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Lecture Notes for Advanced Calculus - supermath.info

Lecture Notes for Advanced CalculusJames S. CookLiberty UniversityDepartment of MathematicsFall 20132introduction 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 algebra is not a prerequisite for this course. However, I will use linear algebra. Matrices,linear transformations and vector spaces are necessary ingredients for a proper discussion of ad-vanced Calculus .

previous experience with calculus. Towards that end, I am including a section or two on series and sequences and a discussion of pointwise verses uniform convergence. These discussions set-up the technique of exchanging derivatives and integrals which is a powerful technique seldom discussed in current calculus courses.

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Transcription of Lecture Notes for Advanced Calculus - supermath.info

1 Lecture Notes for Advanced CalculusJames S. CookLiberty UniversityDepartment of MathematicsFall 20132introduction 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 algebra is not a prerequisite for this course. However, I will use linear algebra. Matrices,linear transformations and vector spaces are necessary ingredients for a proper discussion of ad-vanced Calculus .

2 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 analysis is also not a prerequisite for this course. However, I will present some proofs whichproperly fall in the category of real analysis. Some of these proofs involve a sophistication thatis beyond the usual day-to-day business of the course. I include these thoughts for the sake ofcompleteness. If I am to test them it s probably more on the question of were you paying attentionas opposed to can you reconstruct the monster from my main intent in this course is that you learn Calculus more deeply.

3 Yes we ll learnsome new calculations, but I also hope that what we cover also gives you deeper insight into yourprevious experience with Calculus . Towards that end, I am including a section or two on series andsequences and a discussion of pointwise verses uniform convergence. These discussions set-up thetechnique of exchanging derivatives and integrals which is a powerful technique seldom discussedin current Calculus 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:Xset-theoretic mapping langauge, fibers and images and how to picture relationships in view of the metric topology in concept and application of the derivative and differential of a differentiabilityXinverse function theoremXimplicit function theoremXtangent space and normal space via gradients or derivatives of parametrizations1if you view this as an insult then you haven t met the right babies yet.

4 Baby exercises are for multivariate functions, critical points and the Lagrange multiplier methodXmultivariate Taylor formsXcritical point analysis for multivariate functionsXdual space and the dual dualities and Hodge work and flux form mappings manifold theory (don t let me get too deep, )2 Xvector fields as forms and the exterior derivativeXintegration of formsXgeneralized Stokes s and pull-backsXhow differential forms and submanifolds naturally geometrize differential equationsXelementary differential geometry of curves and surfaces via the method of moving framesXbasic variational Calculus (how to calculate the Euler-Lagrange equations for a given La-grangian)When I sayworking knowledgewhat I intend is that you have some sense of the problem and atleast know where to start looking for a solution.

5 Some of the topics above take a much longer timeto understand deeply. I cover them to spark your interest and seed your intuition if all goes 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 whatR,C,Q,NandZdenote. I assume you know what a subset of a set is. I assume youknow how to prove two sets are equal.

6 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 mathematics4allows 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, mathematically-minded adults, we findthe actual definitions of such objects asRorCare rather involved.

7 I will not attempt to providefoundational arguments to build numbers from basic set theory. I believe it is possible, I thinkit s well-thought-out mathematics, but we take the existence of the real numbers as an axiom forthese Notes . We assume thatRexists and that the real numbers possess all their usual fact, I assumeR,C,Q,NandZall exist complete with their standard properties. In short, Iassume we have numbers to work with. We leave the rigorization of numbers to a different have avoided use of Einstein s implicit summation notation in the majority of these Notes . This hasintroduced some clutter in calculations, but I hope the student finds the added detail helpful.

8 Nat-urally if one goes on to study tensor calculations in physics then no such luxury is granted. In viewof this, I left the morephysicsynotation in the discussion of electromagnetism via differential is the third time I have prepared an official offering of Advanced Calculus . The first offeringwas given to about 10 students, half engineering, half math, it was deliberately given with a com-putational focus. The second offering was intended for an audience of about 6 math students, allbailed except 1 and the course modified into a more serious, theoretically-focused introduction tomanifolds (Spencer 2011). I have taught it off and on as an indpendent study to several students,Bobbi Beller, Jin semester I hope to go further into the exposition of differential forms than I have past attempts, too much time was devoted to developing constructions in basic manifold theorywe didn t really need.

9 So, this time, I take a somewhat formal approach to manifolds. We ll seehow differential forms allow great insight into the shape of surfaces and the geometrization of dif-ferential equations. Finally, at the end of the course I again spend several lectures on the calculusof on editing: ran a little short on time this summer, sorry but only pages 1-224 okfor printing at moment. The remaining 225 and beyond are only about 80% will let you know once those are fixed. Thanks!James Cook, August 18, sets, functions and euclidean set theory .. functions .. vectors and geometry forn-dimensional space.

10 Algebra for three dimensions .. notations for vector arithmetic .. 242 linear vector spaces .. matrix calculation .. linear transformations .. gallery of linear transformations .. matrices .. and isomorphism .. 473 topology and elementary topology and limits .. normed vector spaces .. balls, limits and sequences in normed vector spaces .. intuitive theorems of Calculus .. for single-variable Calculus .. for multivariate Calculus .. 744 the Frechet differential .. partial derivatives and the Jacobian matrix.


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