(x) and...
  48. Z

    Chain Rule for Vector Function

    Homework Statement I'm trying to figure out how to take grad(f(x(t)) where x(t) is a vector. Since it's part of a physics problem, it's assumed x(t) is in 3-dimensional space. The Attempt at a Solution My guess is that grad(f(x(t)) = ((∂f/∂x)(∂x/∂x),(∂f/∂x)(∂x/∂y),(∂f/∂x)(∂x/∂z)) but...
  49. S

    I'm having major difficulties with partial differentiation using the chain rule

    This is another problem than I've been stuck on for a long time and I tried reading and watching videos but I only find first order partial differentiation with more than two variables or higher order partial differentiation with only two variables. (I'm not calling f a variable but I am calling...
  50. S

    Rotational analysis of chain and sprockets system

    Homework Statement Imagine a system consisting of a chain that runs over two sprockets. The chain rotates around the sprockets with a constant linear velocity (i.e. the chain is taut and rigid). The front sprocket has a radius rfront and an angular speed ωfront and the rear sprocket has a...
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