Revisiting the Product Rule

In differential calculus, the product rule is both simple in form and high in utility. As such, it is typically presented early on in calculus courses, and the proof given is almost always the same.

In this post, we’ll explore the merits of a second proof of the product rule using properties of the logarithm, one that I hope presents a motivated and compelling argument as to why the product rule should look the way it does.

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Formal Groups and Where to Find Them

In 1946, S. Bochner published the paper Formal Lie Groups, in which he noted that several classical theorems (due to Sophus Lie) concerning infinitesimal transformations on Lie groups continue to hold when the (convergent) power series locally representing the group law was replaced by a suitable formal analogue. It was not long before this formalism found far-reaching uses in algebraic number theory and algebraic topology.

Unfortunately, few students see more than two or three explicit (i.e. closed form) group laws before stumbling into the deep end of abstract nonsense. In this article, we’ll see in a rigorous sense why this must be the case, providing along the way a complete classification of polynomial and rational formal group laws (over any reduced ring).

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