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Change in variables

  1. Jul 26, 2012 #1
    1. The problem statement, all variables and given/known data
    I am performing a change of variables, s --> t and am wondering can I just write g(s) --> g(t) or do I have to alter the function e.g. g(s) --> a*g(t).


    2. Relevant equations
    s = (a/b)* t
    g(s) is to be found numerically and therefore we do not have it's definition.

    So Option one:
    just rewrite g(s) as g(t)

    Option two:
    g(s) = (a/b)*g(t)

    Thanks
     
  2. jcsd
  3. Jul 26, 2012 #2
    Um, generally, if s -> t, then g(s) does NOT go to g(t). If you know that g is continuous, then you may assume that. Otherwise, you will have to work off the official definition of limits.
     
  4. Jul 26, 2012 #3
    I would assume that if s => ta /b, g(s) => g(ta/b) if a,b are arbitrary / real constants. I could be wrong though.
     
  5. Jul 26, 2012 #4
    That's definitely not true. Consider the function

    g(x) = x + 1 for x > 0 and x - 1 for x <= 0. Then as s -> 0 from the right, g(x) => 1, even though g(0) = -1.
     
  6. Jul 26, 2012 #5

    HallsofIvy

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    You are misunderstanding the notation. The question is not about limits. The OP is asking 'If we change variable s to variable t, can we write f(s) as simply f(t)?'

    MathMonster, there are times when that is acceptable but I would avoid it. Technically if, for example, [itex]f(s)= s^2- 3s+ 2[/itex] then [itex]f(t)= t^2- 3t+ 2[/itex] but I don't believe that is what you are trying to do. If [itex]f(s)= s^2- 3s+ 2[/tex] then replacing s with t= s-2 gives, since s= t+2, [itex]f(s)= f(t+ 2)= (t+ 2)^2- 3(t+ 2)+ 2= t^2+ 4t+ 4- 3t- 6+ 2= t^2+ t[/tex] but notice that is f(t+2), not f(t). If you like you say explicitely that g(t)= f(t+ 2) and then use g(t).
     
  7. Jul 26, 2012 #6
    I see - my apologies. Nevertheless, you should not be moving constants out of expressions, g(at/b) =/= (a/b) * g(t). So unless you would like to simply g(t) with some change of variable (again, this as the above advised, this is not suggested - there are many scenarios where this is invalid - an introductory book on Real Analysis can help clarify the conditions under which you may do so), there isn't too much of a point of doing so.
     
  8. Jul 30, 2012 #7
    Hi!

    Thank you for the reply! Yes I am talking about a change in variables not limits - sorry that's my fault I didn't think of the notation I was using.

    HallsofIvy - thank you! I thought that was the case but was confused how to so it. Basically i've got a function g(s+) and g(s-) and a relation linking s+ & s-. I then have a formula with the two 'g's' in which I need to write as s+ in order to solve it.

    So basically, I will just write g(s-) = g((a/b)*s+) and leave it at that? There's a bit of differentiation that will come in which will take into account the change.

    Thanks!
     
  9. Jul 30, 2012 #8
    Could you possibly post the question you are working on?
     
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