to use wireless to display video the receiver have to support airplay your projector or pc would not support that
you can use the hdmi cable for the projector if it support hdmi
few pc's have hdmi input so not likely that you can use that
doubt you can use the ipad to record video and audio at the
same time as it display something else on the projector if it support hdmi that is
if you are recording with the camera and mic while using the hdmi cable mirror on the projector
it would display what the camera is recording and indicating that it was recording
not good if you wish to record your student, foreign diplomats or other agents without their knowledge
The only way, I believe, to connect an iPad wirelessly to a projector is by connecting an AppleTV to the projector - and I don't think most projectors have the correct input for it.
You can use a number of apps to control the computer through the ipad. There are a few apps that are designed to replicate an interactive whiteboard. One I'm trying is Splashtop Whiteboard - it allows you to use the iPad to annotate over an image on the computer and then project it, or to even use their own flipboard as a background to writing. This would, in theory, allow you to do your work directly on the iPad and send it to the projector through the computer. I say in theory because we are having problems with it, both the app itself and then running it through a whiteboard projector - I've been working with a tech support rep at Splashtop to see if we can resolve it before I can decide whether to recommend to other teachers, as right now I can't - but other people apparently are using it succesfully.
Another app that has similar promise (and higher reviews!) is Doceri Remote. The app is free, but the desktop software is not - $50 (the splashtop app is $20), plus they strongly recommend their proprietary stylus. Because they sell the computer software and not the app, however, you can do a trial - 5 days, I believe, we're going to give it a try after winter break. You can also, I believe, record your presentation using Doceri.
For recording ... while Doceri promises it, it might be easier to use a Screencast software (such as Jing) to record it on the computer - if you're using one of these apps to control the computer and project out, anything on the computer will be recorded by Jing and then you can share it later.
I don't really want to control the computer from my ipad. I'd just basically use the computer as a remote desktop or mirror since whatever is on the computer gets projected. Using Jing on the PC wouldn't work since unfortunately the computer is off on one end of a shoe-box shaped classroom and I will be 15' or so away from it generally. It wouldn't pick up the audio.
Screenchomp does what I want as far as capturing the audio and video on the ipad. I just want to be able to project what's on my ipad.
Thanks for the info on Doceri. Although it is not exactly what I'm looking for, I'm glad to know about it.
Hmm...I can connect my macbook to the projector via wired connection, so I could eliminate the PC piece but still would need a wireless connection from the ipad to my macbook.