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. N

    Analyzing Velocity and Force of a Chain in Motion

    My solution: For the horizontal portion of the chain: let at any instant the length of chain inside the tube is x, and at that instant the chain in the tube is having a velocity v. Then, at any instant: ##F = \frac{\mathrm{d} p}{\mathrm{d} t}## ##p##= mass of the chain in the tube at the...
  2. S

    B Question about transition matrix of Markov chain

    The note I get from the teacher states that for transition matrix, the column part will be current state and the row part will be future state (let this be matrix A) so the sum of each column must be equal to 1. But I read from another source, the row part is the current state and the column...
  3. A

    MHB How to Understand and Solve the Chain Rule Problem in Calculus?

    \[ \frac{\partial \dot{r}}{\partial \dot{q_k}} = \frac{\partial r}{\partial q_k} \] where \[ r = r(q_1,...,q_n,t \] solution \[ \frac{dr }{dt } = \frac{\partial r}{\partial t} + \sum_{i} \frac{\partial r}{\partial q_i}\frac{\partial q_i}{\partial t}\] \[ \dot{r} = \frac{\partial r}{\partial t}...
  4. cwill53

    I Chain Rule in Multiple Variables

    The following link leads to a question I asked on the mathematics Stack Exchange site. https://math.stackexchange.com/questions/3790900/chain-rule-with-a-function-depending-on-functions-of-different-variables/3791017?noredirect=1#comment7809514_3791017 I want to understand how the chain rule...
  5. LCSphysicist

    Finding the Force Applied by a Support on a Falling Chain

    A chain with length L and mass density σ kg/m is held in the position shown in Fig. 5.28, with one end attached to a support. Assume that only a negligible length of the chain starts out below the support. The chain is released. Find the force that the support applies to the chain, as a function...
  6. T

    Higher order derivatives using the chain rule

    Mentor note: Fixed the LaTeX in the following I have the following statement: \begin{cases} u=x \cos \theta - y\sin \theta \\ v=x\sin \theta + y\cos \theta \end{cases} I wan't to calculate: $$\dfrac{\partial^2}{\partial x^2}$$ My solution for ##\dfrac{\partial^2}{\partial...
  7. E

    I can't find my mistake in using the chain rule here

    I literally don't know what's going wrong today, I can't seem to get anything right :oldconfused:. The velocity in S' is easy enough $$v' = \frac{dx'}{dt'} = \frac{\partial f}{\partial t} \frac{\partial t}{\partial t'} + \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}\frac{\partial...
  8. J

    Markov chain on state {1, 2, 3, 4, 5, 6 , 7}

    I need this for a programming project. Could you help?
  9. omega_minus

    A Effective Spring Constant of a Chain Polymer

    Hi All, I'm doing research in magnetic nanoparticles that are coated with chain molecules (oleic acid, I believe) and I am trying to model these molecules' effective spring constant. The basic scenario is this: When a water-based ferrofluid is evaporated, it leaves behind only dried...
  10. karush

    MHB Calculating the Derivative of a Function Using the Chain Rule

    find $F'(x)$ $$F(x)=(7x^6+8x^3)^4$$ chain rule $$4(7x^6+8x^3)^3(42x^5+24x^2)$$ factor $$4x^3(7x^3+8)^3 6x^2(7x^3+4)$$ ok W|A returned this but don't see where the 11 came from $$24 x^{11} (7 x^3 + 4) (7 x^3 + 8)^3$$
  11. B

    I Can the Chain Rule be Applied to Simplify Divergence in Entropy Equation?

    I am looking at the derivation for the Entropy equation for a Newtonian Fluid with Fourier Conduction law. At some point in the derivation I see \frac{1}{T} \nabla \cdot (-\kappa \nabla T) = - \nabla \cdot (\frac{\kappa \nabla T}{T}) - \frac{\kappa}{T^2}(\nabla T)^2 K is a constant and T...
  12. S

    Ehrenfest chain transition probability

    Here is the Ehrenfest Chain that the question is talking about: I was able to solve parts 1 and 2 as shown in the image below. But I'm not really sure how I'd prove part three. Any help would be appreciated, thanks!
  13. redtree

    I Chain rule for denominator in second order derivatives

    Given ## \frac{d^2x}{dy^2} ##, what is the chain rule for transforming to ##\frac{d^2 x}{dz^2} ##? (This is not a homework question)
  14. caffeinemachine

    MHB Regarding mixing time of a Markov chain

    Let $X=X_0, X_1, X_2, \ldots$ be a Markov chain on a finite state space $S$, and let $P$ denote the transition matrix. Assume that there is an $\varepsilon>0$ such that whenever $\mu_0$ and $\nu_o$ are point distributions on $S$ (in other words, $\mu_0$ and $\nu_0$ are Direac masses) we have...
  15. Math Amateur

    MHB The Chain Rule in n Dimensions .... Browder Theorem 8.15 - Another Question ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some further help in order to fully understand the proof of Theorem 8.15 ...
  16. Math Amateur

    MHB The Chain Rule in n Dimensions .... Browder Theorem 8.15

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some help in order to fully understand the proof of Theorem 8.15 ... Theorem 8.15...
  17. C

    Engineering ChemE PhD w/ a Supply Chain Engineer job offer

    Hi, I'm graduating with a PhD in ChemE. I'm interested in a position in industry that is not purely technical and that will have some exposure to the business/leadership side. I have an offer at a F500 company to be a supply chain engineer. The job will entail working with Asian suppliers and...
  18. K

    A Martingale, calculation of probability - Markov chain needed?

    I am trying to estimate probability of loosing (probability of bankrupt ##Pb##) using Martingale system in betting. I will ilustrate my problem on the following example: Let: ##p## = probability of NOT getting a draw (in some match) We will use following system for betting: 1) We will bet only...
  19. S

    I Exploring the Derivative of y(x,t) in Quantum Mechanics

    y(x,t) = 1/2 h(x-vt) + 1/2 h(x+vt) This is from the textbook "quantum mechancs" by Rae. The derivative is given as dy/dt = -v1/2 h(x-vt) + v1/2 h(x+vt) I'm not quite sure how this is? If I use the chain rule and set the function h(x-vt) = u Then by dy/dt = dy/du x du/dt I will get (for the...
  20. Osvaldo

    Lubricate a Chain with Grease & Solvent?

    Does it make sense to mix a petroleum solvent with grease in order to lubricate a chain so the mixture gets well inside the rollers and wait for the solvent to evaporate, leaving a film of grease within the chain components? Will this happen? Lube oils work fine but leave the chain after few...
  21. Gundyam

    Chain Hangs Over a Pulley and Starts Moving

    Homework Statement: A chain hangs over a pulley. Part of it rests on a table, and another part rests on the floor. When released, the chain begins to move and soon reaches a certain constant speed v. Can we find the height h of the table? I think this question need some tricks. I've tried some...
  22. S

    A Chain Rule for Pushforwards

    prove that if ##g:Y→Z## and ##f:X→Y## are two smooth maps between a smooth manifolds, then a homomorphism that induced are fulfilling :## (g◦f)∗=f∗◦g∗\, :\, H∙(Z)→H∙(X)## I must to prove this by a differential forms, but I do not how I can use them . I began in this way: if f∗ : H(Y)→H(X), g∗...
  23. B

    I Basic doubt on chain rule in DAlemberts soln to wave equation

    In D Alembert's soln to wave equation two new variables are defined ##\xi## = x - vt ##\eta## = x + vt x is therefore a function of ##\xi## , ##\eta## , v and t. For fixed phase speed, v and given instant of time, x is a function of ##\xi## and ##\eta##. Therefore partial derivative of x w.r.t...
  24. Physics345

    Understanding the Chain Rule in Multivariable Calculus

    Solution: ##\frac{\partial z}{\partial x} = yx^{y -1}+1## ##\frac{\partial z}{\partial t} =\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}## ##\frac{\partial z}{\partial t} = (\frac{yx^{y-1} + 1}{2\sqrt{s+t}}) +...
  25. M

    Multivariable Chain Rule Question

    For context, we have an equation f(x,y) = \frac{x}{y} and we had used the substitutions x = r \cos\theta and y = r \sin\theta . In the previous parts of the question, we have shown the following result: \frac{\partial f}{\partial x} = \cos\theta \Big(\frac{\partial f}{\partial r}\Big) -...
  26. X

    A Question about crystal structure of long chain fatty acids (XRD)

    Hello, we are studying potato periderm before/after digestion and dewaxing and native vs wounded. One of the techniques is XRD, hence my user name. Among other things we are trying to use XRD to determine the arrangement of fatty acids, esters, and other long-chain aliphatics that are present...
  27. Boltzman Oscillation

    Help explaining the chain rule please

    I had already calculated the first partial derivative to equal the following: $$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}$$ Now the second partial derivative I can use the chain rule to do and get to...
  28. user366312

    Finding the value of ##P(X_3 = 1|X_1 = 2) = ?## in a Markov Chain

    ##P^2=\begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix}= \begin{bmatrix} 1/2 & 1/4 & 1/4\\ 1/4 & 1/2 & 1/4\\ 1/4 & 1/4 & 1/2 \end{bmatrix}## So, ##P(X_3 = 1|X_1 = 2) = 1/4##. Is this solution correct?
  29. user366312

    Finding ##P(X_2 = 2)## of a Markov Chain

    My solution: ##X_1 = \begin{bmatrix} 1/2&1/2&0 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix}## ##X_2 = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0...
  30. user366312

    How can I compute expected return time of a state in a Markov Chain?

    Problem Statement I was watching a YouTube video regarding the calculation of expected return time of a Markov Chain. I haven't understood the calculation of ##m_{12}##. How could he write ##m_{12}=1+p_{11}m_{12}##? I have given a screenshot of the video.
  31. H

    Chain problem - velocity as the chain becomes completely vertical

    To start this problem, I used equation (1) K_i + U_i = K_f + U_f Then, using (2) and (3) and knowing that the initial velocity is 0, I have m_igy_i = \frac{1}{2} m_fv_f^2 + m_fgy_f The mass of the hanging part of the rope is ## \frac{y_0}{L} m ##. Additionally, I set the face of the table...
  32. C

    MHB Vibrations of a Hanging Chain: Modeling Tension with PDEs

    Dear Everyone, I am having trouble with how to start with one part of the question: "In this exercise, we derived the PDE that models the vibrations of a hanging chain of length $L$. For convenience, the x-axis placed vertically with the positive direction pointing upward, and the fixed end...
  33. A

    I Understanding the Chain Rule Equation: Explained with Examples

    If we have an equation ##g (q,w) =f(q,-w)## and we want to find the derivative of that equation with respect to w, we would normally do $$\frac {dg}{dw} = \frac {d}{dw} f(q,-w) = \frac {df}{d(-w)} \frac {d(-w)}{dw} = -\frac {df}{d(-w)} $$ but my friend is saying that $$\frac {dg}{dw}= -\frac...
  34. Robin04

    Solving Chain Rule Problem with Equation (7.8)

    Homework Statement https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1007&context=foundation_wave I'm trying to understand this paper and I'm stuck at equation (7.8). That part of the text is very short so I hope you don't mind if I don't copy the equations here. Homework...
  35. I

    MHB Optimizing Markov Chain Production System Throughput w/ Exponential Rate of 50

    I am new to Markov chain, i want to model this as a continuous-time Markov chain. A wind turbine manufacturer would like to increase the throughput of its production system. For this purpose it intends to install a buffer between the pre-assembly and the final assembly of the wind turbines...
  36. F

    I Chain Rule and acceleration as a function of two variables

    Hello Forum, When the force ##F## and its resulting acceleration ##a## have the most general form, the acceleration can depend on the position ##x##, time ##t## and speed ##v##. Newton's second law is given by ## \frac {d^2x}{d^2t}= a(x,t,v)##. When the acceleration is only a function of...
  37. Math Amateur

    I The Chain Rule for Multivariable Vector-Valued Functions ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ... I need help in order to fully understand Theorem 12.7, Section 12.9 ... Theorem 12.7...
  38. Math Amateur

    MHB The Chain Rule for Multivariable Vector-Valued Functions .... ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...I need help in order to fully understand Theorem 12.7, Section 12.9 ...Theorem 12.7...
  39. V

    Hamiltonian for a 1D-spin chain

    Homework Statement [/B] A 1D spin chain corresponds to the following figure: Suppose there are ##L## particles on the spin chain and that the ##i##th particle has spin corresponding to ##S=\frac{1}{2}(\sigma_i^x,\sigma_i^y,\sigma_i^z)##, where the ##\sigma##'s correspond to the Pauli spin...
  40. S

    A Chain Rule in Lagrangian Transformation

    Hello, I'm trying to follow Goldstein textbook to show that the Lagrangian is invariant under coordinate transformation. I got confused by the step below So ## L = L(q_{i}(s_{j},\dot s_{j},t),\dot q_{i}(s_{j},\dot s_{j},t),t)## The book shows that ##\dot q_{i} = \frac {\partial...
  41. karush

    MHB Find a Solution for $y=\sqrt{7x+\sqrt{7x+\sqrt{7x}}}$

    find y' $$y=\sqrt{7x+\sqrt{7x+\sqrt{7x}}}$$ ok this was on mml but they gave an very long process to solve it don't see any way to expand it except recycle it via chain rule any suggest...
  42. caffeinemachine

    MHB "Two step" Markov chain is also a Markov chain.

    Let $X$ be a compact metric space and $\mathcal X$ be its Borel $\sigma$-algebra. Let $\mathscr P(X)$ be the set of all the Borel probability measures on $X$. A **Markov chain** on $X$ is a measurable map $P:X\to \mathscr P(X)$. We write the image of $x$ under $P$ as $P_x$. (Here $\mathscr P(X)$...
  43. opus

    Chain Rule with Leibniz Notation

    Homework Statement Find the derivative of ##y=cos^3(πx)## *Must be in Leibniz notation Homework EquationsThe Attempt at a Solution (i) $$Let~ w=y^3 , y=cos(u), u=πx$$ (ii) $$\frac{dw}{dy} = 3y^2,~ \frac{dy}{du} = -sin(u),~ \frac{du}{dx}=π$$ (iii) By the Chain Rule, $$\frac{dw}{dx} =...
  44. opus

    Is this a typo? Using Chain Rule and Product Rule

    Homework Statement This isn't really a homework problem, as the entire solution is laid out in the text. My question is in regards to a possible typo, which I have highlighted in blue in the given picture. Usually I don't like to second guess the text, but this one has been absolutely plagued...
  45. sams

    I Chain Rule of Multivariable Calculus

    I am confused when I should use the ∂ notation and the d notation. For example, on http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx, in Case 1, the author wrote dz/dt while in Case 2, the author wrote ∂z/∂t. Could anyone please explain to me when I should use the ∂ notation and the...
  46. starstruck_

    Infinite chain of alternating charges (+/-)

    Homework Statement A crystal is a periodic lattice of positively and negatively charged ions. (a) Consider an infinite one-dimensional crystal of alternating charges +q and −q, separated by distance d...
  47. Mr Davis 97

    Show that union of ascending chain of subgroups is subgroup

    Homework Statement Let ##H_1 \le H_2 \le \cdots## be an ascending chain of subgroups of ##G##. Prove that ##H = \bigcup\limits_{i=1}^{\infty} H_{i}## is a subgroup of ##G##. Homework EquationsThe Attempt at a Solution Certainly ##H## is nonempty, since each subgroup ##H_i## has at least the...
  48. S

    Solving Equal Force Chain System Dynamics

    Homework Statement Imagine exactly same two bodies (same mass, same surface/friction etc) are attached with a string/spring and they suddenly start pulling towards each other with equal force. I can imagine they will move towards the center. If there is a chain (with defined number; not...
  49. navneet9431

    Problem in applying the Chain Rule

    Homework Statement I am facing problem in applying the chain rule. The question which I am trying to solve is, " Find the second derivative of " Homework Equations The Attempt at a Solution So, differentiated it the first time, [BY CHAIN RULE] And now to find the second derivative I...
  50. S

    Polymer Chain in Statistical Mechanics

    Homework Statement A polymer chain consist of a large number N>>1 segments of length d each. The temperature of the system is T. The segments can freely rotate relative to each other. A force f is applied at the ends of the chain. Find the mean distance ##\textbf{r}## between the ends...
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