Derivatives of a volume integral with singular kernel

TO Quy Dong — Sun, 30 Sep 2007 04:49:43 -0400

Hi everybody,

In fact, I encounter this problem in my research and I would be grateful if someone can help. In micro mechanics, there are many problems concerning Green functions, e.g: the displacement is calculated from the distributed force in the domain, etc. Consider the following integral to determine the displacement field.

u(x)=∫A(x,y)dVy where A(x,y) is singular of order r-2 (i.e r2=(x-y)(x-y)).

Now I want to take the derivative of u to derive the strain ε, how can I introduce the derivative after the integral sign?

ε(x)=d/dx∫A(x,y)dVy =∫d/dx A(x,y)dVy + ???.

I have tried to calculate the above derivative considering the derivative of the principal value of A(x,y), but I'm not sure it is correct because I have no references on this.

Thank you

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Derivatives of a volume integral with singular kernel