I also remember taking a class on vector calculus from the same author... which detoured through rudimentary manifold theory and differential forms, and ended with a final week on de Rham cohomology and the Mayer-Vietoris theorem (on vector spaces, to be fair, and not modules in general.)
(And is a very fine K-theorist, too, if I say so myself.)
> I also remember taking a class on vector calculus from the same author... which detoured through rudimentary manifold theory and differential forms, and ended with a final week on de Rham cohomology and the Mayer-Vietoris theorem (on vector spaces, to be fair, and not modules in general.)
Any available references for that that you know of?
My bad – that was a misleading thing to say! Thanks for pointing that out. I figured out what I said wrong. (Caveat emptor, I do biostatistics now.)
The context IIRC was this: one of the key results of the class was generalized Stokes' theorem, but this case (since was a 200-level class) we mostly just looked at differential forms on open spaces in R^n, and then said a few quick things about differentiable manifolds.
At this more concrete level, then, I remember that we constructed de Rham cohomology (fixing an open subset of R^n) beginning with the cochain complex given by vector spaces of k-differential forms and exterior derivatives, instead of working more generally with a cochain complex on modules.
But think I said something wrong here, which I why you were (rightly) confused. I'm not sure that the above distinction matters anyway since IIRC, you can get Mayer-Vietoris by showing that de Rham cohomology satisfies the Eilenberg-Steenrod axioms (stated for cohomology), and the Eilenberg-Steenrod axioms only need abelian groups anyway.
But I'm also 90% sure that TFA did something more direct to get to Mayer-Vietoris that I've forgotten, since we didn't use that much homological algebra.
The first author is well known for teaching "wild ride" undergraduate classes where he compensates by spending a lot of time on their pedagogy.
He once taught an open to all freshman knot theory elective:
https://people.reed.edu/~ormsbyk/138/
I also remember taking a class on vector calculus from the same author... which detoured through rudimentary manifold theory and differential forms, and ended with a final week on de Rham cohomology and the Mayer-Vietoris theorem (on vector spaces, to be fair, and not modules in general.)
(And is a very fine K-theorist, too, if I say so myself.)
> I also remember taking a class on vector calculus from the same author... which detoured through rudimentary manifold theory and differential forms, and ended with a final week on de Rham cohomology and the Mayer-Vietoris theorem (on vector spaces, to be fair, and not modules in general.)
Any available references for that that you know of?
Wait, I know Mayer–Vietoris as a tool for computing homology. What does it mean to compute it on vector spaces or on modules?
My bad – that was a misleading thing to say! Thanks for pointing that out. I figured out what I said wrong. (Caveat emptor, I do biostatistics now.)
The context IIRC was this: one of the key results of the class was generalized Stokes' theorem, but this case (since was a 200-level class) we mostly just looked at differential forms on open spaces in R^n, and then said a few quick things about differentiable manifolds.
At this more concrete level, then, I remember that we constructed de Rham cohomology (fixing an open subset of R^n) beginning with the cochain complex given by vector spaces of k-differential forms and exterior derivatives, instead of working more generally with a cochain complex on modules.
But think I said something wrong here, which I why you were (rightly) confused. I'm not sure that the above distinction matters anyway since IIRC, you can get Mayer-Vietoris by showing that de Rham cohomology satisfies the Eilenberg-Steenrod axioms (stated for cohomology), and the Eilenberg-Steenrod axioms only need abelian groups anyway.
But I'm also 90% sure that TFA did something more direct to get to Mayer-Vietoris that I've forgotten, since we didn't use that much homological algebra.
So, just from the contents ... does anything make this especially different from other discrete math books?