If I have a function(adsbygoogle = window.adsbygoogle || []).push({});

##f(u,u^*) = \int u^* \hat{O} u d^3\mathbf{r}##

both ##u## and ##u^*## are functions of ##\mathbf{r}## where ##\mathbf{r}## position vector, ##\hat{O}## some operation which involves ##\mathbf{r}## (e.g. differentiation), and the star sign denotes complex conjugate. Now I want to find the differential expression for ##f##, namely ##df = \frac{\partial f}{\partial u}du + \frac{\partial f}{\partial u^*}du^*##. I think ##u## and ##u^*## are independent, right? When I have to calculate the partial derivative of ##f## w.r.t. ##u^*##, it seems that I simply need to bring the differentiation inside the integral and differentiate only ##u^*## which gives me

##\int \hat{O} u d^3\mathbf{r}##,

is this also justified? If yes how should I calculate ##\frac{\partial f}{\partial u}##?

Any help is appreciated.

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# Partial differentiation of integral

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