Is it offensive to ask my PhD advisor for permission to record our research/technical discussions? I will be joining a graduate program in the US this fall. To keep track of important technical/research discussions with my advisor, I think it would be a good idea to record conversations with my advisor on a voice recorder/camera. It would be very helpful as I (and my advisor) will not forget/miss anything important. Also, I could listen to my advisor's comments again if I do not understand something in the first go. So I would like to know (from an advisor's perspective) if it is bad etiquette to request my advisor to have our research discussions recorded on my voice recorder/camera. I do not wish to breach anyone's privacy but only wish to keep track of my own research. And I only want to record if my advisor is happy with it.

Update: As mentioned in one of the answers below, sometimes it is hard to keep up with the advisor as my field is highly technical and mathematical (theoretical physics). It has happened to me before that some things my earlier project advisor discussed went completely over my head. I found it hard to even jot down such things since I did not understand what he was trying to say (like lots of mathematics jargon). An example:  If I am just starting as a PhD student in US and I see my prospective advisor (and his collaborators) discussing about something I feel to be very interesting, I do not think I will be able to pick up all the "deep and subtle ideas" that come up during such discussions (being only a beginner). And if I do not pickup what is deep and what is not, I might miss those important ideas and consequently not jot down. It is very important for me to understand those ideas. I do not think my advisor will be able to clear doubts in such situations as I would not be familiar with such advanced stuff at the time of those discussions. This is why I think it is very helpful in such situations to have a record so that I could go back to it later on when I understand all that jargon. It would save a lot of our time as well if my advisor does not have to repeat things multiple times.
You need to ask first, and you need to ask without the recording device in your hand.  

But sure, why not ask?  If I were asked, I would find it a little strange at first, and then I would think about the fact that this shows a real commitment and diligence to keeping track of what is said during our meetings.  One of the biggest issues with student/advisor meetings is that it feels like they should happen very informally, but information is often being provided in a way which the student cannot easily or perfectly understand or even take in.  So there is often a lot of repetition over a period of weeks, months or years.  (Not just between students and advisors, come to think of it.)  With that in mind I would be inclined to say yes.

I also strongly disagree with another answer which suggests that an advisor might drop a student for making such a request.  This seems to contribute to a sentiment that students should be afraid to discuss things with their advisors, which I see so much of on this site and think is very unfortunate.  It would be incredibly unreasonable behavior to drop a student for making this request or really any request which has anything to do with the student's academic or professional life.  

Added: My response is not meant to apply to interactions with other people aside from the OP and the advisor.