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Implicit Function Theorem problem

  1. May 1, 2015 #1
    Part 1. If I want to solve the system;

    u-v = (h-a)e^-s

    w-u = (k-b)e^-t

    ae^s = be^t

    for a, b, u, in terms of the remaining variables using the implicit function theorem...

    If I want to know when I can solve, can I just say f(a, b, u) can not = 0? And if I set a, b, u, = 0

    Than I get k and h can not = 0.



    Part 2. Calculate du/ds, da/ds, db/ds, by exhibiting a linear set of equations.

    So for da/ds for example, solve equation 1 for a giving;

    d(-e^s(u-v)+h)/ds?



    I don’t need to solve the system.
     
  2. jcsd
  3. May 1, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    For Part 1, you don't NEED to solve the system, but the easiest solution is to just go ahead and solve it anyway! Using the notation ##S = e^{-s}## and ##T = e^{-t}##, write your equations as
    [tex] u-v = (h-a)S\\
    (w-u)=(k-b) T \\
    a/S = b/T
    [/tex]
    This is a simple linear system, from which you can easily find ##a,b,u## in terms of ##v,w,h,k,S,T##.

    For Part 2, you can also derive a simple linear system for ##\partial a/ \partial s, \: \partial b / \partial s, \: \partial u / \partial s## and get a solution in terms of ##a,b,u,v,w,h,k,S,T##, using the definitions of ##S,T## in terms of ##s,t##.
     
    Last edited: May 1, 2015
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