Linearization Definition and 72 Threads

In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology.

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  1. H

    First order linearized Euler's equation for a small perturbation

    I'm trying to linearize (first order) the Euler's equation for a small perturbation ##\delta## Starting with ##mna (\frac{\partial}{\partial t} + \frac{\vec{v}}{a} \cdot \nabla ) \vec{u} = - \nabla P - mn \nabla \phi## (1) ##\vec{u} = aH\vec{x(t)} + \vec{v(x,t)}## Where a is the scale factor...
  2. Z

    Linearizing a circuit with a nonlinear element

    I think I managed to solve the entire problem, as I show below. My main doubt is about item (e), the incremental circuit. Part (a) Using the node method and KCL we reach $$\frac{v_I-v_A}{2}=10(1-e^{-v_A/5})\tag{1}$$ Part (b) We can simplify (1) to $$v_A=5\ln{\left ( \frac{20}{v_A+20-v_I}...
  3. F

    I Exploring the Benefits of Linearizing Data - Improve Analysis and Visualization

    Hello, There is a physical phenomenon in which the variable ##X## is related to the variable ##Y## by a cubic relationship, i.e. $$Y= k X^3$$ The data I collected, ##(X,Y)##, seems to fit this relationship well: I used Excel to best fit the data to a power law function (3rd power) and there is...
  4. mcastillo356

    B Linearization of a function error viewed with differentials

    Hi, PF, want to know how can I go from a certain error formula for linearization I understand, to another I do not Error formula for linearization I understand: If ##f''(t)## exists for all ##t## in an interval containing ##a## and ##x##, then there exists some point ##s## between ##a## and...
  5. S

    I Linearization of Lagrange equations

    l am italian student from Milan university, so sorry for my bad english. l am studying lagrange meccanics. We are linearizating lagrange equations. Here l don't understand how you can expand A matrix, how the function f is derivable, how the inverse matrix A is expanded? l am expanding with q0...
  6. V

    MHB Help with Functions - Linearization

    Let f(x) = \sqrt{x} Assume that g is function such that (i) g(c)= c+m(x-1) (ii) f(1) = g(1), and (iii) \lim_{{x}\to{1}}\frac{f(x)-g(x)}{x-1} Answer the following questions. Show all of your work, and explain your reasoning. (a) What are the constants c and m? (b) How does g compare with the...
  7. stockzahn

    A Linearization of nonlinear grad(T) in the diffusion equation

    Dear all, I would like to perform numerical simulations of the heat transfer/temperature field in a static bath of superfluid helium. The heat conduction in superfluid helium can be modeled in two regimes depending on the heat flux. For low heat fluxes ##\dot{q}##, the temperature gradient...
  8. D

    I Problem: perturbation of Ricci tensor

    I am trying to calculate the Ricci tensor in terms of small perturbation hμν over arbitrary background metric gμν whit the restriction \left| \dfrac{h_{\mu\nu}}{g_{\mu\nu}} \right| << 1 Following Michele Maggiore Gravitational Waves vol 1 I correctly expressed the Chirstoffel symbol in terms...
  9. JuanC97

    I Restrictions of 1st Order Perturbation Theory

    Hello guys, I'm wondering if there are some important restrctions on the 'applicability' of first order perturbation theory. I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski...
  10. Jayalk97

    Quick question about linearization using the small angle method

    Hey guys, when you're linearizing a function that has a constant, what do you do to it? An example would be y = x^2 + 3, would you just linearize it using its derivative and get rid of the constant?
  11. C

    B Linearizing a Relation: How to Create a Best Fit Linear Line in Logger Pro

    Hi! Above is a screenshot of logger pro that I'm currently using. I need to linearise this graph and draw a best fit linear line but I have no clue how to do it. What should I do? The table on the left shows the raw data. The first column is showing the values for x-axis, and the second is for...
  12. EEristavi

    An Error Formula for Linearization (involving second Derivative)

    Homework Statement In textbook i was given formula to calculate error. I know that: E(t) = f(t)- L(x) = f(t) - f(a)- f'(a)(t- a) [L(x) is linear approximation]; [Lets call this Formula 1] I understand that, but that I have formula: E(x) = f''(s)/2 * (x-a)^2 [lets call this Formula 2] Here...
  13. M

    I Why Use Nonlinear Polynomials for Linearization?

    Hi everyone. I started to look at different linearization techniques like: -linear interpolation - spline interpolation - curve fitting... Now Iam wondering (and I guess its very stupid) : As polynomials with a degree > 1 are not linear, why can I use them for linearization? With the...
  14. M

    B How Can You Linearize a 1/x Graph in Physics Lab Data?

    So I've been given this practice problem for a test tomorrow and have no clue how to do it. Info: Students Perform a lab and record data on how changes in mass affect acceleration "Derive an equation that you graph to make a linear relationship from the recorded data." Given is a data table...
  15. S

    Homework Problem Regarding Linearization

    Homework Statement Hey Guys, I don't really want to have to post this in of all places due to how advanced everyone else is compared to me, but I went to my second physics class and I have a huge problem. My teacher gave me my homework, but I have no idea on Earth how to do it because she...
  16. G

    MHB Linearization of this equation / Inverse function

    Hello, I need to find the inverse function of the following equation y = a * ((exp(-b * x)) + (c * (1 - (exp(-b * x))))) Where a, b and c are constants. I have experimental points that fit to this equation and I want to use these values in the inverse funtion to linearize it. I have tried...
  17. O

    Curve fitting (Linearization) of functions (and thus graphs)

    Ok, first week of first year of undergraduate physics lab and they explain that we want all our graphs to be linear, and in order to do that we can change our x and y axes to be log(x) or y^2 or whatever. They did some simple examples such as y=(k/x)+c and explained that if the x axes is 1/x we...
  18. karush

    MHB Linearization of sqrt(x+1) +sin x at x=0

    Find the linearization of $f(x) = \sqrt{x+1} +\sin{ x} $ at $x=0$ $L(0)=\frac{3x} {2} + 1$ how is it related to the individual linearization of $\sqrt{x+1}$ and $\sin{x} $ at $x=0$ ? $L_{\sqrt{x+1}} (0 ) = \frac{x} {2}+1 $
  19. I

    Struggling with Linearization and Differentials?

    I am currently taking Calculus 1 and we covered Linearization and Differentials. The title of the section in my textbook is called "Linear Approximations and Differentials," in the book by James Stewart. The book and lecture in this section made absolutely no sense to me. Like, I was COMPLETELY...
  20. Alex734

    Total Stress in the ANSYS Stress Linearization

    Hi! I'm a little bit confused about the meaning of the total stress during stress linearization (by path in the ANSYS). May somebody explain me what's it means or show the formula for this stresses? Thank You!
  21. B

    How Can I Linearize a Velocity vs Force Graph in Circular Motion Analysis?

    Homework Statement In my circular motion lab, I have to find the Varying Force. I have a constant Bob mass, and constant radius. I also have the data recorded of the Hanging mass, Time for 10 circles, period, force, and speed. Then I had a velocity vs force graph which I plotted the data and...
  22. K

    Linearization of f(x,y) = sqrt(29 - 4x^2 - 4y^2) at (2,1)

    Homework Statement Find the linearization of the function ƒ(x,y) = sqrt(29 - 4x2 - 4y2) at the point (2,1) Homework Equations [/B] Point (a,b) L(x,y) = Linearization L(x,y) = ƒ(a,b) + ƒx(a,b)(x-a) + ƒy(a,b)(y-b) The Attempt at a Solution [/B] ƒ(2,1) = 3 ƒx(x,y) =...
  23. A

    How to do Linearization for Non-linear least squares?

    Hy I want to know how to make linearization for some function,...what should by in Non-linear least squares problems. In my book I have only this example how to do: http://i.imgur.com/MUFiHkr.pngSomeone could me help how to do, some receipt of method what I need to do? Non-linear least...
  24. T

    Log Linearization: A Step-By-Step Guide

    Does anyone understand how to log linearize, if so how would I go about doing so? Much Thanks
  25. T

    Linearization of non linear systems

    Homework Statement In my dynamics modelling class, the professor went over an example where we linearize non linear state equations to approximate the behavior. In this case, we are not given an operating point. However, the professor said you can solve for the operating point by setting x' =...
  26. P

    Is It Wrong to Label x-Coordinates as x and (x + Δx)?

    I was reading a chapter on differentials in my calculus book, when I came across the graph shown in the image attached to this post. Two questions came to my mind upon seeing this graph: 1) Isn't it technically wrong to label the x-coordinates as x and (x + Δx)? I mean, wouldn't it be more...
  27. P

    Linearization of Differental system and stabilty

    Homework Statement So I am looking at example where stabilty is determined. And do not understand how linerazation is done. x'=x+2y-ln(1+x)-siny+x^3 y'=4x+2y-sinx+y^2 Homework Equations so the system after linearization looks like: x'=y y'=3x+2y So i thought that higher order x and...
  28. Lebombo

    Local Linear Approximation vs Linearization

    Are Local Linear Approximation, Linear Approximation, and Linearization all the same thing? Question is, I learned about something called Local Linear Approximation in Calc 1. Now in Calc 2, the topic of Linearization from Calc 1 was mentioned. But I never did anything that was referred...
  29. Q

    What is the Correct Linearization of a Function?

    Homework Statement http://i3.minus.com/jbt2vueBfwXvWD.jpg Homework Equations Linearization: f(x) + f'(x)(dx) The Attempt at a Solution The derivative of g(x) using the chain rule is (2lnx)/x. x = e, so that simplifies to 2/e. Linearization: (2/e)x + 1, where 1 is f(e)...
  30. M

    Linearization of Ricatti equations

    In a Riccati equation y'=qo+q1y+q2y^2, if q2 is nonzero then you can make a substitution v=yq2, S=q2q0, R=q1+(q2'/q2) which satisfies a Riccati, v'=v^2+R(x)v+S(x)=(yq2)'=q0q2+(q1+q2'/q2)v+v^2 with double substitution v=-u'/u, u now satisfies a linear 2nd ODE: u''-R(x)u'+S(x)u=0...
  31. J

    What's exactly the meaning of Linearization?

    Hi all, I want to discuss about the assumptions in the linearization. By linearization, I mean the following classic procedure. (1) Original nonlinear governing equation is \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\mathcal{L}(u), RHS is a linear operator (2) introduce...
  32. T

    Linearization and energy equation

    Hey guys. We were given a problem, which was to consider bv^2 as the force acting upwards on a body falling in water. We were asked to find the depth at which a body entering the water at 5 * ve (terminal speed) would end up with a speed of 1.1 * ve. Starting from -bv^2 + mg = ma, the first...
  33. Fernando Revilla

    MHB ^.^'s question at Yahoo Answers (Linearization)

    Here is the question: Here is a link to the question: http://answers.yahoo.com/question/index;_ylt=AhBVp1ZaOojBImS1_MUUqvfFDH1G;_ylv=3?qid=20130409181158AAxlDNJ I have posted a link there to this topic so the OP can find my response.
  34. H

    Flash vaporization balance - ODEs, deviation variables, linearization

    Homework Statement Given the attached figure, a) Develop an ordinary differential equation that describes the dynamic height h(t) in the flash tank in terms of \dot{m}_{i}, \dot{m}_{l},\dot{m}_{v}, \rho_{i}, \rho_{l}, \rho_{v}, and A. b) Given the fact that the process is isenthalpic...
  35. R

    Linearization of Second Order Differential Equations

    I'm having some difficulties figuring out how to linearize second order differential equations for a double pendulum. I have an equation that is in the form of \theta_{1}''\normalsize = function [\theta_{1},\theta_{2},\theta_{1}',\theta_{2}'] (The original equation is found at...
  36. M

    Approximating a number using linearization methods

    Homework Statement The goal is to approximate the number \frac{-1+\sqrt{5}}{2} using linearization methods. Homework Equations This number is a solution to x^{2}=1-x The Attempt at a Solution I was told to use f(x)= x^{2}+x-1 with the Newton method to find x_{1},x_{2},x_{3},x_{4} at...
  37. S

    Finding the intersection of two functions via linearization

    At least I think it's via linearization. Let f(x) = \tan (x^2) - 1 and g(x) = \frac{\ln((x+1)^3)}{3} Find the smallest positive and negative intersection with a relative error of less than 0.001. I don't know. You can linearize one or both, yeah, but you don't have any analytical value to...
  38. N

    Solving non-linear system by linearization

    I have non-linear system of ordinary differential equations to solve by first linearising it. I know it can be linearised by expanding right side of equations by Taylor series and keeping only the linear terms. Then I can solve the linear system of differential equations with given initial...
  39. N

    Even Differentiable Functions and Linearization

    Is there anything special about even differentiable function of x? Give reasons behind your answer. and Find the linearization of g(x)= 3+ ∫sec(t-1)dt at x=-1 It is a definite intergral going from 1 to x^2.. a=1 b=x^2 I understand how to do regular linearization problems but with this...
  40. F

    Solving the Linearization Problem for \dot{x}+√x = 0: Expert Suggestions"

    Hello , I am trying to linearize \dot{x}+√x = 0. The only equilibrium point is at x=0; but the derivative is not defined at this point. Does anybody have a suggestion? Regards.
  41. M

    Linearization of Inverse Proportionality

    Homework Statement Dear All, I currently have a set of data which, when plotted on a scatter diagram, proves an inverse proportionality between two sets of data. (Please see attachment.) I now need to linearize the graph as to estimate the half-life of the foam. Homework Equations...
  42. S

    Linearization Error: Find Approximate Error of f(x) |x| < 1

    Homework Statement I've been given the assignment to make a linearization of the function f(x) = Sin(x) e^(x) by using Taylors polynominal of 3. degree. Next is to find the approximately error of f(x) when |x| < 1 (expressed as a function of x). Homework Equations So finding the...
  43. A

    Linearization of non linear model

    i read that taylor series is used to approximate non linear function at optimal point x0 but i don't understand in which case we use first order approximation and in which cases we use higher order approximations?
  44. N

    Difference between Tangent Plane and Linearization

    What is the difference? According to my text... Tangent Plane: z-z0=fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0) Linearization: L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)
  45. M

    The linearization of the metric of curved space-time

    Why Yab=Xab-kHab+k2HacHcb-... and not Yab=Xab-kHab+(1/2)k2Haccb-...? Y is the curved space-time metric X is the planespace-time metric
  46. M

    How to Linearize a Function without Using Logarithms

    Homework Statement Linearize the following model: y=\alpha*x*e^{\beta*x}Homework Equations The only relevant equations I can think of are the laws of natural logarithms.The Attempt at a Solution I have tried to taking the ln of both sides however that leaves me with an equation that has two...
  47. C

    How to Find the Linearization of the Cosine Function at a Given Point

    Homework Statement Find the linearization L(x) of f(x) = cos(x) at a = π/2 Homework Equations L(x) = f(a) + f'(a)(x-a) The Attempt at a Solution I just want to make sure I did this correctly: L(x) = cos(π/2) + -sin(π/2)(x-(π/2)) L(x) = 0-1(x-(π/2)) L(x) = -x + (π/2)...
  48. Z

    Special Relativity Force and Linearization

    Homework Statement Consider a particle moving at close to the speed of light with v \approx c \ \hat{z}. A small oscillatory force F(t) acts on the particle. Consider F(t) to be a first order (eg. linear) perturbation which will not effect v_o, only v_1, the first order component of v...
  49. G

    Linearization Problem: Solving for Vout in Original and Shifted Coordinates

    Homework Statement Linearize the equation Vout = (10^v)*sin(x) about x=0,0.1, and 1. Write the equation in both the original coordinates and the shifted (linearized) coordinates.Homework Equations y=f(a)+f'(a)(x-a)The Attempt at a Solution dVout/dx = (10^v)*cos(x) evaluating that equation...
  50. M

    Non-Linear First Order ODE: Critical Point Linearization

    Homework Statement dx/dt = x - y + (x^2) - xy dy/dt = -y + (x^2) - Determine the critical points for the equation, - Determine the linearized system for each critical point and discuss whether it can be used to approximate the behaviour of the non-linear system. What is the type and...
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