Spring break is coming up and I plan on taking that week off to go traveling in Europe. Do I ask my supervisor/advisor for permission or do I just inform them that I am doing so?
Do I do it via email or face to face?
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Your contract should include some official vacation time. Doesn't it?
So, it should be your right to take those days off, and your supervisor shouldn't be able to say no, unless it interferes with your work, such as lectures, lab schedules, ...
But it is still nice to keep it cool and ask politely, better in person.
Preferences on this issue vary widely, even within departments. Since it sounds like this is the first time it has come up for you, the simplest thing is to ask your advisor about the preferred approach. Something like "I'm hoping to go away over spring break, I just wondered what the procedure is and whether I need to formally book the time off?" If you feel awkward about asking your supervisor directly (though I don't see why you should), you could try asking others within your research group what they normally do.
The only situation in which I would think you would need to ask permission (as opposed to just let them know) is if you are doing experimental work that is time-dependent and would need to be covered in your absence. But if that was the case I imagine you would have considered those issues when you started the experiments.
Unless your supervisor is very hard to get hold of, I should do it in person.