Chain Definition and 979 Threads

  1. 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...
  2. 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...
  3. 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...
  4. C

    Chain rule or Substitution rule?

    Chain rule or Substitution rule? Homework Statement It appears that a standard result in ODE is the following: if f(x,t) is smooth enough, then the solution \psi(t) to the initial value problem: x(t)=x_{0}+\int_{0}^{t}f(x(s),s)ds x(0)=x_{0} is continuously differentiable with respect to...
  5. J

    Solving Calc Homework: Lifting a Chain to the Ceiling

    Homework Statement Alright, so my calc class isn't getting easier and we started doing 'work' problems, and I'm just not getting it. Here's the question: A 10ft long weighs 25l and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that its level...
  6. C

    Linear first-order diffeq system for radioactive decay chain

    Homework Statement Given the followin[Sg decay chain- X→Y→Z Solve for Nx(t), Ny(t), Nz(t) for the case of Rx(t)=\alphat and assuming Ny(t)=Nz(t)=0 Homework Equations dNx(t)/dt = -\lambdaxNx(t) + Rx(t) dNy(t)/dt = -\lambdayNy(t) +\lambdaxNx(t) dNz(t)/dt = -\lambdazNz(t) +\lambdayNy(t) The...
  7. T

    Chain rule for implicit differentiation

    I have derivative dx/dt = y(u(t)) * z(u(t)) + u(t) Now, what is dx/du ? I know the chain rule should help, but I am stuck :-(
  8. N

    [markov chain] prove that probability equals 6/pi²

    Homework Statement Homework Equations N/A The Attempt at a Solution I'll shortly explain what my reasoning is so far, but please ignore it if it comes across too jumbled: --- Let P denote the markov matrix associated with this problem, then I think I was able to argue that the...
  9. C

    Chain rule and cylindrical coordinates

    I'm trying to understand this one derivation but this one part keeps messing me up; theta = tan^-1 (y/x) r^2 = x^2 + y^2 d theta/ d x = y/ (x^2 + y^2) how did they get this line?
  10. A

    Chain rule violated for arc length?

    ->ds/dt where s is the arc length in cartesian coordinates is ((dx/dt)^2+(dy/dt)^2)^(1/2). -> Therefore by the chain rule ds/dt = ds/dp * dp/dt, but if I substitute dx/dt=dx/dp* dp/dt and dy/dt= dy/dp* dp/dt in the formula above, I get ds/dt=ds/dp * |dp/dt|?? What is happening? ->Even by...
  11. S

    Chain Rule and Multivariable Calculus Question

    I think i found the solution to my problem but i was hoping to have someone check to make sure i did not make a mistake. \xi = x - ct...... (1) u(t,x) = v(t,\xi)......(2) Taking the derivative d[u(t,x) = v(t,\xi)] \frac{\partial u}{\partial t}dt + \frac{\partial u}{\partial x}dx =...
  12. S

    Change of Variables and Chain Rule

    Homework Statement I am trying to solve the transport PDE using a change of variables and the chain rule, and my problem seems to be with the chain rule. The PDE is: \frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x} = 0 ......(1) The change of variables (change of reference frame)...
  13. V

    Prove the chain rule for Jacobi determinants

    Homework Statement Prove the chain rule for Jacobi determinants \frac{d(f,g)}{d(u,v)} * \frac{d(u,v)}{d(x,y)}=\frac{d(f,g)}{d(x,y)} Homework Equations Definition of Jacobi determinant \frac{d(f,g)}{d(u,v)} = \frac{d(f,g)}{d(u,v)} = det \begin{bmatrix} \frac{df}{du}&\frac{df}{dv} \\...
  14. C

    Why does Δx tend to 0 when Δu tends to 0 in the chain rule?

    Hello, Looking through a book on calculus I found the following explanation for the chain rule and I have one unclear thing that I'd like to ask for help on. The canonical example is used, y is a function of u: y = u^{n} and u is a function of x (let's say) u = 3x - 2 therefore by composition...
  15. S

    Markov chain, sum of N dice rolls

    Question : Let Xn be the maximum score obtained after n throws of a fair dice a) Prove that Xn is a markov chain and write down the transition matrix Im having a problem starting the transition matrix im assuming the states are meant to be the sum. then do you write out the transition...
  16. H

    Using Multi-Variable Chain Rule to Prove Equation Involving z = f(x^2 + y^2)

    Homework Statement z = f (x^2 + y^2) prove using multi var chain rule that y * dz/dx - x * dz/dy = 0 Homework Equations The Attempt at a Solution honestly i just need to no how to start it then I am sure i could figure the rest out so i would find dz/dx and dz/dy then...
  17. D

    Finding Derivatives with the Multivariable Chain Rule

    Homework Statement Let f be a differentiable function of one variable, and let z = f(x + 2y). Show that 2∂z/∂x − ∂z/∂y = 0 Homework Equations Multi-variable chain rule The Attempt at a Solution I have no idea where to start with this, any advice would be greatly appreciated...
  18. J

    Solving Markov Chain Question on Two Switches

    Hi, I need help with answering this question. Firstly, I'm not sure what the transition matrix should like. Should there be 2 states? One where both switches are off and one where both switches are on? The question is: Suppose that each of 2 switches is either on or off during the day. On...
  19. J

    Chain Rule For Function of Severable Variables

    Homework Statement I'm trying to follow my textbook on an application of the chain rule. Two objects are traveling in elliptical paths given by the following parametric equation. x1 = 4 cos t x2 = 2 sin 2t y1 = 2 sin t y2 = 3 cos 2t At what rate is the distance between the two...
  20. H

    How do I use the chain rule to differentiate this function?

    Homework Statement Differentiate the functions using chain rule. 2(x3 −1)(3x2 +1)4 Homework Equations Chain Rule = f ' (g(x))g' (x) The Attempt at a Solution I don't know how to do using chain rule, but product rule is easier So using product rule, = f ' (x) g(x) + f (x)g'...
  21. P

    Change in energy for a balled up chain

    Homework Statement A 1.1 m, 4 kg chain is wrapped up in a ball. You grab one of the ends of the chain and apply a constant force of 59 N and it begins to unwrap. When you pull your end of the chain 3.9 m, the chain is completely loose. a) Find the change in energy of the system, before...
  22. H

    How Does a Bug's Temperature Reading Change as It Moves Along a Curve?

    Homework Statement [12] 3.Thetemperature T(x, y) at a point of the xy-plane is given by T(x,y)= ye^(x^2). A bug travels from left to right along the curve y = x^2 at a speed of 0.01m/sec. The bug monitors T(x, y) continuously. What is the rate of change of T as the bug passes through...
  23. N

    Serial link chain manipulator with constrained geometry

    Homework Statement I have a three link revolute manipulator at the origin. I know all the link lengths. The joint angle for the third link is coupled to the second such that Theta3 = k*Theta2. I want to determine the joint angles (thetas) of the manipulator given that the third link should...
  24. N

    Serial Link chain with constrained geometry

    I have a three link revolute manipulator at the origin. I know all the link lengths. The joint angle for the third link is coupled to the second such that Theta3 = k*Theta2. I want to determine the joint angles (thetas) of the manipulator given that the third link should lie on a line an angle...
  25. H

    Mechanics Falling Chain Problem

    Homework Statement A chain of length L and mass density σ kg/m is held in a heap. I grab an end of the chain that protrudes a bit out of the top. The heap is then released so that the chain can unravel with time. Assuming that the chain has no friction with itself, so that the remaining part...
  26. T

    How to Use Chain Rule to Find Second Derivative of Multivariable Functions?

    Suppose I have F(x,y) and y=y(t) and x=x(t) Therefore, Ft = Fx*xt + Fy*yt. Right? Can I write Ftt = (Fxx*xt + Fyy*yt)*xt + Fx*xtt + (Fxx*xt + Fyy*yt)*yt + Fy*ytt ? Basically I'm trying to figure out the second derivative by chain rule.
  27. W

    Whats the most common roller chain norm for bicycles?

    I am designing a bike in Autodesk Inventor for a university project, and I am stuck with the sprockets. Inventor can create them fairly easily when you know the norm of the sprocket and the number of teeth it has, but I don't know the standard of the sprocket I have to design; I merely know that...
  28. S

    Using Chain Rule to Find du/dT & du/dv: Step-by-Step Guide

    If a function is given by u = u(T,v) how to use the chain rule to write how u changes with respect to T & v. Please specify the steps involved. i understand chain rule as \frac{du}{dx} = \frac{du}{dy} \frac{dy}{dx}
  29. T

    Proving Zx + Zy = 0 using Chain Rule

    Homework Statement If Z= F(x-y), show that Zx + Zy = 0 Homework Equations The Attempt at a Solution Suppose I let Q = x-y. Then, by chain rule, Fx(Q) * 1 + Fy(Q) * -1. By identity, this statement must hold for all values x,y. In particular, it must hold for x=y. By x=y...
  30. U

    Chain rule for partial derivatives

    Homework Statement So there is an exercise in which I should "verify" the chain rule for some functions. In other words to do it by substitution, then doing by the formula and checking if the results are the same. (and checking with the book`s answer too) For a few of them, they just don`t...
  31. J

    Alternating linear chain of masses

    Homework Statement A chain of atoms are connected by identical springs of force constant k. Suppose teh atoms of mass m alternate with atoms of mass M. Thus the crystal consists of a sequence ... MkmkMkmMkmk ... which is the periodic repetition of unit cells Mkmk. The size of the unit cell is...
  32. D

    Proof of Multivariable chain rule

    I was wondering how to prove the multivariable chain rule \frac{\mbox{d}z}{\mbox{d}t}=\frac{\partial z}{\partial y}\frac{\mbox{d}y}{\mbox{d}t}+\frac{\partial z}{\partial x}\frac{\mbox{d}x}{\mbox{d}t} where z=z(x(t),y(t)) I don't really need an extremely rigorous proof, but a slightly...
  33. D

    Ease of chain reaction for enriched uranium

    Critical mass is over 50kg so let's say I have 2 halves of a sphere of the isotope U-235, each weighing 30 kg. I drop one onto the other so that they form a supercritical sphere. No doubt a chain reaction would begin, but I assume it would produce energy on the level of a nuclear reactor rather...
  34. J

    Oscilations of a linear chain of masses and springs

    Homework Statement A linear chain consists of N identical particles of mass m are connected by N+1 identical, massless springs with force constant k. The endpoints are fixed to walls on each side. In the static configuration each spring is stretched from its relaxed length l0 to a new length...
  35. M

    The chain rule for 2nd+ order partial differential equations

    Homework Statement w= f(x,y) x = u + v Verify that Wxx - Wyy = Wuv y = u - v Homework Equations The Attempt at a Solution I know how to find Wu or Wv but I have no idea on how to proceed to find the 2nd order derivative (or 3rd,4rth etc.. obviously). I...
  36. N

    When to assign a value to a multiple chain derivative?

    I was doing my homework and I ran into a problem of a chain rule within a chain rule. When do I know what to assign a value? For example: y=e^{-x^2} When I assign u=e^{-x} and y=u^2 I get a wrong value. According to cramster I was supposed to assign y = e^u and u=-x^2. But when am I supposed...
  37. H

    What to use when reverse chain rule doesnt work?

    Hi there, My equation to solve is (xy+(x^2))dx + (-1)dy=0 For method of exact solutions, the partials are not equal to each other so I cannot use exact solutions (reverse chain rule) I don't know how to solve this
  38. J

    Differentiation by the chain rule

    Homework Statement Find the derivative of the following: Homework Equations Y= x^3(5x-1)^4 The Attempt at a Solution 4(3x^2(5x-1)^3)(4(3x^2(3(5x-1)^2)(2(5x-1)(5)
  39. S

    Why Is the Chain Rule Necessary for Differentiating Functions Like e^sqrt(x)?

    Just some general questions as I'm confused with when to use chain rule when not to. For instance, to find the derivative of e^sqrt(x), the right answer is to use chain rule to get e^sqrtx*the derivative of sqrt(x). BUT, isn't there a formula that: d/dx K^x = In(K)*K^x? K for constant and x...
  40. J

    Gamblers Ruin - Markov Chain problem

    This is probably a noob question, background in probability theory isn't great but I was shown this problem in a lecture: "Suppose a gambler starts out with £n, and makes a series of £1 bets against the house. Let the probability of winning each bet be p, and of loosing be q = 1 − p. If...
  41. K

    Partial Differentiation - The Chain Rule

    Homework Statement Calculate ∂f/∂s + ∂f/∂t at s = 2, t = -1. Given: f = f(x,y) x = s - t y = s2 + t2 ∂f/∂x (3,5) = 0.06170 ∂f/∂y (3,5) = 0.06170 Homework Equations ∂f/∂s = ∂f/∂x * ∂x/∂s + ∂f/∂y * ∂y/∂s ∂f/∂t = ∂f/∂x * ∂x/∂t + ∂f/∂y * ∂y/∂t The Attempt at a Solution...
  42. P

    Understanding the Chain Rule: A Derivative Problem Solution

    I did a derivative problem, but my book says that my answer is wrong. f(x)=x2(x-2)4 I didn't see much use in the chain rule so I used the product rule. x2(4(x-2)3) + (x-2)4(2x) =4x2(x-2)3 + 2x(x-2)4 The book says that instead of this, the answer is ... x2(4(x-2)3(1)) + (x-2)4(2x) =...
  43. E

    Simplifying after applying chain rule

    Homework Statement http://images.calcchat.com/solutionart/etf5e/03/d/se03d01063.png Homework Equations The Attempt at a Solution I get to the third row, but can't simplify (Sin2θ)(Cos2θ). I'm looking at the trigonometric double angle formulas, and still can't figure out how the...
  44. S

    Monatomic Linear Chain - Comparison of Numerical and Analytical Results

    Hi all, I'm having a few problems with crystal dynamics of a simple monatomic chain. Taking the dispersion relation: \omega^2 = \frac{4k}{m}\left(\sin^2 \left( \frac{\kappa a}{2}\right)\right) Where k=spring constant, m=mass, \kappa=wavevector, a= lattice constant and \omega=...
  45. T

    Magnitude of force question? Chain link

    A chain consisting of five links, each of mass 0.1 kg, is lifted vertically with a constant acceleration of a = 2.5 m/s2. Find the magnitude of (a)the force on link 1 from link 2 (b)the force on link 2 from link 3 (c)the force on link 3 from link 4 (d)the force on link 4 from link 5...
  46. X

    Chain rule of partial derivatives

    Homework Statement Suppose f(x,y) = 2x^5 + 4xy + 2y^3 g1(u,v) = u^2 - v^2 g2(u,v) = uv h(u,v) = f(g1(u,v), g2(u,v)) Use chain rule to calculate: dh/du (1,-1) and dh/dv (1,-1) Homework Equations The Attempt at a Solution i let h (u,v) = 2(u^2 - v^2)^5 + 4(u^2-v^2)(uv) +...
  47. B

    Solving 2 Chain Rule Problems: Struggling with Derivatives and Need Help

    For some reason I am struggling with these problems. I am lost as a goose trying to fly south for the winter! Homework Statement 9^(5-x2) and another derivative problem using chain rule r/square root of the whole term r^2+5 Homework Equations 1st equation= d/dx= a^x ln a 2nd...
  48. B

    I am having trouble finding trig derivatives using chain rule

    Homework Statement cot^2(Cos\theta)Homework Equations chain rule f prime (x) = f prime(g(x) * g prime (x) The Attempt at a Solution I am not sure if I am just inputting the wrong numbers into webassign or I am just missing and important trig derivative and just completely off of the boat...
  49. S

    Solving Chain Rule & Trig. Power Equations

    I always get muddled when I'm dealing with chain rule of any degree of complexity and also when dealing with powers of trig. functions - this problem contains both: find \frac{\partial n}{\partial A} and \frac{\partial n}{\partial D} of the following function...
  50. W

    Chain rule for commutator (Lie derivative)?

    I'm curious if there's a chain rule for the commutator (I'll explain what I mean) just like there's a product rule ([AB,C]). So, say you have an operator, which can be expressed in terms of another operator, and we know the commutation relationship between x and another operator, y. I'll call...
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