 the main results being left as exercises or problems.The important difference is that exercises have full solutions and problems don't. This approach excites me because I've always wondered what kind of "Moore method" type course I could seriously consider using as a lecturer.When I discussed the matter with several colleagues, the approach I'd always suggest is an example driven one.After all, aren't theorems merely statements of the common truth of an enormous number of specific examples and constructions? Imagine my pleasant surprise finding Erdman making the identical observation in his introduction!!! The result is a deep, broad and far reaching textbook on advanced calculus that strong students will enjoy working through,lecturers will enjoy constructing their lectures from and graduate students will absolutely adore when preparing for their analysis qualifying exams. Very highly recommended!!!
- Advanced Calculus II Math 1530 Piotz Hajlasz University of Pittsburgh Fall 2011 (PG-13) This is another absolutely beautiful set of handwritten notes by Hajlasz. (God, I love this guy's notes. I wish he'd give handwriting classes-I'd have passed more classes if I had his secret to magically clear and legible handwriting!! ) This is a rigorous course on functions of several real variables on metric spaces that goes beyond the usual course on this material as it begins with proofs of function space properties of Banach spaces; such as the Banach contraction mapping theorem and the Arazela-Ascoli Theorem. He then proceeds to give one of the best courses of several variable analysis I've ever seen, covering all the main
expected topics::differentiability, open mapping theorem, implicit function theorem, Lagrange multipliers, submanifolds, integrals, change of variables formula, integration on surfaces: Green's formula, Stokes theorem, Gauss theorem. There are a ton of exercises and beautiful hand-drawn diagrams of surfaces and curves while never abandoning the general perspective. He also includes wonderful sophisticated examples one usually sees in advanced geometry and topology
courses, such as Hutchenson's theorem, the van Koch curve and the Serpinski gasket. Oh, to heck with it, just download and read them. Then read them again and do all the exercises.Let their beauty and brilliance speak for themselves. The only tragedy here is that the first half isn't available any longer. Highest possible recommendation.
- Undergraduate Analysis II Lecture Notes Victor Guillemin Mathematics MIT OpenCourseWare 2005 (PG-13) Strong and very readable set of notes by a major mathematician for a rigorous course on calculus of functions of several variables presuming an advanced calculus course on metric spaces. They're similar in spirit, level and coverage to James Munkres' classic textbook Analysis on Manifolds, which was actually one of the textbooks used for the course. Differentiable maps, inverse and implicit function theorems, n-dimensional Riemann integral, change of variables in multiple integrals, manifolds, differential forms, n-dimensional version of Stokes' theorem. Very readable with careful proofs, but has a big flaw in that it has almost no solved examples. Since the notes are really intended as a supplement and not the main course text, this is disappointing but not surprising. What's quite interesting here is that Guillemin has been teaching two different versions of this course for several years now. He has since 2005 removed and vastly expanded the material on differential forms into an independent,substantial course text which we comment on here. In 2008, he taught a similar version of this course without differential forms using the idea of densities instead in integration and adding material on ordinary differential equations. Those notes are reviewed here.
- Theory of Differential Forms Victor Guillemin MIT Spring 2014 (PG-13) This is a remarkable online textbook draft-and yes, I think it's large and developed enough now that we can safely call it an online textbook draft-that grew out of Guillemin's analysis and differential topology courses over the last decade. Nowadays, as I said elsewhere, it's all the rage to try and teach strong undergraduate students multivariable calculus from a totally modern viewpoint of differentiable manifolds and differential forms. As I also said, I have mixed feelings about this approach. This is Guillemin throwing his hat in this ring to produce a totally modern course in vector analysis. Wisely, he decides to require some mathematical maturity from his students-the minimal prerequisites are a term each of honors calculus/ advanced calculus/ elementary real analysis and modern algebra. Contents: Multilinear algebra,tensors and exterior forms,differential forms on Rn,exterior differentiation, integration of forms on open sets of Rn,change of variables formula revisited,the degree of a differentiable mapping. differential forms on manifolds and De Rham theory, integration of forms on manifolds and Stokes’ theorem.the push-forward operation for forms and much more.Applications to differential topology. This is one of the most careful and lucid sets of lecture notes currently online. The author takes a wonderful middle ground approach which develops forms from both the algebraic and geometric perspective. Both aspects I believe are equally important not only for fully understanding forms, but also for understanding their older and poorly understood siblings, tensors. Most treatments take purely one approach, which is a mistake the author seems well aware of. There are virtually no examples in the first chapter, but that's to be expected as it's merely developing algebraic machinery out of structures the student is assumed to know. Indeed, this chapter can be used as a fine supplement on multilinear algebra in either a graduate or honors undergraduate algebra course. The treatment of forms begins in earnest in the next 3 chapters, which abound with rich examples and wonderful exercises.Sadly, there are not many pictures except in the last chapter on manifolds,but that can be forgiven given how clear the presentation is.There are many deep topics not usually covered in first year graduate differential geometry or algebra courses, which is where one usually learns about forms-such as the pull-back operation,the Poincaré lemma, Thom forms and intersection theory. The real shock and joy here is to find Guillemin thoroughly covering applications to physics of this material in a pure mathematics course, such as a terrific discussions of Maxwell’s equations from the differential form perspective!The multivariable analysis text by Shurman would make an outstanding supplementary source,as it contains a somewhat simpler and very different development of forms that's quite a bit more visual then Gulliemen's. These presentations complement each other very well indeed. I don't know how much longer the author is going to maintain these notes before publishing them as an actual textbook and charging more then the average freshman spends on food in a week for it. I hope it'll remain so for many years to come. I do know that as long as he continues to make it available for free, it will remain one of the very best introductions to multilinear algebra and differential forms that currently exists in any form. The highest possible recommendation!
- Undergraduate Analysis II Lecture Notes Victor Guillemin Mathematics MIT Fall 2010 (PG-13) This is a later version of the earlier supplementary notes for the Analysis II course at MIT we commented on here. The main differences are the removal of the material on differential forms (presumably because Guillemin's comprehensive and highly recommended notes on differential forms now exist), the replacement of forms in the treatment of integration of several variables with densities and the addition of a first chapter on ordinary differential equations in Rn. To be honest, I found the material on ODE's to be the weakest part of the notes and better treatments can be found in many other places. The real breakthrough here is in the treatment of integration. A density is a mapping from a product of vector spaces to the real numbers such that the image of any real valued linear mapping is stretched by a factor of the absolute value of the determinant of its matrix. (This is actually a special case of the tensor density in differential geometry.) It's an interesting alternative presentation, one that emphasizes the geometric interpretation of the determinant in Euclidean spaces. More importantly, it turns out that on manifolds, a density is locally a Lebesgue measure, which of course is why it can be used in place of differential forms to construct a general theory of integration on manifolds. The author does a wonderful job explaining this-many texts at this level completely botch the rigorous explanation of why forms can be used as integrators. This approach using densities is completely clear and works beautifully. As for these specific notes, again, they aren't really complete enough by themselves to function as a full course text. Together with the aforementioned notes on differential forms and the 2005 "edition", the union of the 3 notes comprise the bulk of a wonderful new textbook on analysis of several variables for either first year graduate students or strong undergraduates. Here's hoping the author eventually combines and revises the notes into such a text. Highly recommended as a supplement.
- Undergraduate Analysis Tools Bruce K. Driver University of California at San Diego May 1, 2013 (PG-13) One of a host of lecture notes in progress at Driver's homepage-this is the latest iteration of his undergraduate real analysis notes. They're intended to supplement a real analysis course based on Rudin's Principles of Mathematical Analysis, so as you can imagine, they're not a trifling treatment of the subject. Indeed, they are more then comprehensive enough to act as the sole textbook for the course.I suspect the reason Driver does not yet do so is because he's still in the process of revising it-the notes have changed dramatically in organization since he began writing them in the early 2000's. (I know this because Driver has allowed the various draft versions to remain available at his website.It's a fascinating look at the evolution of a textbook in progress.) Basic properties of the real numbers, complex numbers, metric spaces, sequences and series of real numbers, functions of a real variable and continuity,differentiation, the Riemann-Stieltjes integral, sequences and series of functions, power series, Fourier series, and