1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Multivariable Calculus (1st order approximation)

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Define f:R2→R3 by
    f(x,y,z)=(xy+z)
    ............(x2-yz)

    let p = (1,1,1)T and h=(δ,ε,θ)

    a)what are n and m? evaluate f(p) and f(p+h)
    b)Calculate the Jacobian Matrix Df(x,y,z) and evaluate Df(p)
    c) Calculate the error e(h) in the first order approximation to f(p+h)
    d) show clearly that

    lim h→0 |e(h)| =0
    ....................|h|


    Explain why this is what you expect

    2. Relevant equations



    3. The attempt at a solution

    a)
    n=3
    m=2

    f(p)=(1,1,1) = (2)
    ....................(0)

    f(p+h) = f(1+δ, 1+ε, 1+θ)

    =(2+δε+δ+ε+θ)
    2+2δ-εθ-ε-θ)

    b)
    jac= (y x 1)
    .......(2x -z -y)

    Df(p)=(1 1 1)
    .........(2 -1 -1)

    c)
    f(p+h)≈f(p)+Df(p)h

    only calculation of Df(p)h needed to work out error.

    =(y+x+1)h
    (2x-z-y)

    =(ε+δ+1)
    (2δ-θ-ε)

    e(h)=f(p+h)-(f(p) + Df(p)h)

    =(δε+θ-ε-1)
    2-εθ)

    (not confident on this)

    d)not attempted yet/ dont know how to
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Multivariable Calculus (1st order approximation)
  1. 1st order ODE Question (Replies: 0)

Loading...