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Ulagatin
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Studying mathematics "post-single-variable calculus"...?
My apologies - I posted this in the Abstract and Linear Algebra section earlier, but thought it might be more appropriate in this section...
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Hi everyone,
I'm in Grade 11 this year (in Australia), currently studying from Apostol's first volume "Calculus". I have just recently started working on the theory of integration of trigonometric functions (just giving some information on my background). I am thinking that, perhaps once I've finished the calculus section of Apostol's text, I could move on to either;
(a) Multi-Variable Calculus
(b) Real Analysis
(c) Abstract + Linear Algebra.
I cannot really make a decision at this point in time, as I am unsure about abstract algebra (hence my post here). My interests are in physics, mathematics, computer science and philosophy, and so, I want to make a decision based on relevance to these interests.
A short time back, I e-mailed Professor Apostol himself (giving feedback on his textbook and asking for advice regarding his Mathematical Analysis and Calculus Volume 2 books). He suggested that his second volume isn't necessarily a pre-requisite for his more terse exposition in the Mathematical Analysis text. He also suggested, since I am seriously considering majoring in mathematics at university, that I take a look at his Intro to Analytic Number Theory text.
I would assume this step would come one after that of Real Analysis. But, may I ask, if I decide to pursue pure rather than applied mathematics, how useful would abstract algebra be to this field? What ability does a solid knowledge of abstract algebra present to you: i.e. what type of problems does it allow you to solve, and what is the topic's main idea/motivation?
How difficult is the topic (abstract algebra) in comparison with calculus: is it less visual and geometrically intuitive? My favourite aspect of mathematics is calculus at this stage, and also sequences/series, although I don't understand them as well as many of the integration/differentiation topics and theorems I have covered.
While I may not be the most sophisticated mathematically, I am interested in a rigorous but understandable presentation of the various topics. I'm not afraid to see proofs or try and prove things myself (although for the most part, I'm unsure of how to do that).
Would it be possible to pick up linear algebra as I go, if I decide to study abstract algebra? I'm aware that linear algebra has great importance in computer science and physics, but I've also heard that it's not a greatly interesting branch of mathematics. I'm aware that this is an entirely subjective question, but how does abstract algebra rate on this "interest scale"? Do people find it more fun than calculus?
My ultimate goal, after university, is to become a theoretical physicist, and I've heard that both calculus (analysis?) and algebra are very important to this field. Any information on the basics of abstract algebra, what it is, how it's used, and if possible, recommended texts on this field would be useful, and greatly appreciated.
So, based on all the information I have provided, what would you all suggest for me to move on to? Remember, analytic number theory is potentially an option too, but I fear it may be too advanced for me at this stage. Would abstract algebra be too advanced?
Any enlightenment would be grand.
Cheers,
Davin
My apologies - I posted this in the Abstract and Linear Algebra section earlier, but thought it might be more appropriate in this section...
-----------------------------------------------
Hi everyone,
I'm in Grade 11 this year (in Australia), currently studying from Apostol's first volume "Calculus". I have just recently started working on the theory of integration of trigonometric functions (just giving some information on my background). I am thinking that, perhaps once I've finished the calculus section of Apostol's text, I could move on to either;
(a) Multi-Variable Calculus
(b) Real Analysis
(c) Abstract + Linear Algebra.
I cannot really make a decision at this point in time, as I am unsure about abstract algebra (hence my post here). My interests are in physics, mathematics, computer science and philosophy, and so, I want to make a decision based on relevance to these interests.
A short time back, I e-mailed Professor Apostol himself (giving feedback on his textbook and asking for advice regarding his Mathematical Analysis and Calculus Volume 2 books). He suggested that his second volume isn't necessarily a pre-requisite for his more terse exposition in the Mathematical Analysis text. He also suggested, since I am seriously considering majoring in mathematics at university, that I take a look at his Intro to Analytic Number Theory text.
I would assume this step would come one after that of Real Analysis. But, may I ask, if I decide to pursue pure rather than applied mathematics, how useful would abstract algebra be to this field? What ability does a solid knowledge of abstract algebra present to you: i.e. what type of problems does it allow you to solve, and what is the topic's main idea/motivation?
How difficult is the topic (abstract algebra) in comparison with calculus: is it less visual and geometrically intuitive? My favourite aspect of mathematics is calculus at this stage, and also sequences/series, although I don't understand them as well as many of the integration/differentiation topics and theorems I have covered.
While I may not be the most sophisticated mathematically, I am interested in a rigorous but understandable presentation of the various topics. I'm not afraid to see proofs or try and prove things myself (although for the most part, I'm unsure of how to do that).
Would it be possible to pick up linear algebra as I go, if I decide to study abstract algebra? I'm aware that linear algebra has great importance in computer science and physics, but I've also heard that it's not a greatly interesting branch of mathematics. I'm aware that this is an entirely subjective question, but how does abstract algebra rate on this "interest scale"? Do people find it more fun than calculus?
My ultimate goal, after university, is to become a theoretical physicist, and I've heard that both calculus (analysis?) and algebra are very important to this field. Any information on the basics of abstract algebra, what it is, how it's used, and if possible, recommended texts on this field would be useful, and greatly appreciated.
So, based on all the information I have provided, what would you all suggest for me to move on to? Remember, analytic number theory is potentially an option too, but I fear it may be too advanced for me at this stage. Would abstract algebra be too advanced?
Any enlightenment would be grand.
Cheers,
Davin