N!kofeyn, my apologies for "running off" David Bachman. I think that is inaccurate, but i do recall he asked for corrections on his book and when I took him at his word and pointed out some mathematical errors I had noted, he got angry. I think he stuck around longer than I did however, so you might say he ran me off. I could be wrong.
Anyway I think his book is excellent, the best place I know of to get a real feel for the geometric meaning of differential forms (as measures of oriented volume of "blocks"), and I wish I had refrained from pointing out what were in reality very tiny and subtle errors that would not hinder any student from benefiting from his book. Such errors exist in almost all books, even some of the best and most useful. Lang's famous algebra book abounds with them, as do many other famous and helpful books.
Lesson learned: when people ask for criticism, they usually do not mean it, they really want praise. (Me too.) One of my faults is focusing on the few negative aspects of a situation instead of the many positive ones. This sort of perfectionism makes it hard to actually produce any creative work, and should be avoided as much as possible. Or at least avoided until the last step. After producing a creative work it seems useful to me to go back over it and correct the errors. But when pointing these out in the works of others it helps to be very diplomatic. Producing a creative work takes a lot of effort and displaying it to others afterwards also takes courage, and we should be grateful to these people for helping us learn from them.
Many people including myself, write books which even if they are correct and free from serious false statements, still may have limited usefulness because they are not illuminated by any deep understanding of the subject we are writing about. It is probably thus better to read books by people who really know something worth learning from them even if their treatment contains errors. My notes on the Riemann Roch theorem for example were written before I understood the topic. Still in trying to write up the subject I eventually came to feel I understood it. The main breakthrough was reading Riemann when a translation into English became available.
An outstanding theoretical book on differential forms at least for math students (like myself) is the one by Henri Cartan, all of whose writings are to me a model of perfection, i.e. clear, correct, and succinct. This one is also available in a cheap paperback.
Oh, and Loring Tu is especially famous as a writer whose works are models of clarity. It helps of course to know his clearly stated prerequisites. Still I would suggest one can always learn some thing from Loring.
If you want to share my attempt at explaining something I did not understand at the time fully, there is a section in my free (you get what you pay for) web page algebra notes (math 845-3) near the end, that treats "exterior algebras", i.e. the algebraic aspect of differential forms.
http://www.math.uga.edu/~roy/ (that young kid there was apparently me a longgg time ago.)
