## What’s Next?

Last May we started talking about linear algebra, with a little aside into complex numbers and another into power series along the way. Before that all, long long ago, we were talking about single variable calculus. Specifically, we were studying functions which took a real number in and gave a real number back out, and the two main aspects to this study: differentiation and integration.

The first part studied how a function changed as its input varied near a fixed point by coming up with the best linear approximation of the function near that point. Now that we’ve got an understanding of linear functions between higher-dimensional real vector spaces, we can work towards extending this idea of differential calculus into multivariable functions.

The second part studied how to “add up” a continuously-varying collection of values, each with its own (infinitesimal) weight. Again, our new understanding of higher-dimensional analogues of linear spaces and functions will help us find the right way to generalize the integral calculus.

I’m trying to get access to some of my references again, since I no longer have even as much of a mathematical library down the hall as Western Kentucky University provided. So I’ll pick up when I can.

You are doing a great work here. Do you have plan to make it into downloadable PDF files?

Comment by timur | September 10, 2009 |

Not particularly. I’m turning over this idea of a book, as I mentioned a few weeks ago, but I still think I need something from an assistant to a co-author to help keep me honest on the relevant history of math/philosophy.

Comment by John Armstrong | September 10, 2009 |

Incidentally, are you thinking of continuing your previous series on category theory? Those posts were extremely helpful, and I don’t there are other blogs that seriously discuss categories at a level accessible to beginners.

Comment by Akhil Mathew | September 10, 2009 |

I agree with the others. Nice blog!

Comment by Eitan | September 10, 2009 |

I may return to category theory as such if I have need of it. For now I’m looking at multivariable calculus leading towards manifolds for differential geometry and algebraic topology.

Comment by John Armstrong | September 11, 2009 |

Since I teach at an undergraduate only school – I have a similar problem. Here are some suggestions

1. Arxiv.

2. If you are a member of the MAA, for a few extra dollars you can get on-line access to their journal.

3. You may be able to get academic library access through your school’s alumni organization.

4. If you are near a big city (like Chicago or NYC) they usually have an “academic” public library.

5. Email the author and ask for a copy of the article.

6. The most important thing is access to mathscinet – you can’t even ask for the articles without that.

Comment by Heather | September 11, 2009 |

Heather, I am completely unemployed, so 3 is out. The public libraries in central MD are far from academic standards (even in DC and Baltimore), so 4 is out. And the rest are good for articles, not for

booksas standard background references.Comment by John Armstrong | September 11, 2009 |

Not QUITE unemployed. You are merely currently UNPAID for your wonderful work in your own research, and on the blogosphere, including these superb threads, which should indeed be someday published more widely (perhaps with a collaborator to cover the literature and history).

I rented a car to get to Tuesday’s interview for a high school Math teacher position. I had a 90-minute interview, which I felt went well. I was told that I’d probably get an appointment for another round of discussions, or else an offer, within 24 hours. So I kept the rental car, and waited. And sent 2 follow-up emails. And waited. Today, Friday, I returned the car. This homework assignment cost me $251.49. How much more can I afford to spend, after 2 years as a substitute teacher, and 2 years of College of Education, since I decided to take a pay cut from university professor, and become an inner city high school teacher, because the students and communities needed me so much?

Comment by Jonathan Vos Post | September 11, 2009 |

Semantics and compliment don’t put food down or a roof up.

Comment by John Armstrong | September 11, 2009 |