I'm jumping ahead, because I'm too lazy to recount the explanation of why the definition of vector fields given last time actually deserves the name. Also, I apologize in advance for the incoherence of this post. That's the problem with writing about things I know damn well I don't understand -- I end up chasing my tail as if I was a dog and someone tied a piece of crispy bacon to it. Mmm.... Bacon.... Ahem. So, err, onward!

Say we've got two manifolds, and we call them M and N. Now, let's define a function f: N -> R. That means a function which eats points from N, and produces normal numbers (hence the 'R' part - that just refers to the 'real number line'). Also, let's define a function phi: M -> N. This is a function that takes points on M, and produces points on N.

Now, we can use the above information to get a real-valued function on M. We take a point on M, and feed it to phi. That gives us a point on N, which we can feed to f, which gives us a real number. Hurray. Summarizing, we just did 'f @ phi' (where the @ between f and phi signifies composition of functions, so it's read right to left). Now, we can define something called 'pulling back by phi', because we're pulling f back over from N to M using phi, and we can call phi_* a pullback (phi_* being defined using phi_* f = f @ phi).

Now, notice that something perverse is going on here: phi_* has to be on the 'other side' of f than phi. Because of this, real-valued functions on manifolds (which is what f was, remember) are called 'contravariant'.

"But wait!", you may be thinking. Could it be that we're stuck with this perverse backwardness simply because we *asked* for it? We defined f on N, and we've got a function phi that goes from M *to* N. So *of course* we had to go 'backwards' to get f to work on M! This is the line of thought I was preoccupied with while meditating upon a family-sized package of TP in the Chamber of Reflection.

To resolve this existential crisis, let's define f: M -> R, and now try to get a real valued function on N. If our concern above is justified, this shouldn't exhibit any 'backwardsness' (mind you, we're keeping phi defined just as it was before, from M -> N). So, off we go.

We need to somehow persuade f to eat points of N. It, at the moment, only eats points of M. Now, to go from M -> N, phi is just what we need. But we, unfortunately, need to go the *other* way - we need points of N disguised as M. To get that, we need the inverse of phi, that is, phi^-1. Uh-oh. We're getting backwardsy again. But, now we're in business, because f @ phi^-1 will do what we want: N -> R. So how the bloody hell do we define a pullback now? phi^* f = f @ phi^-1 ? I dunno, but probably something like (phi^-1)^* f = f @ phi^-1 is better.

Heh, well, if I haven't done anything stupid in the above, I just demonstrated that we still get the backwardness in this case. Hmm. Good. Moving on, then...

Not all things in life are contravariant, according to Baez and Munian. Tangent vectors, for one. Say we again have two manifolds, M and N, and phi: M -> N. Now, for a point p in M, we've got a tangent vector v [- T_pM (The [- being the 'element of' symbol). Our aim is to get transfer this tangent vector to N. That is, to get phi_* v [- T_phi(p)N. This ends up being called a pushforward.

At this point, I must confess not actually understanding pushforwards. I don't understand the bloody definition of a pushforward, quite. I can quote it, and even use it, to a degree, but I don't *grok* it. I don't understand why, exactly, aside from "it's nice that it be so", (phi @ y)'(t) = phi_* y'(t). (y(t) being a curve in M, that is, y(t): R -> M, and y'(t) being a tangent vector to that curve, and so to points the curve goes through). I'm thoroughly confused, and it's all the more frustrating because I think I've the 'gist' of things at this point, I don't feel I really understand them. Hopefully some more stewing and marinading will cure the confusion.

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