So this is beginning to feel like the beginning of the 4th movement of Beethoven's Ninth: it is all coming together. Manifolds,Lie Algebra, Lie Groups and Exterior Algebra. And now I have another simple question that is more linguistic in nature. What does one mean by "Calculus on Manifolds"? (Yes,I have seen other posts on this topic in this forum, but please allow me to state it in my terms.) I now feel I was severely mis-educated as an engineer. And perhaps I do not really appreciate calculus. Thus... Can you give me an physical (but engineering) example of calculus being done on manifolds? Why do I want to do calculus on a manifold? What does DOING calculus mean? (FORGIVE ME for this one, but... you have a function of variables, you take derivatives... what's the big deal? what is the DOING part of the calculus on manifolds? What are examples of it? What can I do in R3 that I CANNOT do in.... another place... unless that place is like a manifold? Assuming that a Lie group is a continuous manifold, how does one DO calculus on a Lie Group? I assume one uses Lie Algebraic structures. How is that related... IN WORDS... not equations... to the Lie Derivative. And regarding differential forms... it now seems that topic matters to me ONLY if I want to understand the distinction between vectors and covectors... but if my TARGET is dynamics, then I need not spend so much time with differential forms and exterior algebra as much as I should spend with manifolds, Lie groups and Lie algebras. yes?