Sal Balkus – Advanced Calculus II Blog

Although we mostly covered the conclusion to proving that all continuous functions are Darboux integrable in class, I wanted to take part of this week to summarize what I learned and put it in my own words. This way, I will have a satisfying conclusion to the first integration section that we’ve been building up to this whole time through our discussions of the Bolzano-Weierstrauss theorem and uniform continuity. I also worked out a very simple exercise in that section of the instructor’s notes that we did not cover.

Integration (1)

I also began estimating the integrability of the Riemann function. Although I still have more work to do, I have included my initial plotting results in a Mathematica Notebook.

Riemann Function

 

Last week, I suppose I “jumped the gun” on working on the Bolzano-Weierstrauss theorem, which was assigned in class this week. So, I added an extension to my proof to argue that the fact that all sequences have a montone subsequence demonstrates the Bolzano-Weierstrauss theorem for bounded sequences. Based on feedback in class, I also included a discussion section reflecting on my proof and including a summary of the “standard” method of proof,

Integration.

This week, we discussed the accumulation and the foundations of the Bolzano-Weierstrauss. To extend this idea, I used an exercise from Lebl’s Basic Analysis to expand my understanding of the proof and work out an alternate method of said proof that is specified as an exercise. I also continued working through the extra content of 1.5, with a proof that a partition and its norm as defined in the class text form a directed pre-ordered set.

 

Week_1___Integration (5)

This week I added proofs regarding uniform continuity of functions. I also included a discussion of partition refinement and norms of partitions, along with a simple proof exercise. Finally, I include a Mathematica notebook to provide solutions to two computational exercises posed in class.

 

Week_1___Integration (4) 1-5-2

This week, I cleaned up the proofs of the Darboux integration, added the x^2 case, and started work on an exercise involving Riemann integration.

Week_1___Integration (1)

Welcome to my MTH312 blog! Here I will be posting all of the writing, mathematical proofs, and computational demonstrations that I complete in this course.

This first week of class, I started out by summarizing our reading on Darboux integration and completing some basic epsilon-delta proofs involving Darboux integrals. See the attached document for my complete work.

Week_1___Integration

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