what is advance mathematics?

bogosort

Joined Sep 24, 2011
459
"Advanced" is in the eye of the beholder, but in terms of a university curriculum, a reasonable criterion is that advanced math courses are based on proofs rather than calculation. Your homeworks and tests in an advanced math class will very likely include more words than numbers. Some examples: linear algebra, abstract algebra, analysis (real/complex/functional), topology, geometry (algebraic/differential), number theory.
 

wayneh

Joined Sep 9, 2010
16,390
"Advanced" is in the eye of the beholder, but in terms of a university curriculum, a reasonable criterion is that advanced math courses are based on proofs rather than calculation. Your homeworks and tests in an advanced math class will very likely include more words than numbers. Some examples: linear algebra, abstract algebra, analysis (real/complex/functional), topology, geometry (algebraic/differential), number theory.
I recall just one time in my calculus course when our problem could actually be answered with an actual number, and I did so. The professor marked it down, even though he admitted my answer was correct AND I had shown all my work in getting to the answer. He explained that I should have stopped a step earlier, that mathematicians find numerical answers repugnant and devoid of meaning.
 

bogosort

Joined Sep 24, 2011
459
I recall just one time in my calculus course when our problem could actually be answered with an actual number, and I did so. The professor marked it down, even though he admitted my answer was correct AND I had shown all my work in getting to the answer. He explained that I should have stopped a step earlier, that mathematicians find numerical answers repugnant and devoid of meaning.
Sounds like you actually had a mathematician teaching the class; did you do proofs? My calculus courses were purely mechanical; we were told the theorems, shown the techniques, and plugged-and-chugged our way to the answer. (To be fair, my Calc I teacher usually included an extra credit question asking us to prove, or at least justify with words, some theorem or procedure.)
 

MrChips

Joined Oct 2, 2009
21,150
Like the difference between a scientist and an engineer.

An engineer just wants to get there. He doesn't care how you got there.

A scientist doesn't care where you got to. He wants to know how you got there.
 

wayneh

Joined Sep 9, 2010
16,390
Sounds like you actually had a mathematician teaching the class; did you do proofs? My calculus courses were purely mechanical; we were told the theorems, shown the techniques, and plugged-and-chugged our way to the answer. (To be fair, my Calc I teacher usually included an extra credit question asking us to prove, or at least justify with words, some theorem or procedure.)
He was definitely a true mathematician. I don't really recall doing proofs although I'm sure we were shown them. Working to get the answers is all I can remember. There were some doozies.
 

killivolt

Joined Jan 10, 2010
780
Working to get the answers is all I can remember. There were some doozies.
This reminds me way back when. Did you do them with a slide rule like I did?

I remember some Heat / Latent Heat / Super Heat Calculations we had to do in College, which I never used as a Heating Air repair Contractor, I never calculated anything, I just went into the repair end so all the Engineering went out the window.

I barely remember them, all I remember is you start with a few number on one side of a 8x11 which grew to fill the width at the bottom then began on the reverse side width at the top then back down to a few numbers. How I completed them I have no idea I just followed the rules without knowing the real why about how they work, all done with a slide rule.

kv
 

wayneh

Joined Sep 9, 2010
16,390
This reminds me way back when. Did you do them with a slide rule like I did?
Certainly not in math class - no numbers allowed. But we did plenty of calculations in Chem Engineering. I was amongst the first classes to use calculators instead of slide rules. I had both and knew how to use both, but of course quickly dropped the slide rule.
 

BR-549

Joined Sep 22, 2013
4,938
Advanced math is advanced ratios. All math is just ratios. Just a tool.

Advanced math can ratio apples and oranges. Nature can not.
 

Papabravo

Joined Feb 24, 2006
13,743
I have two candidates for Advanced Mathematics and I'd be surprised if anyone had studied either one:
  1. Stochastic Caclulus, including Ito's Lemma and the Feynman-Kac theorem
  2. Calculus of Variations
I learned the former and was completely humbled by the latter.
 

WBahn

Joined Mar 31, 2012
25,776
I have two candidates for Advanced Mathematics and I'd be surprised if anyone had studied either one:
  1. Stochastic Caclulus, including Ito's Lemma and the Feynman-Kac theorem
  2. Calculus of Variations
I learned the former and was completely humbled by the latter.
I never did stochastic calculus explicitly. I took a course on stochastic processes and stochastic signal processing. I think that is related, but probably as a subtopic.

I did calculus of variations in a couple of my senior physics courses, one dealing with mechancis and the other with electromagnetics. It was pretty interesting, but I would have to agree it was initially quite humbling.

So... surprise!!
 

wayneh

Joined Sep 9, 2010
16,390
The most advanced math I recall, and I call it that only because I never really ‘got it’, was differential equations. Oh wait, that stuff in P Chem was even worse. I got an A in both those classes but only because everyone else was more lost than I was.
 

WBahn

Joined Mar 31, 2012
25,776
The most advanced math I recall, and I call it that only because I never really ‘got it’, was differential equations. Oh wait, that stuff in P Chem was even worse. I got an A in both those classes but only because everyone else was more lost than I was.
I can relate. I got placed into P-chem the first semester of my freshman year but was only is second semester calculus. So I had never really dealt with differential equations (and was a year out of taking Diffy-Q) when all of a sudden we were having to deal with partial differential equations in P-chem. I definitely struggled in that course. Most of the other folks did as well, but I was way out on a limb in comparison to the majority. Fortunately one of the chem majors took some pity on this floundering freshman and we studied together. I ended up with a B in the course and was happy to take it.
 

killivolt

Joined Jan 10, 2010
780
Advanced math is advanced ratios. All math is just ratios. Just a tool.

Advanced math can ratio apples and oranges. Nature can not.
Can you explain why Nature cannot ? I can see that Advanced Math is dealing with Ratio's but how? is "All math just ratios"

I'm no Math Wizard so, speak in terms that a High School Student could understand.

kv
 

Papabravo

Joined Feb 24, 2006
13,743
I never did stochastic calculus explicitly. I took a course on stochastic processes and stochastic signal processing. I think that is related, but probably as a subtopic.

I did calculus of variations in a couple of my senior physics courses, one dealing with mechancis and the other with electromagnetics. It was pretty interesting, but I would have to agree it was initially quite humbling.

So... surprise!!
If it was going to be anybody, you would be at the top of the list.
 

jgessling

Joined Jul 31, 2009
82
You all left out symbolic logic. That’s a branch of math that consumed a year or so of my life back in the day. Bottom line to all that is that not everything is provable and I can prove it.

And speaking of slide rules, I like to think of myself as one of the last math teachers that covered slide rule methods in my class. That would be in Nkroful, Ghana. Fifth form maths preparing the students for their “O” level exams. 1976 and they all passed. Other fun topics we covered that are rarely seen in the US were combinatorics and tesselations. Combinatorics like there are 15 people at a party, 3 named Smith and 2 named Jones. If everyone shakes hands except people with the same name, how many hanshakes occur?
 

wayneh

Joined Sep 9, 2010
16,390
Other fun topics we covered that are rarely seen in the US were combinatorics and tesselations. Combinatorics like there are 15 people at a party, 3 named Smith and 2 named Jones. If everyone shakes hands except people with the same name, how many hanshakes occur?
There's not much time spent on probability or statistics of any kind. I got a big dose of it once I went to graduate school, and made good use of it on the job. That experience convinced me that it would have been nice to learn it much earlier. Maybe less time studying poetry or French history or whatever the hell else they tried to shove in our heads.
 

Papabravo

Joined Feb 24, 2006
13,743
I never did stochastic calculus explicitly. I took a course on stochastic processes and stochastic signal processing. I think that is related, but probably as a subtopic.

I did calculus of variations in a couple of my senior physics courses, one dealing with mechancis and the other with electromagnetics. It was pretty interesting, but I would have to agree it was initially quite humbling.

So... surprise!!
The Stochastic Calculus deals with functions that are everywhere continuous but nowhere differentiable. In common terms they are made up of corners like the one at the origin for the absolute value function. A Wiener process is an example.

https://en.wikipedia.org/wiki/Wiener_process
 
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#12

Joined Nov 30, 2010
18,210
I don't really recall doing proofs although I'm sure we were shown them.
I must have seen some proofs somewhere, but I can't remember where. At one point, my Calculus teacher asked me where I got a certain number and I replied in one word, "Identity". I think he didn't expect me to know that. In fact, I didn't expect me to know that because it had been at least 10 years since I learned it! All I'm sure of is that some proofs wandered through the class room somewhere in the 1960's.
 
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