You use it like a conceptual dictionary. Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with (e.g. a submanifold). You’d look up “submanifold” and see that it is “ subset of a manifold that is itself a manifold, but has smaller dimension.” Okay, that seems to fit the intuition of a “sub”-something. But I don’t know what a “manifold” is. So I’d look that up.
“A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in R^n)”
At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done. If not, you either keep digging deeper into the various terms or you start seriously considering reading one or more of the curated reference books listed at the end of each entry.
Over time you develop the “mathematical maturity” that you don’t need to do a deep dive into the books and can mostly just use the reference.
“A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in R^n)”
At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done. If not, you either keep digging deeper into the various terms or you start seriously considering reading one or more of the curated reference books listed at the end of each entry.
Over time you develop the “mathematical maturity” that you don’t need to do a deep dive into the books and can mostly just use the reference.