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Elements of differential geometry George D. Parker, Richard S. Millman
Publisher: Prentice Hall

I've looked at a variety of SDG papers but none of them seem to make the connection with AD. It is only later on, when calculus became more algebraic in outlook that one can begin to make a meaningful separation between the subjects of calculus and differential geometry. Before moving on to the next section, it might be worth pointing out that, when speaking of manifolds, it is customary to refer to the elements of the set as points and the real numbers which label them as points. The big secret about DEC is that it's literally nothing more than the good-old fashioned (continuous) exterior calculus we've been learning about, except that we integrate differential forms over elements of our mesh. Early differential geometers studied such .. Elements of differential geometry. Elements of artificial neural networks. My own formulation of the technique is more algebraic than the description that is usually given, and recently it's begun to dawn on me that all I've done is rediscover Synthetic Differential Geometry (SDG). For the second derivtive you need an element such that d^3=0 but d^2=0. Such an element can be found in R⊗R, as you've discovered. Libigl is a simple c++ geometry processing library.