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Math Help - Prove that this is a total derivative

  1. #1
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    Prove that this is a total derivative

    I have:

    f=\frac{bt}{m}L-\frac{tx^2}{2}k'(t)x^2e^{\frac{bt}{m}}

    L is a lagrangian so I can be confident that it can be integrated with respect to time.

    I want to show that this quantity is the total derivative of some function F. I.e. a function F exists such that:

    \frac {dF}{dt}=f

    1. How would I make such an argument?
    2. Can I make an argument without actually fnding F?
    3. What features would/could a function f have if this were not true?
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  2. #2
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    Re: Prove that this is a total derivative

    Hey Kiwi_Dave.

    You could show that the function f is Integrable over some region (Riemannian Integrable) and that would suffice. Showing Riemannian Integrability can be shown if you have two functions that are Riemann Integrable then the product is Riemann Integrable (I've seen this proof on a pdf somewhere so you can search for it if you want).

    I suggest this way to split up the Langrangian with bt/m and the other term.

    Once you do this, you will have shown that an anti-derivative exists, and you're done.
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  3. #3
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    Re: Prove that this is a total derivative

    Ahhh! If I expanded t*k'(t) in a Laurent series and found that it had a term in t^n where n<0 then it might not be integrable at t=0.

    To flip that on its head I might say:

    "Provided k(t) expanded as a Laurent series has no term t^n with n<-1 then I can be assured that F(t) exists"????
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  4. #4
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    Re: Prove that this is a total derivative

    You also need to supply the range of integration as well but if k'(t) is Riemman integrable over that range then that term will be integrable (I'm also assuming x is independent from t as well).
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