Shifrin Math 3500

[Azalee Freas]

I'm Professor Emeritus of Mathematics at . I received the FranklinCollegeOutstandingAcademic Advising Award for 2012. I received the Lothar Tresp Outstanding Honors Professor Award in 2002 and 2010, as well as the Honoratus Medal in 1992. Iwas one of five recipients of the 1997 JosiahMeigsAward for Excellence in Teaching at The University of Georgia. I was the 2000 winner of the Award for Distinguished College or University Teaching of Mathematics, Southeast section, presented by the Mathematical Association of America. My research interests are in differential geometry and complex algebraic geometry.

If you'd like to see the "text" of my talk at the MAA Southeastern Section meeting, March 30, 2001, entitled Tidbits of Geometry Through the Ages, you may download a .pdf file.

I am the Honors adviser for students majoring in Mathematics at The University of Georgia. I also advise Honors freshmen and sophomores majoring in Computer Science, Physics, Physics & Astronomy, and Statistics. If you would like to see how the Honors Program at The University of Georgia has recently garnered national attention, you might try the cover story of the September 16, 1996 issue of U.S. News & World Report, p. 109. (I have a personal stake in this, of course.)

Long ago, I wrote a senior-level mathematics text, AbstractAlgebra:A Geometric Approach, published by Prentice Hall (now Pearson) in 1996. You might want to refer to the list of typos and emendations. Please email me if you find other errors or have any comments or suggestions.

Malcolm Adams and I recently completed the second edition of our linear algebra text, LinearAlgebra:AGeometric Approach, published by W.H. Freeman in 2011. Our approach puts greater emphasis on both geometry and proof techniques than most books currently available; somewhat novel is a discussion of the mathematics of computer graphics. As we find out about them, we will be maintaining a list of errata and typos.

My textbook MultivariableMathematics:Linear Algebra, Multivariable Calculus, and Manifolds was published by J. Wiley & Sons in 2004. The text integrates the linear algebra and calculus material, emphasizing the theme of implicit versus explicit. It includes proofs and all the theory of the calculus without giving short shrift to computations and physical applications. There is, as always, the obligatory list of errata and typos; please email me if you have any comments or have discovered any errors. Click here if you want a list of errata in the solutions manual.

With gracious thanks to Patty Wagner, Eric Lybrand, Cameron Bjorklund, Justin Payan, and Cameron Zahedi, my lectures in Multivariable Mathematics (MATH 3500(H)MATH 3510(H)) are available, for better or for worse, on YouTube. We are currently recording the first semester (covering through the basics of linear algebra and differential calculus); the second semester (covering integration, manifolds, and eigenvalues) is already posted.

I have written some informal class notes for MATH 4250/6250, Differential Geometry: A First Course in Curves and Surfaces. They are available in .pdf format, and, as usual, comments and suggestions are always welcome. I have recently revised the notes. If you're interested in using them as a class text, all I ask is that the students incur at most a copying fee. I am always happy to hear from people who have used the notes and have comments and suggestions to improve them.

I'm taking multivariable calc this fall. I began self-studying on my own a couple months ago, using Salas's calc text. Then I stumbled on Ted Shrifin's MTH3500/10 incredible lecture series on Youtube. His text, Multivariable Mathematics, arrived in the mail yesterday evening! It's a freakin' gorgeous book, and I'm super excited to start. I have three questions.

(1) Question for anyone that's worked or taught from this book: In terms of coverage, how does this book compare with something like Munkres, Calculus on Manifolds? Is there significant overlap? Will I be prepared for Munkres after reading Shifrin?

(2) I decided I had to have the book after bing-watching the lectures. The ideas there are just so lovely, and so nicely explained. The idea of linear maps is a beautiful one, and I'm amazed at how it generalizes the results of single-variable calc. Matrices, matrix multiplication, and the like can seem so unmotivated and pointless, until one sees that matrix multiplication is the algebra behind the composition of linear maps. Historically, was it the need to put multivariable calc on a sound footing that motivated the development of linear algebra?

(3) I'll be taking linear algebra in the fall, too. I wonder: Why (or how!) would anyone successfully teach linear algebra without using multivariable calculus and the geometry of linear maps to reify matrices and their symbol-shunting? I mean, Shifrin strives to show the connection between linear algebra and multivariable calc, and this is an unusual approach, right? But then how else would linear algebra be taught?

In Ted's book he teaches you about the topology on $\mathbbR^n$, so you will have this as a reference when looking at more abstract examples of topological spaces. The space $\mathbbR^n$ becomes a metric space with the euclidean metric and metric spaces are topological spaces. So yes, you will be prepared for Munkres. Generally, the only hard part out point-set topology is the set theory and extracting useful information from pictures.

The concept of merging calculus and geometry has been around for sometime. In differential geometry they study non-linear objects i.e manifolds by linear ones i.e the differential (and Hessian) which you've seen is a linear map. This is beautiful because the theory of linear maps i.e linear algebra is very developed. It was after I seen linear algebra used in other fields that I thought, "my god, this should never be taught by itself." However, it was the exploration of this subject independent of these other fields that allow us to use it's theory at great lengths.

With the help of Patty Wagner and a few of my then colleagues I helped record and uploaded lectures of the MATH 3500/3510 course at UGA. That course was incredibly influential on my career as a mathematician. You may find those lectures at the following YouTube page.

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