Wikipedia:Reference desk/Archives/Mathematics/2012 February 2

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February 2

good introduction to abstract algebra from very basics

Hello, I am looking for a good introduction to abstract algebra (group theory, linear algebra, etc.) that builds from the very basics and does not assume a prerequisite of set theory. I have no real requirements on rigour other than that it not be so rigorous as to be unreadable in a reasonable amount of time as well as not so unrigorous that it is not useful. My main concern is that I cannot get bogged down in proofs because I have to preferably finish it within a month or two and I don't really have large stretches of time to devote to reading about math. Does anyone have any recommendations? Thanks. 24.92.85.35 (talk) 04:51, 2 February 2012 (UTC) PS: I have some experience in real analysis for what its worth.

Just a personal preference here, but Fraleigh's 'First Course in Abstract Algebra' gives a gentle introduction to group theory, and goes quite far. It doesn't give any treatment of linear algebra, however. It does have exercises, with some solutions, though. Icthyos (talk) 11:38, 3 February 2012 (UTC)

Herstein has a little book for absolute beginners entitled "Abstract algebra" that's worth a read, but doesn't cover linear algebra. A slightly more comprehensive (but also more advanced) book is Maclane and Birkhoff, which is very well-written and includes quite a bit of linear algebra. Sławomir Biały (talk) 13:25, 3 February 2012 (UTC)

I don't know what you are using it for, but if "have to finish it in a month or two" means for background for a PhD, listen to the people above. If it is for undergrad exams (maybe as a supplement anyway) I always use the Schaum's guides when I want to strengthen my mathematical problem solving, rather than know my way around a topic in detail. The Schaum's guide to Abstract Algebra is pretty good, but doesn't have heaps of material on groups etc. (two chapters on groups, one on rings, one on division rings/fields/integral domains combined), nor much linear algebra (one chapter on vector spaces, one on matrices, does cover linear equations and determinants, though). Heaps of solved problems rather than gobs of proofs. I guess it depends on whether you want to turn yourself into a mathematician, or do better in an exam. IBE (talk) 16:11, 4 February 2012 (UTC)

Regular or partial (definite integral) notation (i.e., “d” or “∂”)?

Where $C'(\mathrm {A} ,\sigma )={\sqrt {\cos ^{2}(o\!\varepsilon )+(\cos(\mathrm {A} )\sin(\sigma )\sin(o\!\varepsilon ))^{2}}}^{\color {white}|}\,\!$, if

${\begin{aligned}\int _{\mathrm {A} _{s}2}^{\mathrm {A} _{f}2}\int _{\sigma _{s}}^{\sigma _{f}}&{\color {white}=}{\sqrt {\cos ^{2}(o\!\varepsilon )+(\cos(\mathrm {A} )\sin(\sigma )\sin(o\!\varepsilon ))^{2}}}\mathrm {d} \sigma \mathrm {d} \mathrm {A} ,\\&=\int _{\mathrm {A} _{s}2}^{\mathrm {A} _{f}2}\int _{\sigma _{s}}^{\sigma _{f}}C'(\mathrm {A} ,\sigma )\mathrm {d} \sigma \mathrm {d} \mathrm {A} ,\end{aligned}}\,\!$,

does/should the more basic

${\begin{aligned}\int _{\sigma _{s}}^{\sigma _{f}}&{\color {white}=}{\sqrt {\cos ^{2}(o\!\varepsilon )+(\cos(\mathrm {A} )\sin(\sigma )\sin(o\!\varepsilon ))^{2}}}\mathrm {d} \sigma ,\\&=\int _{\sigma _{s}}^{\sigma _{f}}C'(\mathrm {A} ,\sigma )\mathrm {d} \sigma \quad {\mbox{or}}\quad \int _{\sigma _{s}}^{\sigma _{f}}C'(\mathrm {A} ,\sigma )\partial \sigma \;?{\color {white}.}\end{aligned}}\,\!$

 ~Kaimbridge~ (talk) 20:22, 2 February 2012 (UTC)

I don't think the distinction makes sense. The variable you integrate with respect to has to be a dummy variable, so A can't depend on sigma. That means the distinction between partial and total differentiation (ie. whether or not you take the indirect dependence into account) doesn't apply. I've only ever seen a straight d used. --Tango (talk) 01:33, 5 February 2012 (UTC)

That certainly sounds reasonable.

In this case, $\theta =\arcsin(\cos(\mathrm {A} )\sin(\sigma ))\,\!$ ("latitude"), where $\mathrm {A} \,\!$ is the azimuth at the equator, or arcpath (a "great circle"), and $\sigma \,\!$ is a point along an arcpath. Usually one is just looking for the distance between $\sigma \,\!$s,$\int _{\sigma _{s}}^{\sigma _{f}}C'(\mathrm {A} ,\sigma )\mathrm {d} \sigma ,\,\!$ or even just $\theta \!\!:\int _{\sigma _{s}}^{\sigma _{f}}C'(0,\sigma )\mathrm {d} \sigma =\int _{\theta _{s}}^{\theta _{f}}C'(\theta )\mathrm {d} \theta ,\,\!$ but there is the case (e.g.) where you might want to find a (transverse) annulus, in which case you would find the distance between two $\sigma \,\!$s, $\sigma _{s},\sigma _{f}\,\!$ for the full, 0 to 90° range of $\mathrm {A} \,\!$:

$\int _{0}^{90}\int _{\sigma _{s}}^{\sigma _{f}}C'(\mathrm {A} ,\sigma )\mathrm {d} \sigma \mathrm {d} \mathrm {A} ;\,\!$

I just wasn’t sure if you presented the integrand as a two variable argument——even if one of the potential variables is a constant——that it was necessarily partial or not.  ~Kaimbridge~ (talk) 14:11, 5 February 2012 (UTC)