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Time derivative of definite integral

  1. Sep 8, 2003 #1

    Physics books gloss over math. Sometimes it bothers me.

    Given a separable function of time and position f = h(x)g(t) then

    d / dt of [inte] h(x)g(t)dx = [inte] h(x) dg(t)/dt dx

    Where the derivative in the second integral is a partial deriv.

    Why the chain rule does not apply is glossed over by the statement that the definite integral of f(x,t) with respect to x results in a new function that is a function of t only. I believe that, but since x can be a function of t I can't see why the chain rule does not apply. Can someone please explain this to me?

    To be more specific the book says:

    d/dt of [inte] x * psi(x,t)dx = [inte] x * dpsi(x,t)/dt dx
    where again the second integral has partial deriv. w.r.t. time t. The '*' above just means times here.
  2. jcsd
  3. Sep 8, 2003 #2


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    x cannot be a function of t because x is a dummy variable of integration.

    (However, you do pick up some extra terms that bear resemblance to the chain rule if the limits of integration depend on t)
  4. Sep 8, 2003 #3
    Thanks Hurkyl,

    The limits are definite ( plus/minus infinity and psi(x,t) goes to zero as x goes to infinity). If x can't be a function of t, then the chain rule says x*dpsi/dt + dx/dt * psi and the second term is zero. Surely the book would have said so, so I must have messed something up in the presentation.

    Can you confirm that if f(x,t) is separable into h(x)g(t) then it makes no sense for x to depend on t? It would not be separable in fact if x were a function of t?

    Sorry to be dense but if everything I've written is correct the author has made a mountain out of a molehill.
  5. Sep 9, 2003 #4


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    If your integral is F(t)= [int](from a to b) f(x,t)dx then
    dF/dt= [int](from a to b) df/dt dx where "df/dt" is the partial derivative. (In your "separable" example, you don't need partial derivative.)

    More generally "LaGrange's Formula" is:
    If F(t)= int(from a(t) to b(t)) f(x,t)dx the
    dF/dt= int(from a(t) to b(t))df/dtdx+(db/dt)f(t,b(t))-(da/dt)f(t,a(t))

    Notice that, here, the limits of integration are also functions of t.
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