Sups and Sandwiches

[pteromys]

The phrase of the night: "Analysis is about sups and sandwiches."

This post isn't about analysis, though. It's about math words. Today I want to draw your attention to the words "algebra", "field", and "exact".

First, "algebra". In the most general sense an algebra just refers to a set with some operations. Here are two specific cases that could otherwise seem pretty unrelated:

1. An algebra over a ring is something like a vector space with a multiplication law and with scalars taken from the ring.
2. An algebra over a set is a collection of subsets that is closed under complements, finite intersections, and finite unions.

The confusing part is how these connect to the word "field":

1. If an algebra over a ring is a field, its multiplication is commutative, and it also has division.
2. If an algebra over a set is a Borel field, it has countable unions and arises from some topology on the underlying set.

Meanwhile, a field of sets seems to be merely another way to refer to an algebra over a set, and the notion of a vector field appears completely unrelated. Then again it seems that all of these usages of "field" are mostly unrelated to each other.

Moving on, I did manage to relate usages of "exact"... and I wonder how much was intentional and how much was a fortunate accident:

1. The exact derivative of a function F(x(t),y(t)) is written as:
   ^dF/_dt = ^∂F/_∂x^dx/_dt + ^∂F/_∂y^dy/_dt
2. An exact equation is a differential equation that, for some F(x,y), can be written as:
   ^∂F/_∂x dx + ^∂F/_∂y dy = 0
   Equivalently, P dx + Q dy = 0 is exact if the vector (P,Q) is the grad (gradient) of some function F.
3. An exact differential form is something vaguely like a vector or scalar field that is the grad, curl, or div of something.
4. Finally, recall the following identities from multivariable calculus:
   • F = grad G implies curl F = 0
   • F = curl G implies div F = 0
   In spaces where the converses hold, the grad, curl, and div operations give you what is called an exact sequence.