# Differentiation and integration

1. Mar 31, 2016

### davon806

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):

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. Mar 31, 2016

### ShayanJ

The general formula is:

But you don't need this. You only need to use the properties of Dirac delta, i.e. $\int \delta(x-y) f(y) dy=f(x)$.

3. Mar 31, 2016

### davon806

Thanks :) Is there a name for this formula?

4. Mar 31, 2016