Hello I am a mechanical engineer who is teaching himself the math of exterior algebra and differential forms. It is not easy for me and I have had many SIMPLE stumbling blocks due to my not respecting algebra. May I ask for help on some simple aspects? (Please be patient with me.) My question is less a mathematical question and more one of semantics (perhaps stemming from ignorance). (As a caveat: I am not entirely sure I am even asking what I want to ask properly... so there might be followups.) In some places I have read: one PULLS BACK a FORM and PUSHES FORWARD a VECTOR. And the authors make an issue of this. In other places I have read one can pull back both and push forward both. First: what gives? Can I assume the SECOND case is ONLY possible when the mappings themselves are readily reversible? Could it be that? Why then do most authors make an issue of this? What is the historical reason? Also, many of the mappings are from M to N to R. And they... I really do not know how to say this without really revealing my ignorance, so here goes.... they go from left to right or in one direction. Is THAT the ORIGIN of the phrase PULL BACK or push FORWARD Now, if is true, WHY do books and theories focus MORE on the pull back? If one is pulling something BACK, then one must assume you are in the first space and yanking it back to you. If you are pushing it forward, you are in that same space and pushing it to a place where you are not. What is it about the space were you are that you are not happy with things THERE and things HERE: God I know I must sound like an idiot, but WTF. And now comes the odder questions… What is the big deal? HOW does pulling back FORMS matter to an engineer? How does pushing forward vectors matter? Yes, I understand FORMS are dual to Vectors. But what IS it about forms and what IS it about vectors that we pull one BACK and push the other FORWARD. What are we pulling them back TO? How does pulling them back help me? Could you make a comment in SIMPLE language? Also, I am getting the feeling that this thing I have come to know as the Material derivative in fluid mechanics, is intimately connected to this idea of a pullback or pushforward, but I cannot see it through the fog right now. Could you comment on that? Note that I am putting in hours to teach myself topology, manifolds, etc and learning a lot about how my educatino failed me. But for now, I really could use WORD responses that get to the core of the issue, qualitatively.