Currently, I'm applying for PhD positions at several universities in pure mathematics and computer science. Despite the fact that these two fields are content-wise different and do not have too much in common, I'm open to do a PhD in both of them. However, if I ask a professor to write a letter of recommendation for each them, I'm not sure if the professor would negatively understand it. He may think that I should choose only one of these fields for PhD positions.
My question are the following and may be based on common (best) practices/experiences:
Is it recommendable/valid to apply PhD positions in different fields at different universities?
Does a professor perceive it as inappropriate/negative when he is
asked to write recommendation letters for two different fields?
Furher recommendations regarding this issues are very welcome.