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Differentiation and integration

  1. Mar 31, 2016 #1
    1. The problem statement, all variables and given/known data
    Hi,I saw a statement in my physics notes like this(Anyway it is a maths problem):

    diff.png

    where L is a general differential operator.G is a green's function(I guess it is irrelevant)
    My question is related to the red line:
    Suppose we have this:
    ∂/∂x ∫ f(x-y)g(y) ∂y
    is it generally true that
    ∂/∂x ∫ f(x-y)g(y) ∂y = ∫ ∂/∂x [ f(x-y)g(y) ] ∂y ?
    2. Relevant equations
    Please answer it as simply as you can...Since I have not done multivariable calculus...(Though I would be happy to check it out if there is a theorem related to this.)

    3. The attempt at a solution
    If the above is simplified to ∂/∂x ∫ f(x)g(y) ∂y ,then ∂/∂x [f(x) ∫ g(y) ∂y]
    ⇒ ∫ ∂/∂x [ f(x) ] g(y) ∂y
    ⇒ ∫ ∂/∂x [ f(x)g(y) ] ∂y

    But I dont know what to do with the (x-y) term..

    Thanks
     
  2. jcsd
  3. Mar 31, 2016 #2

    ShayanJ

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    Gold Member

    The general formula is:
    deaf6cf147e2c72ca92f02fbe5bb17ef.png

    But you don't need this. You only need to use the properties of Dirac delta, i.e. ## \int \delta(x-y) f(y) dy=f(x)##.
     
  4. Mar 31, 2016 #3
    Thanks :) Is there a name for this formula?
     
  5. Mar 31, 2016 #4

    ShayanJ

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    Gold Member

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