What Does Advanced Calculus Include and What Are the Best Books for Learning It?

[relinquished™]

I'm a BS Math Student Majoring in Actuarial Science, finished with math subjects like Differential Calculus, Statistics, and Linear Algebra, currently studying Integral Calculus, Math Logic, and Probability Theory in my third year at College. By the time I graduate, I've already have finsihed Real and Numerical Analysis, Life Contengiencies and the like. My new professors suggest that I take up graduate school before I take up any Actuarial Science Exams because the current curriculum isn't sufficient enough for us to passthe exams to become Actuaries. I was thinking I should just do some self study on the subjects I'm not familiar with, particularly math subjects involving "Advanced Calculus". I'm not sure what Advanced calculus means because the term Advanced may be different to other people.

1.) Can anyone tell me what the scope of "Advanced Calculus" is?
2.) Are there any good books out there that can help me learn "Advanced Calculus"? I'm particularly interested in books that show definitions and proofs that are "easy to digest" with moderate exercises to solve.

 

[Gokul43201]

This thread belongs in General Math (or Calculus). Can someone move it there ?

 

[Kalimaa23]

Advanced calculus can mean just about anything.
Are you familiar with variational calculus and complex analysis? If not, I'd work on that first.

 

[bfd]

I'm a BS Math Student Majoring in Actuarial Science, finished with math subjects like Differential Calculus, Statistics, and Linear Algebra, currently studying Integral Calculus, Math Logic, and Probability Theory in my third year at College. By the time I graduate, I've already have finsihed Real and Numerical Analysis, Life Contengiencies and the like. My new professors suggest that I take up graduate school before I take up any Actuarial Science Exams because the current curriculum isn't sufficient enough for us to passthe exams to become Actuaries. I was thinking I should just do some self study on the subjects I'm not familiar with, particularly math subjects involving "Advanced Calculus". I'm not sure what Advanced calculus means because the term Advanced may be different to other people.

1.) Can anyone tell me what the scope of "Advanced Calculus" is?
2.) Are there any good books out there that can help me learn "Advanced Calculus"? I'm particularly interested in books that show definitions and proofs that are "easy to digest" with moderate exercises to solve.

Hello, I'm a recent graduate and studying for the first actuarial exam myself. If you are currently in an actuarial science program then you should have more than enough background to take the first exam. I would highly suggest that you buy a study manual for whatever exam you are interested in taking. I believe the actuarial bookstore still sells such manuals. (www.actuarialbookstore.com)

As for the rest of the exams I'm not to sure. At my old university, somewhere on the department of statistics website (www.pstat.ucsb.edu) they give a chart that matches up current classes offered and what actuarial exams they correspond to. If I remember correctly there wasn't any graduate classes on that list. Hope this helps

 

[relinquished™]

First, I would like to say sorry for placing the thread in the wrong topic/forum.

Dimitri, I am not familiar with variational calculus. As for complex analysis I'm sure we'll take that up as a subject next year.

bfd, thanks for the useful info. I just checked up the site www.pstat,ucsb.edu[/URL] and I realize I need a whole lot of relearning to do. Additional question: May I know what was the scope of your actuarial science program, i.e., the subjects you took up as an undergraduate?

 

[HallsofIvy]

"Advanced Calculus" often gets used in two distinctly different ways. Some courses in Advanced Calculus are aimed at applications (I've seen books titled "Advanced Calculus for Engineers". They tend to go deeper into multi-variable calculus (Green's and Stokes' theorems), differential equations (including partial differential equations), and "special functions" (Bessel functions, etc.). Those tend to be at colleges that have strong engineering schools.

Colleges that tend to be more "liberal arts" tend to have "Advanced Calculus" courses that are more "Mathematical Analysis"- that is, emphasizing the theory of calculus. Those courses would include basic metric spaces, compact sets, proof of the intermediate value theorem for continuous functions, proof of the extreme value theorem for continuous functions, theorems on Riemann integration, Riemann-Stieljes integration, and possibly Lebesque integration.

Quite often, colleges that have a more "applications oriented" Advanced Calculus course will have a separate "Analysis" course.

 

[Wong]

Hello, relinquished. Are you taking the SOA exams? If you are, I reallly doubt there is any need for "advanced" calculus...SOA exams are famous/infamous for its generally "uncreative" questions. They won't test you on how to derive/prove equations. Most often, you just need to remember a bunch of equations, plug x into here and y there and you get the answer.

For course 1, you rarely need anything beyond the calculus you learn in high-school. Maybe you need to do a bit of double integrals, but that is easy to handle.

 

[bfd]

First, I would like to say sorry for placing the thread in the wrong topic/forum.

Dimitri, I am not familiar with variational calculus. As for complex analysis I'm sure we'll take that up as a subject next year.

bfd, thanks for the useful info. I just checked up the site www.pstat,ucsb.edu[/URL] and I realize I need a whole lot of relearning to do. Additional question: May I know what was the scope of your actuarial science program, i.e., the subjects you took up as an undergraduate?[/QUOTE]

Well I was actaully an econ/math major though it was not a double major. It was simply a combination of both subjects. I remember reading somewhere that some of the same books I used for one of my upperdivision econ classes was used as a reference book for the fourth actuarial exam. That class was unofficially labled a grad school class so it would make a some sense to me why your professors would suggest taking such advanced classes.

But even if you don't take them you're more prepared than many other people out there who take such exams. I've heard of many people who even hold degrees in such subjects as music and history who become actuaries. The point is that its recommended you have a somewhat good exposure to calculus and some upperdivision probability courses. Given your situation I'd say you're right on track.

 

[gravenewworld]

Advanced Calculus is where the men are separated from the boys. AC is mostly working with proofs, you will not be taking derivatives and integrals of polynomials etc. The concept of the derivative is one of the last things you will see in AC. First you might work with the idea of a metric space, then you will explore the ideas behind the real numbers. After that, you will be PROVING limits, not simply finding a limit. The thing that everyone always remembers about AC are epsilon delta proofs. I don't know if there really are AC books with proofs that are very "easy" to digest since all the ideas are very deep. It all depends though on what is "easy" to you.

 

[relinquished™]

First, I would like to thank everyone for their advice and replies.

HallsOfIvy, from the two "categories" of "Advanced" Calculus, it seems our college is more of the "liberal art" type, since we concentrate on the theories of Calculus. In fact, our math subjects are under "Mathematical Analysis"

Wong, I have no plans of taking the SOA exams, or any actuary exam for that matter, yet, not until I'm at least in fourth year, when I finally take up those subjects concentrated in actuarial research and the like (the only subject that focuses on that that I'm taking up right now, to me, is Theory of Probability). I don't want to rush things yet until I know I'm fully prepared. Also, from what you said, it looks like the SOA questions are "plug" and "chug" kind of questions.

gravenewworld, the idea of Advanced Calculus you told me sounds interesting. If I ever decided to try to take up graduate studies, I'll look foward into that "Advanced" Calculus.

bfd, it's an amazing fact to hear that professionals holding music and history degrees can take up and pass the actuary exam. It's kind of like a ray of hope ^_^. But my professors are still worried about the "insufficient" curriculum of math we students have. You see, I'm currently studying in the Philippines, so I'm not really sure if the curriculum in the United States or anywhere else in the world is better than the actuarial program here. It's not that I'm insulting or degrading my college, it's just that if I ever decide to study abroad (which I am contemplating at), I might go into a sort of "educational shock" because there might be so many things I have yet to learn . I fear that kind of scenario, so while I'm still an undergrad I feel like I should at least study in advance. That's the main reason why I asked what your curriculum was. ^_^

Thanx again for the replies.

 

[Kalimaa23]

Advanced Calculus is where the men are separated from the boys. AC is mostly working with proofs, you will not be taking derivatives and integrals of polynomials etc. The concept of the derivative is one of the last things you will see in AC. First you might work with the idea of a metric space, then you will explore the ideas behind the real numbers. After that, you will be PROVING limits, not simply finding a limit. The thing that everyone always remembers about AC are epsilon delta proofs. I don't know if there really are AC books with proofs that are very "easy" to digest since all the ideas are very deep. It all depends though on what is "easy" to you.

They call this advanced calculus ??
I did this stuff in high school...

 

[fourier jr]

I say "Schaum's Outline of Advanced Calculus", "Div Grad Curl & All That" and "Intermediate Calculus" by Protter & Morrey are 3 good books on advanced calculus.

advanced calculus = Sequences and series of real numbers; sequences and series of real valued functions; uniform convergence; Fourier series; differentiation and integration of series of real valued functions; power series; Taylor series; Taylor's formula with remainder; multivariate calculus; implicit function, Stokes and divergence theorems. (according to my school anyway; that's the calendar entry ^ )