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

    Evaluating Partial Derivative of g w.r.t. Theta at (2*sqrt(2),pi/4)

    use chain rule to evaluate partial derivative of g with respect to theta at (r,[theta])=(2*sqrt(2),pi/4), where g(x,y) = 1/(x+y^2), x=rsin[theta] and y=rcos[theta] [b] r^2=x^2+y^2 and tan[theta]=y/x [b] The Attempt at a Solution I understand how to use the chain rule for partial...
  2. M

    Polymerase Chain Reaction (PCR)

    Hey! I have learned that one of the first steps in this technique involves heating the DNA sample to around 95 degrees. Now as far as I am aware the only thing holding the two Sugar Phosphate backbones together is the Hydrogen bonding between base pairs. Why is such a high temperature needed...
  3. quasar987

    What's wrong with this proof of the chain rule

    Can anyone see where the flaw is in the development below, where I prove that (g o f)'(x)=g'(f(x)) instead of g'(f(x))f'(x), as it should be. Consider the usual hypothese under which the chain rule for real-valued function applies. Consider \epsilon>0. Since g is differentiable at f(x_0)...
  4. P

    Mechanics Question - Atoms Modelled As A Chain of Masses Connected By A Spring

    For a chain of masses lying on a horizontal frictionless surface, with each mass connected to its neighbour mass by a spring of force constant s, the equation of motion for the nth mass is: m(x_n)'' = -s[2(x_n) - x(n-1) - x(n=1)] Where: x_n is the displacement of the nth mass from its...
  5. C

    Chain rule Product rule Derivative

    Homework Statement (2x^2 - 3x + 1)(4x^3 + 4x -3)^5 Homework Equations The Attempt at a Solution
  6. B

    How to Differentiate Complex Functions Using Product and Chain Rules?

    Hi 2 questions having a mental block and can't figure them out any help would be apprieciated Q1 differentiate f(x)=ax(2x+b)^7 where a and b are constants Q2 differentiate f(x)=(x^2+cos^3(x^4))^10 thanks for any help cheers
  7. A

    How Does the Chain Rule Apply to Homogeneous Functions in Calculus?

    Homework Statement A function is called homogeneous of degree n if it satisfied the equation f(tx,ty) =t^(n) f(x,y), for all t, where n is a positive integer and f has continuous 2nd order partial derivatives. If f is homogeneous of degree n, show that df/dx (tx,ty) = t^(n-1) df/dx(x,y)...
  8. H

    Multivariable functions - chain rule

    Homework Statement Since both my questions are on the same topic, i'll throw them both in here 1. Find dz/dt for z=(x^2)(t^2), x^2+3xt+2t^2=1 2. Show that if u=xy, v=xy and z=f(u,v) then: x.dz/dx-y.dz/dy=(x-y)dz/dv Homework Equations The Attempt at a Solution 1. I only...
  9. F

    Solve for g(8) and g'(8) in f(g(x)) = x using the Chain Rule

    ok so f(g(x)) = x, for all x. f(3)=8 f'(3)=9 what are the values of g(8) and g'(8) ok, so g(8) = 3 because f(g(8)) must equal 8, and f(3) = 8, so g(x) must equal three. however, i have NO idea how to do g'(x) i was thinking of using the chain rule, but this gets me nowhere..help...
  10. D

    Calculate Maximum Velocity with Swing Chain Length and Angle

    Homework Statement Using the length of a swing's chain (1.8m) and using the angle the swing starts at relative to the vertical (30 deg.) devise a method to calculate the max veloc. of the swing at the bottom. Assume mass of person+swing=72 kg Homework Equations Ui+Ki=Uf+Kf K=1/2 mv^2...
  11. C

    Markov chain - stationary distribution -

    Homework Statement -A taxi is located either at the airport or in the city. From the city, the next trip is to the airport with 1/4 probability, or to somewhere else in the city with 3/4. From the airport, the next trip is always to the city. (a) find the stationary distribution (b)...
  12. G

    Solving Hamiltonian with chain of charge centers?

    I was thinking of how to solve the single particle Hamiltonian H=...+\sum_i \frac{1}{\vec{r}-\vec{r}_i} where \vec{r}_i=i\cdot\vec{a} Transforming it into second quantization k-space I had terms like H=...+\sum_G...c^\dag_{k+G}c_k But it seems to me that for the method of trial wavefunctions any...
  13. K

    Weird Issue with the Chain Rule

    Homework Statement I'm working on a quick problem regarding a presentation that I'm giving, but I've come across an issue that I can't seem to resolve. Namely \displaystyle \left. \frac{d}{dt} \right|_{t=0} f(\phi^p (t+t_0) ) = \left( \phi^p \right) ^\prime (t_0) f Does anybody see...
  14. E

    Is the chain rule applied in the derivative of cos(t^3) ?

    How would you compute the derivative of cos(t^3)? Would you use the chain rule? Does anyone have a good way of recognizing when to use chain rule and when not to?
  15. ?

    Aperiodicity of a Markov Chain

    Homework Statement Transition matrix is 0 0 1 0 0 1 (1/3) (2/3) 0 "argue that this chain is aperiodic" Homework Equations definition of aperiodicity - there must exist a time n such that there is a non-zero probability of going from state i to state j for all i & j The...
  16. ?

    Aperiodicity of a markov chain

    my transition matrix is 0 0 1 0 0 1 (1/3) (2/3) 0 I'm supposed to argue that this chain is aperiodic, A markov chain is aperiodic iff there exists a time n such that there is a positive probability of going from state i to state j for all i and j...
  17. G

    Is My Calculation of Derivatives Using the Chain Rule Correct?

    I am trying to find the first and second derivative using the chain rule of the following: u sin(x^2) This is what I have but I don't think it is correct. Can someone pls let me know? first derivative: u * 2x cos(x^2) + sin(x^2) u' second derivative: u * 2( x * -2sin(x^2) +...
  18. G

    Is My Chain Rule Derivation Correct for u*sin(x^2)?

    I am trying to find the first and second derivative using the chain rule of the following: u sin(x^2) This is what I have but I don't think it is correct. Can someone pls let me know? first derivative: u * 2x cos(x^2) + sin(x^2) u' second derivative: u * 2( x * -2sin(x^2) +...
  19. P

    Stationary distribution of Markov chain

    Hi all, I'm given a Markov chain Q_k, k>0 with stationary transition probabilities. The state is space uncountable. What I want to show is that the chain is asymptotically stationary, that is it converges in distribution to some random variable Q. All I have at hand is an k-independent upper...
  20. 7

    How do I use the chain rule to derive certain functions?

    id like some help deriving certain functions using the chain rule the way our teacher does it is different from what the textbook says he derives the outermost functions before getting to the innermost functions, this is where i get confused =( for example f(x) = sincos(5x) i get...
  21. T

    Finding max/min of 3d graphs with chain rule

    http://math.berkeley.edu/~theojf/Midterm2Practice.pdf can someone please help me on problem number 2 of the link above? apologies for the bad handwriting. my professor is just horrible with that. i've done max and min with multivariables before and I've done chain rule , but I've never...
  22. S

    Implicit differentiation and chain rule-

    The question and my workings are attached:
  23. P

    Applying the Chain Rule to the Derivative of y=2x^{sinx}

    y=2x^{sinx} i know i should use the product rule within a chain rule. but how can i use chain rule with sinx is the anwser y=-2x^{cosx} can anyone give me pointer to this easy problem and tell if am forgetting something.
  24. I

    Master the Chain Rule: Derivative of f(x) = \sqrt{5-x^{2}} and Its Composition

    Homework Statement A: Write f(x) = \sqrt{5-x^{2}} as a composite of two functions. B: Use the Chain Rule to find the derivative of f(x) = \sqrt{5-x^{2}} Homework Equations Chain Rule: y`= \frac{dy}{du} \frac{du}{dx} The Attempt at a Solution A: y = \sqrt{u} u = 5 -...
  25. nicksauce

    Find Time for Chain to Pass Thru Tube: Lagrange Eqn Solution

    Homework Statement A uniform chain of length L and mass M is constrained to move in a frictionless tube. Initially a fraction (1-f0) of the chain rests in a horizontal section of the tube. The remaining fraction f rests in a section of the tube that is inclined downward from the horizontal at...
  26. P

    Exploring the Chain Rule in Calculus: A Simple Example

    [SOLVED] another chain rule: easy one y=xe^{-x^2} i have no i dea how to start. f'= x^{x^2} or -2x^blah blah blah just get me started and i'll promise you i will finish it myself
  27. P

    First derivative: chain rule: easy for you guys

    [SOLVED] first derivative: chain rule: easy for you guys Y=E^(-mx) f= E^x g= -mx f'= E^x g'= o E^(-mx) * 0(E^(-mx)) i think, not sure though Y'= 0 which is wrong someone help
  28. B

    Higher order partial derivatives and the chain rule

    Hi guys, please see attachment Basically, could somebody please explain to me how I find {\varphi}_u_u, I really don't understand how it's come about. Apparantly I need to use the chain rule again and the product rule but I don't understand how to, if somebody could show me explicitly how to...
  29. 8

    Basic physics- chain strength question.

    A chain always breaks ath the weakest link. Right? Hypothetical: Where would a chain break if all the links were of equal strength, shape, shape, size... equal in every way. Ignore all tolerance rules, this is hypothetical! Thanks!
  30. D

    How Do You Determine When to Use the Chain Rule or Product Rule in Calculus?

    I have a question more than a problem to answer. I'm having a difficult time recognizing when to use the product rule and when to use the chain rule. How do you recognize when to use each, especially when you have to use both in the same problem. Problems like y+x^4y^3-5x^6+3y^8-42=0 tend...
  31. J

    Chain rule with functional derivative

    This is supposedly the chain rule with functional derivative: \frac{\delta F}{\delta\psi(x)} = \int dy\; \frac{\delta F}{\delta\phi(y)}\frac{\delta\phi(y)}{\delta\psi(x)} I have difficulty understanding what everything in this identity means. The functional derivative is usually a derivative...
  32. I

    Chain rule problem with partial derivatives

    [SOLVED] Chain rule problem with partial derivatives Homework Statement Suppose that z = f(u) and u = g(x,y). Show that.. \frac{\partial^{2} z}{\partial x^{2}} = \frac{dz}{du} \frac{\partial^{2} u}{\partial x^{2}} + \frac{d^{2} z}{du^{2}} \frac{(\partial u)^{2}}{(\partial x)^{2}}...
  33. I

    Lowest freq mode of fixed-end linear chain oscillations

    Hi, (All oscillations I'll be talking about here are longitudinal.) For coupled oscillations of 2 masses between 3 identical springs (ends held fixed by walls), I think it was a standard textbook mechanics problem to show that the lowest-frequency mode is the symmetric one (where the masses...
  34. N

    Chain consisting of five links

    So the first part of this question asks: A chain consisting of 5 links, each of mass .19kg, is lifted vertically with a constant acceleration of a=2.8 m/s^2. Find the magnitude of the force that link 3 exerts on link 2. I found this answer to be 4.7 N with the following formula: F(link 3...
  35. T

    Markov Chain/ Monte Carlo Simulation

    Let \bold{X} be a discrete random variable whose set of possible values is \bold{x}_j, \ j \geq 1 . Let the probability mass function of \bold{X} be given by P \{\bold{X} = \bold{x}_j \}, \ j \geq 1 , and suppose we are interested in calculating \theta = E[h(\bold{X})] =...
  36. M

    Calculating tension on chain fence with someone sitting on it.

    Hi, I'm hoping someone can show me a simple formula to calculate the tension or force on a chain fence with someone sitting on it. I imagine that the variables are; - the distance between the fence posts - the arc of the chain - the persons weight Thanks Munga
  37. M

    Center of mass / momentum chain

    Homework Statement Given: A uniform flexible chain whose mass is 4.1 kg and length is 5 m. A table whose top is frictionless. Initially you are holding the chain at rest and one-half of the length of the chain is hung over the edge of the table. When you let loose of the chain it falls...
  38. D

    Chain on table - calculate work

    1. A chain is help on a frictionless table with one-fourth of its length hanging over the edge. If the chain has length L and mass M, how much work is required to pull the hanging part back on the table? i don't think i could have W = mgL/4 will the work change each time a new chain ring is...
  39. E

    Partial derivative chain rule question

    Homework Statement Given z= square root of xy, x = 2t - 1, y = 3t +4, use the chain rule to find dz/dt as a function of t. Homework Equations The Attempt at a Solution dz/dt = partial derivative of z with respect to x multiplied by dx/dt + (partial derivative of z with respect...
  40. A

    Ladder network.attenuator chain?

    Homework Statement I apologize for my crappy diagramming. http://img100.imageshack.us/img100/4427/ladderwd4.jpg The problem text exactly: "Some important kinds of networks are infinite in extent. The figure shows a chain of series and parallel resistors stretching off endlessly to the...
  41. K

    Higher Partial Derivatives & Chain Rule

    Higher Partial Derivatives & Chain Rule (urgent) I'll have a test this evening, and I don't want to fail on a question like this, so please help me out! I will greatly appreciate for any help provided. The question: http://www.geocities.com/asdfasdf23135/advcal11.JPG My attempt...
  42. N

    Chain shape (Euler-Lagrange equations)

    A chain with uniform linear density d and length L is tied at two ends to the ceiling. How to find its shape using Euler-Lagrange equations? (I know it can be done with other methods, but I want to know how to do it using E-L).
  43. P

    Messy partial differentials with chain rule.

    Homework Statement the problem asks: Find \deltaf/\deltax and \deltaf/\deltay at x=1 and y=2 if z=f(x,y) is defined implicitly by 2x^{}2y/z + 3z/xy - xy\sqrt{}z = 3. Note that (1,2,4) is a point on the surface. Homework Equations Im not really sure how to approach this one. The...
  44. P

    Messy partial differentials with chain rule.

    Homework Statement the problem asks: Find \deltaf/\delta/x and \deltaf/\deltay at x=1 and y=2 if z=f(x,y) is defined implicitly by 2x^{}2y/z + 3z/xy - xy\sqrt{}z = 3. Note that (1,2,4) is a point on the surface. Homework Equations Im not really sure how to approach this one. The...
  45. K

    Multivariable Chain Rule & Variable-Dependence

    I am having a terrible hard time with the multivariable chain rule and its related stuff (I read my textbook many times, but it doesn't help that much because the explanations are very limited). I hope that someone can help me to withdraw from this darkness of confusion. 1) (Differentiation...
  46. G

    Quantum Spin Chain Problem: An Overview

    what is the quantum spin chain problem? From my research on the net I can see that its solved through exact diagonlization of some sort of matrix, but I can't work out exactly what the problem is...
  47. V

    Differentiate using chain rule

    Hi everyone, I'm new to this forum... I hope I've posted in the right section... How do I differentiate y=2e(2x+1) using the chain rule? I let u= (2x+1) so du/dx = 2 but how do I differentiate y= 2eU ? thank you :)
  48. M

    Applications of Partial Derivatives and chain rule

    Homework Statement The lengths a,b,c of a rectangle are changing with time. At the instant in question, a=1m, b=2m, c=3m and da/dt = db/dt = 1m/sec, and dc/dt = -3m/sec. At what rate is the box's volume changing at this instant? Homework Equations Chain rule for partial derivatives...
  49. P

    Crystal Vibrations: Impurity in a monoatomic chain

    Homework Statement Hello everyone, I'm a bit stuck on this particular problem and i was hoping somebody could give me a couple tips or hints! "consider a linear monoatomic chain where all atoms have a mass M except one which has a small mass m. The force constant is C." First I'm asked...
  50. F

    Understanding the Chain Rule in Theorem Proof

    I'm trying to understand the proof for this theorem, and I can't see what they did to get from one step to the next. THEOREM: Suppose F(z) is an analytic function and that f(z) = F'(z) is continuous on a domain D. Then for a contour C lying in D with endpoints z1 and z2...
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