What is considered Calculus on Manifolds?

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You can do your derivative there. Integration is a rather different beast.

There's been a lot of work in the last 15 or so years on how best to do numerical integration on a manifold, Lie groups in particular.

Naively use the numerical integration that work so nicely on R^n on on a Lie group as if it were R^n and you'll quickly run into trouble. The very first step takes you off the manifold, and subsequent steps make things that much worse. For a long time, the standard kludge was to pull each step back on to the manifold.

This is a kludge, and it is essentially invalid. That it happens to work (kinda, sorta) is a feature of the fact that locally a manifold looks like R^n.

Much better is to use some technique that keeps the integration on the manifold. There has been a lot of work on such techniques as of late, at least for Lie groups. All of them use the exponential map. Some also use the commutator, some use this nasty thing called [itex]d\exp^{-1}[/itex], or dexpinv.