# Prove that this is a total derivative

• Apr 18th 2013, 07:18 PM
Kiwi_Dave
Prove that this is a total derivative
I have:

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

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:

$\displaystyle \frac {dF}{dt}=f$

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?
• Apr 19th 2013, 04:41 AM
chiro
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.
• Apr 19th 2013, 07:25 PM
Kiwi_Dave
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"????
• Apr 19th 2013, 07:36 PM
chiro
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).