### Sups and Sandwiches

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:

- An
*algebra over a ring*is something like a vector space with a multiplication law and with scalars taken from the ring. - 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":

- If an algebra over a ring is a
*field*, its multiplication is commutative, and it also has division. - 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:

- The
*exact derivative*of a function F(x(t),y(t)) is written as:^{dF}/_{dt}=^{∂F}/_{∂x}^{dx}/_{dt}+^{∂F}/_{∂y}^{dy}/_{dt} - An
*exact equation*is a differential equation that, for some F(x,y), can be written as:Equivalently, P dx + Q dy = 0 is exact if the vector (P,Q) is the grad (gradient) of some function F.^{∂F}/_{∂x}dx +^{∂F}/_{∂y}dy = 0 - An
*exact differential form*is something vaguely like a vector or scalar field that is the grad, curl, or div of something. - Finally, recall the following identities from multivariable calculus:
- F = grad G implies curl F = 0
- F = curl G implies div F = 0

*exact sequence*.