S*****n
发帖数: 16 | 1 How to do the following partial derivative of indefinite integral?
Given/Define:
Q= int {g(x)*f(x)dx}
How to get:
partial derivative of Q with respect to g(x)?
Thanks! | e**********n
发帖数: 359 | 2 This is not differetiation but variation instead. Formally, you can assume a
small local change in g(x), i.e. g(x)-> g(x) + c\delta(x-x'), where \delta(
x-x') is the Dirac-delta function. Calculate the integral Q(c) and take the
limit Q(c)/c with c->0. What you get is f(x') in the end. There is rigorous
mathematics (distribution theory and generalized function) behind this, but
the above reasoning is sufficient for physics. | S*****n
发帖数: 16 | 3 Thanks eventhorizon!
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the
rigorous
but
【在 e**********n 的大作中提到】
: This is not differetiation but variation instead. Formally, you can assume a
: small local change in g(x), i.e. g(x)-> g(x) + c\delta(x-x'), where \delta(
: x-x') is the Dirac-delta function. Calculate the integral Q(c) and take the
: limit Q(c)/c with c->0. What you get is f(x') in the end. There is rigorous
: mathematics (distribution theory and generalized function) behind this, but
: the above reasoning is sufficient for physics.
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