16.2

1. Pardon my ignorance, but how can we take a derivative of a function (implicit or not) when there is no (obvious) notion of continuity. Don't we use difference quotients or something? Is that the same thing in this case?

2. I am consistently amazed at the connections and applications that seemingly obscure branches of math have. First, the recognition that many of the results of, say, topology come not from a construct itself (say, the complex numbers), but from the properties that construct has (field properties). It then allows one to interpret things like fractions in finite fields where all members are representable by integers. More pertinently, the investigation of arc length of ellipses led to a more intense study of elliptic curves which turn out to be extremely fascinating. I must say apologetically that I have nothing of interest to say about elliptic curves over Z mod n. This is all new to me.