[SOLVED] Analysis, Topology, and Differential Geometry too broad?

[Klaas van Aarsen]

I've been somewhat confused about the categories in university math.

The category Analysis, Topology, and Differential Geometry seems a bit too broad to me.
I consider Real, Complex, and Functional Analysis as quite different from Topology and Differential Geometry.
In my opinion Analysis is much closer to Calculus than to Topology or Differential Geometry.
It makes me confused as to where a new thread should be placed.

How do you feel about revising those categories?

 

[Jameson]

How do you feel about revising those categories?

I'm always for revising/tweaking/improving!

So the question now is what is the proposed revision? Add "Analysis" onto "Calculus" or give it its own subforum?

 

[MarkFL]

I would give Analysis its own sub-forum, as a means of keeping elementary calculus and advanced calculus separate.

 

[Klaas van Aarsen]

Both would be good.

I'm reserving my opinion depending on more responses.
At this time I'm not so sure we should make a difference between elementary calculus and advanced analysis.
Don't we already have the difference between pre-university and university?

 

[Fantini]

I'd be in favor of giving Analysis its own sub-forum. Typically, while there are pretty hard Calculus' problems, they involve messy and lengthy calculations, whereas Analysis questions tend to center around controlling epsilons, majoring functions and working with inequalities, and that's a different skill. While they are both essential, they are distinct enough to warrant different places for discussion. It also keeps the forum neater.

Cheers.

Fantini

 

[Klaas van Aarsen]

I'd be in favor of giving Analysis its own sub-forum. Typically, while there are pretty hard Calculus' problems, they involve messy and lengthy calculations, whereas Analysis questions tend to center around controlling epsilons, majoring functions and working with inequalities, and that's a different skill. While they are both essential, they are distinct enough to warrant different places for discussion. It also keeps the forum neater.

Currently the description of Calculus contains "Advanced Limits".
Doesn't that imply epsilons?

 

[MarkFL]

...
At this time I'm not so sure we should make a difference between elementary calculus and advanced analysis.
Don't we already have the difference between pre-university and university?

We have a Pre-Calculus forum in the Pre-University Math area, but I tend the think of the Calculus forum in the University Math area as being solely for Calc I - Calc III topics.

 

[Klaas van Aarsen]

We have a Pre-Calculus forum in the Pre-University Math area, but I tend the think of the Calculus forum in the University Math area as being solely for Calc I - Calc III topics.

To be honest, I do not know what that means...
I'm not a native American, does that matter?

 

[Fantini]

Currently the description of Calculus contains "Advanced Limits".
Doesn't that imply epsilons?

For limits, but what about other kinds? In measure theory there's plenty of results and techniques involving epsilons that aren't "limits". Some of the most important results in measure theory, which belong to real analysis, show the necessity of mastering inequalities and 'epsilons', such as the Monotone Convergence Theorem and the Dominated Convergence Theorem. I believe it's pretty vast enough to deserve its own home.

We have a Pre-Calculus forum in the Pre-University Math area, but I tend the think of the Calculus forum in the University Math area as being solely for Calc I - Calc III topics.

Yes, but should we be so strict? I mean, there are many problems in other "areas" that could belong to the Calculus merely because it requires techniques and mindset of those, even if they didn't come from that specific setting. For example, let's assume you had to do a fluid mechanics problem and you had to use Stokes' theorem, in $\mathbb{R}^3$. You got the physics figured out, you're just having trouble doing the calculations. I'd say it belongs in the Calculus forum (even citing it as one good application of things we learn in usual courses).

Cheers!

Fantini

 

[MarkFL]

My apologies...perhaps my post was assuming U.S. educational trends be followed.

When I was a student, the course outlines were basically:

Calculus I:

• Functions and limits (including the epsilon-delta definition)
• The derivative
• Applications of the derivative
• The integral
• Applications of the integral
• Logarithmic and exponential functions

Calculus II

• Inverse trigonometric and hyperbolic functions
• Techniques of integration
• Indeterminate forms and improper integrals
• Sequences and series
• Analytic geometry in the plane
• Parametric equations and polar coordinates

Calculus III

• Vectors and 3-space
• Vector-valued functions
• Differential calculus of functions of several variables
• Integral calculus of functions of several variables
• Vector integral calculus
• Survey of differential equations

These are the topics I have come to think of as fitting the Calculus forum, but of course we cannot be rigid or strict about it, there must be leeway.

 

[Klaas van Aarsen]

Erm... all of these are pre-university where I come from, except for multi-variable calculus...

I guess it does matter where I come from.

 

[MarkFL]

Many of those topics are studied in high school (secondary school) in the U.S. as part of AP (advanced placement) courses, but most students here do not take any calculus until they are in college.

 

[Fantini]

Here in Brazil we don't even have pre-university calculus courses (or any pre-university courses for that matter), neither AP. Also, the Calculus I-III courses I had were a bit different from Mark's.

Calculus I comprised all of his points up to indeterminate forms and improper integrals. Calculus II was the whole multivariable calculus, including parametric equations, polar, cylindrical and spherical coordinates. Finally, Calculus III works with sequences, series, lots of differential equations (Riccati, Euler, Bernoulli, etc), an introduction to PDEs (basically the heat equations), Laplace Transforms, Frobenius and Fourier series.

Analytic geometry in the plane and in space were handled in a separate subject, done concurrently with Calculus I.

 

[Klaas van Aarsen]

In high school I had Math I and Math II.
Math I included all the calculus topics except multi-variable and 3D.
Math II was all about 3D linear algebra.

In university we had Analysis I, II, III, IVa, and IVb.
Analysis I was a repetition and extension of what we had in Math I.
Analysis II was about multi-variable calculus, Green, Gauss, and Stokes.
Analysis III was about complex analysis.
Analysis IVa was about measure theory (specifically for math students).
Analysis IVb was about Fourier transforms (specifically for physics students).
Being the disturbed person that I am, I followed both Analysis IV courses.

 

[Klaas van Aarsen]

To extend, in university I got:
- Algebra I
- Algebra II
- Linear Algebra I
- Linear Algebra II

 

[Bacterius]

Well from where I come from (a weird france/new zealand hybrid) we have:

high school:
- algebra, functions, sequences and series
- introductory calculus (derivatives, integrals up to integration by parts, limits, l'Hopital's rule, etc..)
- complex numbers (though no complex calculus)
- extremely basic linear algebra (solving 2 or 3 simultaneous linear equations, optimization problems, that sort of thing)

university:
- hyperbolic functions
- differential equations
- linear algebra (introductory, and then advanced)
- more difficult integrals, and then vector calculus, and multivariate calculus
- all the rest that's too advanced for me to even know about

So pretty much the same as MarkFL's outlines, with the difference that we were introduced to sequences and series much earlier for some reason.

 

[Klaas van Aarsen]

For me l'Hôpital's rule (yes, I'm close to France) and complex numbers came at university.
Introductory linear algebra and differential equations came at high school.

 

[Jameson]

This is all great information and now the question is how to best use it.

On one hand it might be tempting to just do away with the labels of "pre-university math" and "university" math but I think that would be a bad idea. It's nice to have some separation between more basic topics and more advanced topics and I fear that without the current two main categories more people would incorrectly place their threads. For example many students might think that "Linear Algebra" is something related to "basic algebra" that is just learning to solve for some amount of unknowns without going into the theory of it that linear algebra does. The same thing could happen (and has happened here) with basic versus advanced statistics.

Now that said I'm still open to disagreement. Should we still consider changing the main distinction of "pre-university" and "university" topics for all subforums? Or should we consider adding a third section for topics that can be considered both pre-university and university? Or should we go around route?

Let's make a plan on how to address this issue and probably make MHB a lot more internationally friendly.

 

[Fantini]

This is all great information and now the question is how to best use it.

On one hand it might be tempting to just do away with the labels of "pre-university math" and "university" math but I think that would be a bad idea. It's nice to have some separation between more basic topics and more advanced topics and I fear that without the current two main categories more people would incorrectly place their threads. For example many students might think that "Linear Algebra" is something related to "basic algebra" that is just learning to solve for some amount of unknowns without going into the theory of it that linear algebra does. The same thing could happen (and has happened here) with basic versus advanced statistics.

Now that said I'm still open to disagreement. Should we still consider changing the main distinction of "pre-university" and "university" topics for all subforums? Or should we consider adding a third section for topics that can be considered both pre-university and university? Or should we go around route?

Let's make a plan on how to address this issue and probably make MHB a lot more internationally friendly.

I don't believe we should change from "pre-university" and "university" for all subforums. Besides, the original question (and still the most important) is the if "Analysis, Topology and Differential Geometry" is too broad, which the question remains opened if we should split Analysis and give its own subforum (I say yes). Perhaps we could reconsider the pre-university/university on a later debate, should it arise again. For now, I point out that Analysis is big enough to warrant its own place.

 

[Klaas van Aarsen]

I don't believe we should change from "pre-university" and "university" for all subforums. Besides, the original question (and still the most important) is the if "Analysis, Topology and Differential Geometry" is too broad, which the question remains opened if we should split Analysis and give its own subforum (I say yes). Perhaps we could reconsider the pre-university/university on a later debate, should it arise again. For now, I point out that Analysis is big enough to warrant its own place.

That would be fine by me.

 

[MarkFL]

I don't believe we should change from "pre-university" and "university" for all subforums. Besides, the original question (and still the most important) is the if "Analysis, Topology and Differential Geometry" is too broad, which the question remains opened if we should split Analysis and give its own subforum (I say yes). Perhaps we could reconsider the pre-university/university on a later debate, should it arise again. For now, I point out that Analysis is big enough to warrant its own place.

This works for me too, my main issue was keeping analysis separate from calculus.

 

[Jameson]

There are 13 pages of threads in the current Analysis, Topology and Differential Geometry subforum. If I create a new subforum, Analysis, we will need at least a couple of volunteers to help move the current threads over.

Any takers?

 

[Klaas van Aarsen]

There are 13 pages of threads in the current Analysis, Topology and Differential Geometry subforum. If I create a new subforum, Analysis, we will need at least a couple of volunteers to help move the current threads over.

Any takers?

Sure!

 

[ZaidAlyafey]

There are 13 pages of threads in the current Analysis, Topology and Differential Geometry subforum. If I create a new subforum, Analysis, we will need at least a couple of volunteers to help move the current threads over.

Any takers?

Willing to help , if my position permits !

 

[MarkFL]

Count me in too!