Lecture Notes for Advanced Calculus - supermath.info

1 Lecture Notes for Advanced CalculusJames S. CookLiberty UniversityDepartment of MathematicsFall 20112introduction and motivations for these notesThere are many excellent texts on portions of this subject. However, the particular path I choosethis semester is not quite in line with any particular text. I required the text on Advanced Cal-culus by Edwards because it contains all the major theorems that traditionally are covered in anAdvanced Calculus focus differs significantly. If I had students who had already completed a semester in realanalysis then we could delve into the more analytic aspects of the subject. However, real analy-sis is not a prerequisite so we take a different path.

2 Generically the story is as follows: a linearapproximation replaces a complicated object very well so long as we are close to the base-pointfor the approximation. The first level of understanding is what I would characterize asalgebraic,beyond that is theanalyticunderstanding. I would argue that we must first have a firm grasp ofthe algebraic before we can properly attack the analytic aspects of the covers both the algebraic and the analytic. This makes his text hard to read in placesbecause the full story is at some points technical. My goal is to focus on the algebraic. That said,I will try to at least point the reader to the section of Edward where the proof can be algebra is not a prerequisite for this course.

3 However, I will use linear algebra . Matrices,linear transformations and vector spaces are necessary ingredients for a proper discussion of ad-vanced Calculus . I believe an interested student can easily assimilate the needed tools as we go so Iam not terribly worried if you have not had linear algebra previously. I will make a point to includesome baby1linear exercises to make sure everyone who is working at this course keeps up with thestory that the homework is doing the course. I cannot overemphasize the importance of thinkingthrough the homework. I would be happy if you left this course with a working knowledge of: set-theoretic mapping langauge, fibers and images and how to picture relationships diagra-matically.

4 Continuity in view of the metric topology in n-space. the concept and application of the derivative and differential of a mapping. continuous differentiability inverse function theorem implicit function theorem tangent space and normal space via gradients1if you view this as an insult then you haven t met the right babies yet. Baby exercises are extrema for multivariate functions, critical points and the Lagrange multiplier method multivariate Taylor series. quadratic forms critical point analysis for multivariate functions dual space and the dual basis. multilinear algebra . metric dualities and Hodge duality. the work and flux form mappings for 3. basic manifold theory vector fields as derivations.

5 Lie series and how vector fields generate symmetries differential forms and the exterior derivative integration of forms generalized Stokes s Theorem. surfaces fundmental forms and curvature for surfaces differential form formulation of classical differential geometry some algebra and Calculus of supermathematicsBefore we begin, I should warn you that I assume quite a few things from the reader. These notesare intended for someone who has already grappled with the problem of constructing proofs. Iassume you know the difference between and . I assume the phrase iff is known to assume you are ready and willing to do a proof by induction, strong or weak. I assume youknow what , , , and denote.

6 I assume you know what a subset of a set is. I assume youknow how to prove two sets are equal. I assume you are familar with basic set operations suchas union and intersection (although we don t use those much). More importantly, I assume youhave started to appreciate that mathematics is more than just calculations. Calculations withoutcontext, without theory, are doomed to failure. At a minimum theory and proper mathematicsallows you to communicate analytical concepts to other like-educated of the most seemingly basic objects in mathematics are insidiously complex. We ve beentaught they re simple since our childhood, but as adults, mathematical adults, we find the actual4definitions of such objects as or are rather involved.

7 I will not attempt to provide foundationalarguments to build numbers from basic set theory. I believe it is possible, I think it s well-thought-out mathematics, but we take the existence of the real numbers as an axiom for these Notes . Weassume that exists and that the real numbers possess all their usual properties. In fact, I assume , , , and all exist complete with their standard properties. In short, I assume we havenumbers to work with. We leave the rigorization of numbers to a different format of these Notes is similar to that of my Calculus and linear algebra and Advanced calculusnotes from 2009-2011. However, I will make a number of definitions in the body of the text. Thosesort of definitions are typically background-type definitions and I will make a point of putting themin bold so you can find them with have avoided use of Einstein s implicit summation notation in the majority of these Notes .

8 Thishas introduced some clutter in calculations, but I hope the student finds the added detail if one goes on to study tensor calculations in physics then no such luxury is granted, youwill have to grapple with the meaning of Einstein s convention. I suspect that is a minority in thisaudience so I took that task off the to-do list for this content of this course differs somewhat from my previous offering. The presentation of ge-ometry and manifolds is almost entirely altered. Also, I have removed the chapter on Newtonianmechanics as well as the later chapter on variational Calculus . Naturally, the interested student isinvited to study those as indendent studies past this course. If interested please should mention that James Callahan sAdvanced Calculus : a geometric viewhas influenced mythinking in this reformulation of my Notes .

9 His discussion of Morse s work was a useful addition tothe critical point was inspired by Flander s text on differential form computation. It is my goal to implement someof his nicer calculations as an addition to my previous treatment of differential forms. In addition,I intend to encorporate material from Burns and Gidea sDifferential Geometry and Topology witha View to Dynamical Systemsas well as Munkrese Analysis on Manifolds. These additions shouldgreatly improve the depth of the manifold discussion. I intend to go significantly deeper this yearso the student can perhaps begin to appreciate manifold plan to take the last few weeks of class to discuss supermathematics. This will serve as a sidewaysreview for Calculus on.

10 In addition, I hope the exercise of abstracting Calculus to supernumbersgives you some ideas about the process of abstraction in general. Abstraction is a cornerstone ofmodern mathematics and it is an essential skill for a mathematician. We may also discuss some ofthe historical motivations and modern applications of supermath to supersymmetric field To BE DELETED:-add pictures from 2009 equations to numbered set theory .. vectors and geometry for -dimensional space .. algebra for three dimensions .. notations for vector arithmetic .. functions .. elementary topology and limits .. 252 linear vector spaces .. matrix calculation.