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
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...
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...
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...
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...
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?
->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...
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 =...
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)...
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...
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...
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...
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...
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...
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...
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'...
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...
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...
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...
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...
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...
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.
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...
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}
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...
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...
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...
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...
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...
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...
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...
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...
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
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)
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...
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...
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) =...
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...
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=...
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...
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...
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...
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...
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...
Suppose X is a random walk with probability
P(X_k=+1)=p and P(X_k=-1)=q=1-p
and S_n=X_1+X_2+...+X_n
Can anyone explain why does line 3 equal to line 4?
P(S_k-S_0≠0 ,S_k-S_1≠0 ,…,S_k-S_{k-1}≠0)
=P(X_k+X_{k-1}+⋯+X_1≠0 ,X_k+X_{k-1}+⋯+X_2≠0 ,…,X_k≠0)
=P( X_k≠0 ,X_k+X_{k-1}≠0...
Homework Statement
For the van der Waals equation of state, confirm the following property:
(∂P/∂T)V (∂T/∂V)P (∂V/∂P)T = -1
Homework Equations
The van der Waals equation of state is:
P = nRT/(v-nb) - an2/V2
*R, n, a, b are const.
The Attempt at a Solution
I...
It's been years since I've done maths properly so I'm rusty with it. I'm helping out a colleague at work who is studying maths for an OU course.
Homework Statement
Part 1: Differentiate function.
f(x) = e^(0.5x+cos(x))
Part 2: Use answer from part 1 to show.
g(x) =...