Should I Become a Mathematician?

[mathwonk]

ok. but you have an intuitive idea of continuity, and the three big theorems just say:

1) if f is continuous on an interval, then the values of f form an interval also.

2) if f is continuous on a closed bounded interval, then the values also form a closed bounded interval.

well ok i got them to two theorems.

so then the question becomes, how do i define continuity precisely so as to make these intuitive theorems actually true?

the answer is to say that small changes in the inputs produce only small changes in the outputs. but this has to be made very precise, with letters for the degree of change in input (delta) and output (epsilon).

 

[Darkiekurdo]

But why do they use the absolute value?

 

[quasar987]

Try to answer that question for yourself by proving that |x|<a iff -a<x<a.

So writing |x|<a is just another more compact way of expressing the fact that x is btw -a and +a.

 

[Darkiekurdo]

Aha. Thank you! I understand.

 

[mathwonk]

if x is between -a and a, then since sin is always smaller than 1 in absolute value, then x sin(anything) is also between -a and a.

 

[Darkiekurdo]

Is it strange that after I read a chapter of Spivak's book I have no idea how to do the problems? And after I read the chapter again I still can't do the problems.

 

[symbolipoint]

Is it strange that after I read a chapter of Spivak's book I have no idea how to do the problems? And after I read the chapter again I still can't do the problems.

Which of Spivak's books? Some or many of us have not seen any of his books (although others of us certainly have - not me, though); we may like to know which is the one you are finding trouble understanding, in case we might like to obtain a copy to use for study. Is it his Calculus book which you find difficult?

All enthusiasts of Spivak's books, please give your discussions about this.

Also, something interesting - do a web search and you can find a wikipedia article on Michael Spivak.

 

[quasar987]

It is a bit strange, but not totally surprising since it's your first time dealing with higher math.

It could be that you are reading the book like a novel. You have to read it as slow as it takes for the material to sink in.

It could be that you are just not used to the level of the problems; as soon as a problem begins with "show that" or involves something more original than a direct application of one theorem in the book, you are lost. I think the cure to this for me was that there were hints at the end of the book. So after thinking hard for a while about each problem and getting nowhere, I would look at the hint and try again. The key is really to genuinely try hard to solve the problem on your own though. Because I believe the best time to learn is when you are convinced that you have explored all the possible ways to approach a problem and they all failed. In this situation, I found that after I look at the solution, I sticks.

As you progress, you will find that there is in fact a finite number of methods/tricks to solving problems that appear again and again. I found it helpful to make a list out of these tricks and systematically try them out on every problem.

Here's my list.

-multiply by the conjugate
-add 0
-multiply by 1
-exploit the properties of exp and log
-can I use an identity? (Gauss' sum, Bernoulli inequality, etc)
-factorisation
-triangle inequality
-put fractions on the same denominator
-decompose in a sum of partial fractions
-suppose W.L.O.G. (without loss of generality)
-change of variable
-proof by contradiction
-prove something that is equivalent but easier
-decompose the problem in a sum of smaller problems
-complete the square
-by induction

These will begin to make sense to you when you encounter them again and again.

 

[d_leet]

Which of Spivak's books? Some or many of us have not seen any of his books (although others of us certainly have - not me, though); we may like to know which is the one you are finding trouble understanding, in case we might like to obtain a copy to use for study. Is it his Calculus book which you find difficult?

All enthusiasts of Spivak's books, please give your discussions about this.

Also, something interesting - do a web search and you can find a wikipedia article on Michael Spivak.

I believe he was using Spivak's Calculus book. About the problems, they are not all easy and computational problems, many require proofs using theorems and techniques learned in the chapter. Some problems are harder than others, but I believe this is the case with practically any textbook. Most of Spivak's problems require more thought than those in books like Stewart's where the problems are almost entirely computations.

I do not find Spivak's problems overly difficult, but that is not to say that I understand how to do all of them (right now I am having difficulty with a problem in the continuity chapter). I would say not to take one person's issues the problems to be necessarily true for everyone, and that you should judge for yourself whether or not you like his book, just be warned that you should not expect Spivak's Calculus book to be like other such books, it is more rigorous and if you have not had experience with mathematical proofs before than you may have some difficulties doing the problems.

Forgive me if that made no sense at all, I'm sure I repeated myself about five times or so, hopefully that helped you out some.

 

[Darkiekurdo]

It is a bit strange, but not totally surprising since it's your first time dealing with higher math.

It could be that you are reading the book like a novel. You have to read it as slow as it takes for the material to sink in.

It could be that you are just not used to the level of the problems; as soon as a problem begins with "show that" or involves something more original than a direct application of one theorem in the book, you are lost. I think the cure to this for me was that there were hints at the end of the book. So after thinking hard for a while about each problem and getting nowhere, I would look at the hint and try again. The key is really to genuinely try hard to solve the problem on your own though. Because I believe the best time to learn is when you are convinced that you have explored all the possible ways to approach a problem and they all failed. In this situation, I found that after I look at the solution, I sticks.

As you progress, you will find that there is in fact a finite number of methods/tricks to solving problems that appear again and again. I found it helpful to make a list out of these tricks and systematically try them out on every problem.

Here's my list.

-multiply by the conjugate
-add 0
-multiply by 1
-exploit the properties of exp and log
-can I use an identity? (Gauss' sum, Bernoulli inequality, etc)
-factorisation
-triangle inequality
-put fractions on the same denominator
-decompose in a sum of partial fractions
-suppose W.L.O.G. (without loss of generality)
-change of variable
-proof by contradiction
-prove something that is equivalent but easier
-decompose the problem in a sum of smaller problems
-complete the square
-by induction

These will begin to make sense to you when you encounter them again and again.

Yes, this is the first time I am learning from a rigorous book. All the others just took the properties for granted. I like it, but it is hard.

And indeed, I have difficulty with problems where I have to prove something or show that something works like that. But I am able to do the problems after I get a hint. I just don't know how to start.

I will try to see if I can use your tips on problems.

Thank for you for all your responses!

 

[Darkiekurdo]

Does not being able to solve problems that involve proving something mean I am not going to succeed in mathematics/physics?

 

[quasar987]

Does not being able to do smashes the first time you play tennis mean you're never going to be able to play?

 

[Darkiekurdo]

No, but isn't proving a creative thing?

 

[quasar987]

IMO, the creativity part comes in when inventing new theories and figuring out results not yet known. In 99% of cases, proving propositions is usually simply a matter of applying definitions and manipulating logic to show your proposition is indeed consistent with the definition and theorems you previously proved.

"If only I had the theorems. Then I should find the proofs easily enough." --Riemann

 

[ekrim]

No, but isn't proving a creative thing?

Art is a creative thing, but you need to learn the fundamentals of brush strokes, color mixing, etc. before painting masterpieces. In the same way you need to development the proof fundamentals as tools through which you can express creativity

that being said, I picked up Rudin from the library today and have been trying to get some fundamentals by working his proofs. i haven't decided if its more arduous or humbling yet

 

[mathwonk]

try some of my books.

 

[ekrim]

try some of my books.

I actually looked up artin, but couldn't find him in the library, so i reached for the familiar name. any others you might recommend? the rudin isn't impossible though, i think id have the same trouble with anything theory based

 

[threetheoreom]

Is it strange that after I read a chapter of Spivak's book I have no idea how to do the problems? And after I read the chapter again I still can't do the problems.

Which of Spivak's books? Some or many of us have not seen any of his books (although others of us certainly have - not me, though); we may like to know which is the one you are finding trouble understanding, in case we might like to obtain a copy to use for study. Is it his Calculus book which you find difficult?

All enthusiasts of Spivak's books, please give your discussions about this.

Also, something interesting - do a web search and you can find a wikipedia article on Michael Spivak.

Yes he is using Spivak's Calculus book.

Maybe a bit strange, but please consider that spivak himself said some of the problems are so difficult ( marked by the asteriks ) that even the brightest students will have to be really interested to continue trying to solve them. One problem in the second chapter he says " if you have figured or looked up the answer" That said it is worth working through these problems because they not only enhance your conception of the topic being discussed but also teach you problem solving.


I recommend that you work out even the examples that he shows ever so clearly, because that clarity can fool you into think you understand such and such.

P.s i use the same book.

 

[morphism]

I like Herstein's view on the matter: "Many [problems] are introduced not so much to be solved as to be tackled. The value of a problem is not so much in coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver." (Taken from the preface to the first edition of Topics in Algebra.)

I think the same applies to Spivak's problems (many of which come from Courant, by the way); in fact, some are notoriously difficult. So don't feel disheartened if it takes you a lot of time and effort to do a small portion of them.

 

[threetheoreom]

I like Herstein's view on the matter: "Many [problems] are introduced not so much to be solved as to be tackled. The value of a problem is not so much in coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver." (Taken from the preface to the first edition of Topics in Algebra.)

Well said

(many of which come from Courant, by the way)

really i am not aware, also i think the larger amount of problems in the later editions transcended from the first in the 1960's. I am not familiar with the problems of Courant's book nor the publication date but i think Spivak's preceeds R. Courant book, again i am not sure.

 

[quasar987]

Courant is an old folk... When Spivak received his doctorate, Courant was 76.

http://en.wikipedia.org/wiki/Michael_Spivak
http://www.genealogy.ams.org/html/id.phtml?id=15162
http://en.wikipedia.org/wiki/Richard_Courant

 

[Darkiekurdo]

Thanks guys. I felt stupid because I couldn't do the problems after reading the chapter several times.

 

[threetheoreom]

Courant is an old folk... When Spivak received his doctorate, Courant was 76.

Okay thanks

 

[mathwonk]

From studying both books, (and having been a Harvard student myself), I believe Spivak may have learned from Courant, probably as a student in honors calc at Harvard, around 1960. Courant's book of course dates from the 1930's.And my suggestion to try reading some of "my books" for proofs, was a suggestion to try some of my free books from my webpage, which is visible in my public profile.

 

[teleport]

Does anyone know of "Undergraduate Algebra" by Lang, and "Advanced Calculus" by Taylor? These will be my 'main subject' books for the upcoming year. I think I will particularly like the first one. I read yesterday as much as I could (I was not able to avoid it!), and the explanations and proofs seem very clear. The solution of many exercises seem to use techniques used in the proofs of the theorems, a thing that I just love. It actualy makes me feel I learned a cute trick.

The size of the calculus book scares me. Would u recommend sticking to it or to look for another book? Note though, the first statement does not completely imply the question. Thanks

 

[ekrim]

From studying both books, (and having been a Harvard student myself), I believe Spivak may have learned from Courant, probably as a student in honors calc at Harvard, around 1960. Courant's book of course dates from the 1930's.


And my suggestion to try reading some of "my books" for proofs, was a suggestion to try some of my free books from my webpage, which is visible in my public profile.

ahh, ill definitely try that, thanks dr. mathwonk

 

[teleport]

Mathwonk, I will definitely use your elementary algebra notes as reference during the next semester. Thank you. Are your other algebra notes designed for an introduction into abstract algebra? This is what I'll have next year.

 

[mathwonk]

my math 843-844-845 notes are a detailed introduction for students who have studied matrices and determinants.

 

[threetheoreom]

I am using the 4000 notes, particularly the notes on polynomials.

 

[mathwonk]

the 4000 notes are for our intro to algebra course, but we precede that course by an intro to proofs course (3200) where i tend to pre teach much of the same stuff, at least the elementary number theory part.

they also have usually had alinear algebra course from adams and shifrin beforehand (math 3000). then there is 4010 course introducing groups.
afterwards students should be ready for my 843 notes, but my 843 notes are pretty self contained on groups, so could be an introduction to them.

i do use matrices though, and only teach them later, in the 845 notes.

how are the 4000 notes going? those were actual class notes as handed out, and not rewritten, so may lack some organization or editing.

heres the catalog descrioption:
MATH 4000/6000. Modern Algebra and Geometry I. 3 hours.
Oasis Title: MOD ALG & GEOM I.
Undergraduate prerequisite: (MATH 3000 or MATH 3500) and (MATH 3200 or MATH 3610).
Abstract algebra, emphasizing geometric motivation and applications. Beginning with a careful study of integers, modular arithmetic, and the Euclidean algorithm, the course moves on to fields, isometries of the complex plane, polynomials, splitting fields, rings, homomorphisms, field extensions, and compass and straightedge constructions.
Offered fall, spring, and summer semesters every year.
MATH 4010/6010. Modern Algebra and Geometry II. 3 hours.
Oasis Title: MOD ALG & GEOM II.
Undergraduate prerequisite: MATH 4000/6000.
More advanced abstract algebraic structures and concepts, such as groups, symmetry, group actions, counting principles, symmetry groups of the regular polyhedra, Burnside's Theorem, isometries of R^3, Galois Theory, and affine and projective geometry.
Offered spring semester every year.

 

[teleport]

Abstract algebra, emphasizing geometric motivation and applications. Beginning with a careful study of integers, modular arithmetic, and the Euclidean algorithm, the course moves on to fields, isometries of the complex plane, polynomials, splitting fields, rings, homomorphisms, field extensions, and compass and straightedge constructions.

Yeah, we will have almost all of this plus groups. Unfortunately, we will not have constructions. Damn, when will someone show me with detail why the circle can't be squared! My problem is that I'm not seeing any, not even convergence of classical geometry with other modern subject. I've looked into my future courses, and classical geometry seems to be ignored. Is this becoming common in many universities, or is it just mine?

BTW, we will also cover Galois theory! (I think at a fundamental level, but don't know)

 

[mathwonk]

since circles and lines have equations of degres 2 or 1, it turns out that the coordinates of points obtained by intersecting them, satisfy equations of degree 2 or 1 over the field generated by the coordinates of the points determining the lines and circles themselves.

thus constructible points have coordinates which lie in an extension of Q which is composed of successive quadratic extensions. by multiplicativity of degree of field extensions, this means they lie in field extensions of degree 2^n for some n.

hence points whose coordinates satisy irreducible cubics/Q for instance cannot be constructed. this is why an angle of 20degrees (i believe) cannot be constructed, so one cannot trisect a 60degree angle.

similarly a point whose coordinates do not satisfy any rational equations at all, such as pi, cannot be constructed. this is why a circle cannot be squared. the detailed proof is in jacobson's algebra book, complete with a proof that pi is transcendental.

 

[teleport]

Wow what a mouthfull! Well, if part of the general purpose of providing the explanation is to motivate us in the studies of abstract algebra to get answers to such problems, then u got me there. I would absolutely love to understand all of this which was mentioned.

 

[ekrim]

Hey Mathwonk, I'd greatly appreciate if you could tell me what's the standard/classic text for PDE's. Unfortunately my professor teaches by copying directly from the book (McOwen) to the board. It's a little weak on theory and leaves me unsatisfied.

 

[mathwonk]

i am ignorant in pde, but i myself like vladimir arnol'd's books, and i personally have his text on pde.

i gather there is no systematic theory of pde's as there is for ode's, so one studies the classically important special cases, like: heat equation, wave equation, and laplace equation.

i myself have studied the (several variables complex) heat equation quite a bit, and of course the laplace equation is important in all complex analysis, since both real and imaginary parts of holomorphic functions satisfy it. harmonic functions are also important in geometry.

but i know nothing about the wave equation.

but i recommend arnol'd for auxiliary reading in any course.