What exactly is Advanced Calculus?

[MathWarrior]

What exactly is "Advanced Calculus?"

My school lists this class with the description of:

Advanced Calculus I
"real numbers, completeness, continuity, sequences/series, uniform theorems, derivative, pointwise/uniform convergence, Riemann integral, Taylor's theorem."

as the description what can i expect from it?

 

[fss]

...is this a trick question? The course description doesn't seem very ambigious.

 

[micromass]

What kind of answer do you expect from us? Advanced calculus seems to be like a bridge between calculus and real analysis...

 

[holomorphic]

My school lists this class with the description of:

Advanced Calculus I
"real numbers, completeness, continuity, sequences/series, uniform theorems, derivative, pointwise/uniform convergence, Riemann integral, Taylor's theorem."

as the description what can i expect from it?

I would find out what textbook they use for the class. That will tell you most of what you need to know.

In some places, Advanced Calculus is actually multivariable calculus. In others, it's an introduction to real analysis. In others (for instance, my institution) there is an easier "introduction to analysis course," and then advanced calculus is what you take if 1) you've taken an introductory analysis course or 2) you don't need the introductory course.

EDIT: from the unambiguous description, yours sounds like an "introduction to analysis" variety.

 

[mathwonk]

I agree. That stuff is taught in elementary calculus some places (like my honors calc class). Real advanced calculus is (ACCORDING TO SPivak) three theorems:

1) the inverse/implicit function theorem,
2) Fubini's theorem, and
3) stokes theorem

/as i recall. oh yes and it helps a lot to know about differential forms.

 

[MathWarrior]

What kind of answer do you expect from us? Advanced calculus seems to be like a bridge between calculus and real analysis...

...is this a trick question? The course description doesn't seem very ambigious.

Yes, I felt the same way that is why I asked.

I would find out what textbook they use for the class. That will tell you most of what you need to know.

Looks like they use a book on intro to analysis.

 

[hitmeoff]

Looks like they use a book on intro to analysis.

So in Calc I-III you learned how to do calculus (ie. you learned how to take a limit, take a derivative and take an integral). Now you learn how calculus actually works. You begin with building up the real number from the integers, then you learn how sequences work, learn what the limit of a sequence is (rigorously), how to prove (not just calculate) if a sequence has a limit, learn about continuous function, proving limits of continuous functions and eventually leading up to differentiation and integration.

You do all these things in the context of Axioms, Theorems, and Proofs. Basically, instead of calculating you now work on PROVING.

 

[Skrew]

So in Calc I-III you learned how to do calculus (ie. you learned how to take a limit, take a derivative and take an integral). Now you learn how calculus actually works. You begin with building up the real number from the integers, then you learn how sequences work, learn what the limit of a sequence is (rigorously), how to prove (not just calculate) if a sequence has a limit, learn about continuous function, proving limits of continuous functions and eventually leading up to differentiation and integration.

You do all these things in the context of Axioms, Theorems, and Proofs. Basically, instead of calculating you now work on PROVING.

A significant number of books don't bother building the reals from the integers.

 

[Ryker]

You do all these things in the context of Axioms, Theorems, and Proofs. Basically, instead of calculating you now work on PROVING.

I thought this is what you do in Calc I - III anyway

 

[Dembadon]

My school lists this class with the description of:

Advanced Calculus I
"real numbers, completeness, continuity, sequences/series, uniform theorems, derivative, pointwise/uniform convergence, Riemann integral, Taylor's theorem."

as the description what can i expect from it?

My university has a course with a description that's very similar; it's called Introduction to Analysis (Math 310). It is a recommended pre-requisite for many of the junior and senior level mathematics courses.

 

[Shackleford]

My university has Intermediate Analysis and Advanced Multivariable Calculus.

MATH 3333: Intermediate Analysis
Cr. 3. (3-0). Prerequisite: MATH 2433. Properties of real number system, properties of continuous functions, and sequences of functions.

MATH 3334: Advanced Multivariable Calculus
Cr. 3. (3-0). Prerequisite: MATH 3333. Topology of n-space; continuity, uniform continuity and uniform convergence; extremal problems; Taylor's theorem for multivariate functions; multiple integrals; transformations and their derivatives; implicit functions.

MATH 4331;4332: Introduction to Real Analysis
Cr. 3 per semester. (3-0). Prerequisite: MATH 3334 or consent of instructor. Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals.