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Have a math degree, need to refamiliarize with advanced math

  • #1
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Main Question or Discussion Point

I've been out of school for a while and working as a programmer. I want to start taking some masters courses for applied math (PDEs, numerical analysis, etc) and need to become familiar again with the advanced math I used to use in undergrad. I took two semesters of real analysis as an undergrad, but don't remember a lot of the rigor, or how to properly use concepts for proofs (MVT, partitions of an interval, etc). I remember the intuition behind the stuff that stumps a lot of undergrads, like epsilon-N/delta, but [your deity here] knows if I could still properly write proofs.

So should I just go back through a book like Spivak to practice proofs, etc? Or is there a better/more advanced book I should be using that anyone can recommend?

Same question with Algebra, even though it's tagged as Analysis. I had two semesters of algebra in which we did Group Theory (first semester), and then second semester was Ring and Field theory, finishing up with a bit of an intro to Galois theory. We worked through Charles Pinter's A Book of Abstract Algebra, skipping only the 2 "optional" chapters on number theory and geometry. I have a copy of Gallian, that I may go back through, but was wondering if something like Lang may be better.
 

Answers and Replies

  • #2
Stephen Tashi
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I've been out of school for a while and working as a programmer. I want to start taking some masters courses for applied math (PDEs, numerical analysis, etc) and need to become familiar again with the advanced math I used to use in undergrad.
Don't go overboard on studying rigorous proofs. You might not be asked to do many proofs in an applied math program. The best plan would be to look at the texts used in the applied math program that you intend to enter. You might be able to find the course syllabus online and you might even find the homework assignments and class notes.

In advanced courses, there is usually the pain of having to learn or re-learn things that you aren't familiar with. The oft-pursued dream is to master such things by self-study before you take an advanced course. However, this is , in my experience, impractical. For example, if you review calculus and try to attain "mastery" of every topic then you do things like spend 50 hours studying the material in Chapter 3 of a calculus book and by the time the year is over, you still have 20 chapters more to do. Yes, it is less painful and more satisfying to pursue studies in a series of sure and comfortable steps, but it takes forever.

How decayed is you knowledge of calculus? Can you still do the computational aspects ? - find limits, compute partial derivatives, do integration by parts ?
 
  • #3
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Don't go overboard on studying rigorous proofs. You might not be asked to do many proofs in an applied math program. The best plan would be to look at the texts used in the applied math program that you intend to enter. You might be able to find the course syllabus online and you might even find the homework assignments and class notes.

In advanced courses, there is usually the pain of having to learn or re-learn things that you aren't familiar with. The oft-pursued dream is to master such things by self-study before you take an advanced course. However, this is , in my experience, impractical. For example, if you review calculus and try to attain "mastery" of every topic then you do things like spend 50 hours studying the material in Chapter 3 of a calculus book and by the time the year is over, you still have 20 chapters more to do. Yes, it is less painful and more satisfying to pursue studies in a series of sure and comfortable steps, but it takes forever.

How decayed is you knowledge of calculus? Can you still do the computational aspects ? - find limits, compute partial derivatives, do integration by parts ?

The computation part I am not so rusty with. I should review and practice, but in general I remember how to do a lot. I'm more concerned about the theory (since I know real analysis will be classes I should take even in an applied program)
 
  • #4
Stephen Tashi
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I'm more concerned about the theory (since I know real analysis will be classes I should take even in an applied program)
Is a graduate course in real analysis given as part of the applied math program you intend to enter?
 
  • #5
FactChecker
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The courses you have had look like theoretical, pure math classes. The classes you take for applied math might be significantly different. PDE and numerical analysis are good applied subjects. Are you also considering some probability and statistics? If I were you, I would look at the text books of the classes you plan to take before you get too concerned about your background. If you understood the proofs before, it will probably come back fairly quickly.

PS. You should not assume that you need real analysis for your future classes. Check the prerequisites. Same for abstract algebra.
 
  • #6
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Is a graduate course in real analysis given as part of the applied math program you intend to enter?
Not required. But it is available as an option, and I also figured that other courses like Numerical Analysis woild be more proof based at the masters level than what I took in undergrad.
 
  • #7
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The courses you have had look like theoretical, pure math classes. The classes you take for applied math might be significantly different. PDE and numerical analysis are good applied subjects. Are you also considering some probability and statistics? If I were you, I would look at the text books of the classes you plan to take before you get too concerned about your background. If you understood the proofs before, it will probably come back fairly quickly.

PS. You should not assume that you need real analysis for your future classes. Check the prerequisites. Same for abstract algebra.
I took plenty of applied courses... PDEs, numerical analysis, etc. But as I said in my other reply, I was expecting that those courses at a masters/graduate level would be rather proof based and a good base in Real Analysis would be required.
 
  • #8
S.G. Janssens
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I took plenty of applied courses... PDEs, numerical analysis, etc. But as I said in my other reply, I was expecting that those courses at a masters/graduate level would be rather proof based and a good base in Real Analysis would be required.
Yes, so would I.

Graduate programs in PDE or numerical analysis that wave the need for a solid basis in real analysis look extremely suspect to me. Applied mathematics is, well, applied, but also still mathematics so it does require rigorous proofs.

Perhaps you could consider having a look at Cheney's book "Analysis for Applied Mathematics" and perhaps also "Theoretical Numerical Analysis" by Atkinson and Han. (The latter approaches the theory of ##L^p## spaces using completions, rather than Lebesgue integration.) Atkinson's book on numerical analysis is more basic, but also very good.
 
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  • #9
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Yes, so would I.

Graduate programs in PDE or numerical analysis that wave the need for a solid basis in real analysis look extremely suspect to me. Applied mathematics is, well, applied, but also still mathematics so it does require rigorous proofs.

Perhaps you could consider having a look at Cheney's book "Analysis for Applied Mathematics" and perhaps also "Theoretical Numerical Analysis" by Atkinson and Han. (The latter approaches the theory of ##L^p## spaces using completions, rather than Lebesgue integration.) Atkinson's book on numerical analysis is more basic, but also very good.
Interestingly enough, the PDE book used for recent semesters at the program I'm looking at is the same we used for my undergrad PDE course.

I'll take a look at those. You say it still requires rigorous proofs -- but my question is more along the lines of if it requires you to understand the proof or write the proof yourself for exams. Theres a big difference, I think...
 
  • #10
S.G. Janssens
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Interestingly enough, the PDE book used for recent semesters at the program I'm looking at is the same we used for my undergrad PDE course.
I had the same with functional analysis, but the depth of treatment was different and the graduate course focused on the more advanced chapters.
I'll take a look at those. You say it still requires rigorous proofs -- but my question is more along the lines of if it requires you to understand the proof or write the proof yourself for exams. Theres a big difference, I think...
Fortunately I was required to do both. Of course the focus was on proofs for those results that were relevant for applications. In my opinion it is very difficult to contribute to the rigorous design of new methodology without an active understanding of proof writing. Applied mathematics is more than just calculation and quoting some theorems. I am very well aware that other applied mathematicians may disagree, but such is my viewpoint.
 
  • #11
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In my opinion it is very difficult to contribute to the rigorous design of new methodology without an active understanding of proof writing.
Agreed, but technically I'm not looking to go into academic research
 

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