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/∂xdx/dt + ∂F/∂ydy/dt
- An exact equation is a differential equation that, for some F(x,y), can be written as:
∂F/∂x dx + ∂F/∂y dy = 0Equivalently, P dx + Q dy = 0 is exact if the vector (P,Q) is the grad (gradient) of some function F.
- 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