Yes, you can. However, I would suggest using a paper pad for notes initially and then later retyping them or photocopying them into your device. Basically use your device like a book and use the traditional paper notebook for notes. Taking notes from things you've read mean flipping back and forth on the screen or using a split screen with reduced font sizes which can get tiring. Also you should use airplane mode to eliminate all distractions or at least turn off certain apps that text for alert on email...
You will also need to backup your device periodically and perhaps even email your notes to yourself to preserve them in the cloud in case your device gets hacked or stolen or destroyed.
While I've not used it in a classroom setting, some apps come with built-in microphone support to record lectures. One such app, is Notability on the iPad. You can take notes while its recording the lecture and your notes are tied to the recording ie is when reviewing your notes later on tapping a particular note will cause the app to playback from that point in the lecture. Notability can also display pdf files and import and resize images for your notes.
For ebooks, there's GoodReader on the iPad which allows you to annotate and organize your ebooks.
For math oriented work, there's Pocket CAS which can generate 2D and 3D graphs and handle varios calculations. Its not MATLAB but it can do a lot of cool things.
Apple also provides a Office suite of Pages (word processor), Numbers (spreadsheet) and a point like tool. MS also provides its suite for iPad. For me though I tend to use Textastic (programmer editor) or Editorial for markdown support which is faster than using a word processor and all the bells and whistles to make your document look good.
For programming, there's Textastic for viewing and editing code and Pythonista for programming python directly on the device and Codea for programming in Lua on the device. Apple doesn't support Java or C/C++ programing on device though you can use Textastic to edit and view source code.
For fancy drawing, there's Procreate, a multi-layered drawing tool with many bells and whistles to create stunning graphics if you know how to use the features.
Anyway, it would be a great adventure to try all this out in a classroom setting and see how efficient it could be. However, remember that your best strtegy is to attend all lectures no matter how boring, record and take minimal notes so as to focus on what the instructor is saying use a separate phone to take blackboard shots in case you need it for your notes.Realize that any device fidgeting will distract you from learning and that not what you want in the long run.
When I went to school, things were much simpler, paper notebook, don't write too much and listen well. I tried taping a lecture once but never replayed it as I got too busy. I didn't rewrite my notes to make them more complete and to find what I'd missed. What worked for me was simply listening to the instructor.and taking down what I could. One one Theoretical Mechanics class,
I remember the instructor writing some equations on the board and he'd constantly erase and correct what he wrote making my notes a mess. In addition, I misread his greek letters confusing ##\psi## and ##\zeta## in his stress/strain tensor equations. it was only after a week or so after browsing a book by Landau that I realized my confusion.
Erasing and rewriting equations or erasing whole blackboards to write more stuff can mke notetaking less efficient as you try to capture the instructors every word which means you should be instead listen take a few notes and review and rewrite what you've written. Who knows someday those notes will be a portal to a book or to you teaching that same course, erasing boards, correcting things, enjoying your captive audience all in an attempt to foil student notetakers to get them to listen more and learn a lot.
Lastly, devices can affect your sleep cycle:
http://www.webmd.com/sleep-disorders/news/20141222/tablets-and-e-readers-may-disrupt-your-sleep#1
meaning you should turn them off at least a half hour before sleeping and get a GOOD nights rest don't party like its1999.