Preparing for Analysis: Is My Background in Calculus Sufficient?

[stefan10]

I have been accepted to Carnegie Mellon University, and I plan to major in Physics and possibly Mathematics as well.

I am interested in a specific Honors course. Information can be found on page 9 of this pamphlet.

http://www.cmu.edu/mcs/undergrad/advising/forms/SchedulingInfo-fall.pdf

This is a course labeled as Analysis, which uses the textbook Calculus, Volume 1, Second Edition, by Apostol. I expect to receive a 5 on my AP Calculus AB(currently I score 4s on practice tests, and I'm only a few points away from a 5.) I also plan to study the last three chapters of my textbook, so that I can answer the rest of the integral calculus questions on Carnegie Mellon's Calculus placement exam. Is this a sufficient amount of background for this course's rigor? I want to start proof-based classes as soon as possible. Since I am pretty good at doing calculations, I'm sure a more theoretical approach would be fun and entertaining. I am a hard worker, I am very enthusiastic about mathematics, and I will study a lot. Although I'm not anybody special, and it does take me time to learn something, but once I learn something it sticks for good -- I rarely forget it. Can anybody reveal the rigor of this textbook and whether or not I should be prepared with my current abilities in AP Calculus AB? I want to know now so that I can decide whether or not to buy it and start studying it a bit on my own over the summer break. Thank you!

 

[Broccoli21]

Apostol's Calculus is a great book. I myself learned all my calculus from it. If you have done well in AP calc, then you should have no problem with the class. Apostol is usually used as a stepping stone into more advanced analysis (the next book is usually baby Rudin). To get used to proofs, it might be a good idea to pick up the book early and take a look. Have fun!

 

[DeadOriginal]

It will be difficult as writing proofs the first time around is usually very tough especially when you are fresh out of high school. Mathwonk in his thread called who wants to be a mathematician? talked about this. A lot of students do amazingly in AP calculus in high school and all of a sudden they find themselves failing university mathematics.

I'm a firm believer in nothing is impossible though so if you set your mind to it, I think you'd do great.

 

[stefan10]

Alright, so regardless of whether or not I take the class, buying Apostol's book would be a good investment.

I'm leaning towards taking the course at this point. I figure, I'll have to learn how to construct proofs sooner or later. I'd rather learn it sooner than later so I have more time to develop my abilities. I find this aspect of mathematics much more interesting than the plug in and solve methods learned in high school anyway, and usually when I find something very interesting I work more efficiently to learn it. So I'm confident I can do well. This is especially since I expect to put almost all of my time into this and my intro physics course. I just want to make sure that this is realistically possible to do well in. I don't care about if it will be hard to do well in, but if I can possibly do well in it. I expect all my classes will be at least moderately difficult, and I also expect to study many hours, actually I prefer to study many hours - it is fun to do.

Is this generally what I'll be doing in this type of class? Writing proofs like these?

http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx

Do you think I'll be expected to come up with unique proofs? Or just to know what the ones we are taught mean, and how to write them?

In the pamphlet this is within the course description:

Thus, the motivation to master the construction of mathematical proofs is a key pre-requisite.

I assume that means I will be taught to construct proofs, rather than be required to learn how on my own or in a previous class? I just need to understand this is a requirement, and I must give all of my efforts to mastering this skill, correct?

Edit: Looking at their website, I think this is outdated. It seems as if they changed their entire course structure. Instead there is a class specifically for learning how to write proofs and an advanced honors course called "Matrix Theory" which I assume is similar to an introductory Linear Algebra course. Thanks though, I think I might get the book anyway just to learn for fun. :)

 

[micromass]

Yes, it is realistically possible to do well in the course. However, if you never did proofs before, then there might be some rough bumps along the way.

You will probably be expected to:
- be able to read and understand quite complicated proofs.
- be able to construct a not-too-complicated proof.
- be able to write the proof in a mathematical way.

You will likely be taught how to do proofs along the way. You might however do some reading during the summer to help ease the process.

In general: the more proofs you read and really understand, the better you will be at it. And with reading, I mean: active reading. When reading a proof - any proof - I always ask myself the following:
- Where did I use every hypothesis?
- Is the theorem true if I weaken a hypothesis?? Find a counterexample if not.
- What are the main techniques of the proof?
- How can I use the techniques in later proofs?
- Is the converse of the theorem true?
- What previous results were important in the proof?
- Where do I use the theorem later on? Is it just theoretical importance?
- Can I apply the theorem on an example??
etc.

If you read every proof with these questions in mind, then you will learn how to deal with rigorous mathematics very fast! Learning will go quite slow however, but it's really worth it.

 

[Mépris]

Yes, it is realistically possible to do well in the course. However, if you never did proofs before, then there might be some rough bumps along the way.

Have you heard of the book, "How to Prove It: A Structured Approach" by Daniel J. Velleman? If yes, I would be interested in knowing your opinion about it. :-)

I was told that it would help ease one's way into proof-based mathematics. My question was to familiarise myself and be able to do Olympiad Mathematics questions but my educated guess would be that this book might help the OP.

 

[micromass]

Have you heard of the book, "How to Prove It: A Structured Approach" by Daniel J. Velleman? If yes, I would be interested in knowing your opinion about it. :-)

I was told that it would help ease one's way into proof-based mathematics. My question was to familiarise myself and be able to do Olympiad Mathematics questions but my educated guess would be that this book might help the OP.

I'm not a real fan of proof books. A proof book separates proofs from its applications in mathematics. I don't like this. I feel that one should do proofs while actually doing it in a mathematical theory.

I don't believe for a second that this book well help you with olympiad questions.

I would always substitute this book for a decent logic/set theory book.

Other people liked this book however, so my opinion is simply an opinion.

 

[Mépris]

Deleted my former post, as I thought of something else some time after.

Well, I was quite off put because of that as well and I find what you said to be reasonable. Now, I've taken a look again at the book's table of contents and it seems to do what you described would be good: "proofs while actually doing it in a mathematical theory".

There are seven chapters in the book. Namely: Sentential Logic, Quantificational Logic, Proofs, Relations, Functions, Mathematical Induction and Infinite Set. Does this not look like a "logic and set theory" book, then?

Note: I only ask for I am not familiar with any of these things and trying to make sense of them by just looking through the table of contents would not be possible until I've actually studied the said topics from a few different sources. Sorry if my question sounds stupid or something.

 

[920118]

Now, I've taken a look again at the book's table of contents and it seems to do what you described would be good: "proofs while actually doing it in a mathematical theory".

There are seven chapters in the book. Namely: Sentential Logic, Quantificational Logic, Proofs, Relations, Functions, Mathematical Induction and Infinite Set. Does this not look like a "logic and set theory" book, then?

Velleman isn't a book on set theory or logic. It just introduces basic logical and set theoretical concepts so the reader can try to prove really rudimentary things. It's on a very basic level. I would say Eccles' "An Introduction to Mathematical Reasoning" is a better introduction to proofs. It's pretty basic as well, but is more mathy and less "this is yet another example of how you prove that there is no function from A onto P(A)".

 

[stefan10]

Well it seems this Analysis course has been replaced(pamphlet was from 2008) with these courses. Then during spring one takes Vector Analysis, and in sophomore year one takes Honors Real Analysis and Algebra courses with two semester parts. Other than the bulkiness of this in my course-load with the addition of my physics courses and general requirements, I find this to seemingly be a better structure.

Concepts of Mathematics
Fall or Spring: 9 units
By the end of this course the student should be able to

Construct logically correct proofs using basic proof techniques such as proof by contradiction.
Deploy basic problem solving strategies.
Truth values, connectives, truth tables, contrapositives. Quantifiers. Proof by contradiction. Sets, intersections, unions, differences, the empty set. Integers, divisibility. Proof by induction. Primes, sieve of Eratosthenes, prime factorization. Gcd and lcm, Euclid's algorithm, solving ax + by =c. Congruences, modular arithmetic. Recursion. Linear recurrences. Functions and inverses. Permutations. Binomial coefficients, Catalan number. Inclusion-exclusion. Infinite cardinalities. Binary operations. Groups. Binary relations, equivalence relations. Graphs. Euler characteristic, planar graphs, five color theorem, rationals, reals, polynomials, complex numbers.

and

Matrix Theory
Spring and Fall: 10 units
An honors version of 21-241 (Matrix Algebra and Linear Transformations) for students of greater aptitude and motivation. More emphasis will be placed on writing proofs. Topics to be covered: complex numbers, real and complex vectors and matrices, rowspace and columnspace of a matrix, rank and nullity, solving linear systems by row reduction of a matrix, inverse matrices and determinants, change of basis, linear transformations, inner product of vectors, orthonormal bases and the gram-schmidt process, eigenvectors and eigenvalues, diagonalization of a matrix, symmetric and orthogonal matrices, hermitian and unitary matrices, quadratic forms. 3 hrs. lec., 1 hr. rec.