Plan to revise my learned math & start learning pure math

[AfterSunShine]

Hello everyone,
I'm planning to review learned mathematics from high school & start leaning pure math.

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My math level : Graduates as engineer 10 years ago. Got A in Calculus 1 & 2 (Those were single variable calculus), Got B+ in multivariable calculus. A in ODE. Cannot remember linear algebra grade but was in B range.

Whats exactly my current "math level" : I reviewed first 4 chapters from stewart calculus, easy to recall information & do problems, found weak points in geometry & trigonometry. Cannot recall geometry at all. For trig, found myself only knowing basic identities but still able to do related calculus problems with trig with no issues.

What do I want : Planning to self study previous math & move on to pure math. I really want to start again . Aiming on starting calculus with Spivak directly without stewart (single variable).

What do I want to reach : Reaching Real & Complex analysis level.

Why ? I really enjoy studying math & I have a lot of free time.

Draft plan to be reviewed & recommend textbook :
Stage 1 : Basic tools (Algebra, trigonometry & geometry).
For algebra & trig, thinking about combining both & study "A Graphical approach to algebra & trigonometry". Or I can go with Gelfand trig + algebra books separately ? I have no issue to study algebra & trig in rigorous books.
Geometry, my deepest weak point. Need full study again from zero. Googled a bit & recommendation of books mostly about moise geometry (not the advanced book) & Jacobs geometry. What do you think?

Stage 2 : introductory proof book (discrete math) + 2 small books i liked & willing to read.
Proof book (Discrete), needed to build math maturity & proofing ability : Here where I got lost. So many books! That book supposed to include logic, set theory, number theory introduction, proof methods. Need simple book as introductory. Will study advanced rigorous book in stage 3.

Two books I liked : (1) Introduction to inequalities by bellman (Will read it after finishing chosen proof book). (2) Basic mathematics by Lang. (Worth for quick reading as a refresher what I studied in Stage 1?)

Stage 3 : Rigorous proof book :
After studying & building solid stage 1, reading introductory proof book, focusing on inequalities in separate book, quick reading of Lang (Maybe?), Am thinking about 2nd advanced proof book as rigorous book to solidify what learned in 1st chosen proof book. So simple proof book in stage 2 & more rigorous book in Stage 3, and this the most part I need your book recommendation guys in these 2 books.

Stage 4 : Calculus by Spivak (Single variable) :

Stage 5,6 & 7 : Multivariable calculus, Linear Algebra & ODE : Need book suggestions.

Stage 8 : What next? to reach real & complex analysis ? Can I go directly to these books or something needed between them & previous stages ?Time is not an issue at all. It is self-study & am not putting any time limits at all.

Thanks.

 

[MidgetDwarf]

For Geometry I like the book by Edwin E Moise, "Geometry." Do not confuses it with Elementary Geometry From An Advance Standpoint, which is a university level geometry book.

For proofs, there is the free legal download of Hammock: Book of Proofs. You can obtain a physical copy cheap.

After Spivak , you can go into Analysis. Or you can skip Spivak, and learn Analysis from something like Abbot: Understanding Analysis. Which sticks to Analysis on the Real Line (in R), and proves most of the theorems in general calculus course. From here you can learn multivariable analysis. There are many choices.

Some readable books are Munkres: Analysis on Manifolds, Bartle: Elements of Real Analysis, and the books by Hubbard and Hubbard and Shifrin, respectively.

It does not hurt to learn Linear Algebra, since it will help further understand Multivariable Analysis and Complex Analysis.

You can read a baby book on Complex Analysis after learning Real Analysis on the real line. A good intro book, is the one by Churchill: Complex Variables. Then move onto a more theory based book after learning Multivariable Analysis.

Remember, it all depends on mathematical maturity...

There is also the Real Analysis book series by Peterson titled Basic Analysis. Which covers Real Analysis up to functional analysis in 5 volumes. I am currently reading the first volume (chapter 7), and find it a pedagogical masterpiece. However, the problems so far are a bit easy, and a bit expensive. Try to see if you can view it before you purchase when you are ready to learn Analysis.

I find it more clear than Abbot, where Abbot is probably the clearest single variable analysis book available.

 

[MidgetDwarf]

oh you can learn ODE at any time. It requires only calculus 2 (Integration techniques). A bit simple. Now, if you want to learn the theory of ODE, then lots more math courses are needed.

 

[AfterSunShine]

Thank you.
What about algebra and trigonometry (Stage 1)? Go for separated gelfand books or do you have other rigorous recommendations?
Am really interested in rigorous books for algebra & trig.
Any recommendation is appreciated to start plan instead of just thinking about book & waste time.

 

[MidgetDwarf]

I think using a general pre-calculus book for US universities will meet most of your high school algebra/trig needs. Cohen Pre-Calculus: A problem... etc.

You can also supplement with Serge Lang Basic Mathematics.

As for Trig, I remember reading Gelfand, but its been many years. It was not a bad book, but needed to be supplemented. There is always Loney:Plane Trigonometry. Lots of gems in here, but the language (archaic) may make it a bit hard to read.

My advice, is to do Pre-Calculus/Geometry simultaneously. While reading a proofs book, if time permits. No need to start at the basics, since you have an engineering degree...