Updated: Extreme math / physics plan

[Mike in SA]

Would love to get some input from our physicists (RR???).

We think we have a budding cosmologist. He is pressing us to deliver a working knowledge of general relativity. Question is this: what would you absolutely expect to see changed in the below (please ignore questions of feasibility - we are prepared to adjust for that):

Year 1
AOPS intermediate algebra
Stereometry (3d and nonEuclidean geometry)
Gentle introduction to math of relativity (basic math for conceptual understanding)

Year 2
Aops precalculus and additional trig / analytic geometry
Formal nonEuclidean geometry
Astrophysics (minimal calculus)

Year 3
Calculus 1-3
Projective geometry
Physics AP C

Year 4
Differential geometry
AOPS counting & probability
MIT-OCW Relativity

Year 5
General Topology

Vector Calculus
MIT-OCW Relativity II (Exploring Black Holes)

 

Year 6+

Probably at the local uni (don't really think we'll be able to hold off beyond this point)...

Gut check? Anything blatantly missing for creative cosmology?

Edited April 18, 2016 by Mike in SA

[lewelma]

All I will say is that my older ds would love to live at your house! :thumbup1:

 

Ruth in NZ

[8filltheheart]

No idea what the actual answer to your question is, but I thought I would share that my ds did 2 semesters of classical mechanics after modern.  He also self-designed a course on dark matter and black holes that he absolutely loved.  He also self-studied 2 astronomy courses based on the cosmos and the solar system and his advisor this semester placed him into a 400 level class b/c he had already covered their lower level content.

 

Don't know if that is at all helpful or not.

 

 

[arliemaria]

I just have to say that I love seeing your individualized curriculum map.  Robby is currently drawing diagrams of his payload to launch a scientific weather balloon with a GoPro camera, etc.  I might need to talk to you in a few years.  :)

[Mike in SA]

All I will say is that my older ds would love to live at your house! :thumbup1:

 

Ruth in NZ

Welcome to come visit if a trip to Texas is on the horizon! ;)

[Mike in SA]

No idea what the actual answer to your question is, but I thought I would share that my ds did 2 semesters of classical mechanics after modern. He also self-designed a course on dark matter and black holes that he absolutely loved. He also self-studied 2 astronomy courses based on the cosmos and the solar system and his advisor this semester placed him into a 400 level class b/c he had already covered their lower level content.

 

Don't know if that is at all helpful or not.

Actually trying to limit mechanics, believe it or not. It's a local approximation, you know? Unusual perspectives on spacetime topologies would be right up our alley, though.

[AEC]

Unusual perspectives on spacetime topologies would be right up our alley, though.

 

This will be WAY off the beaten path, but the connection to relativity is much more than just marketing - the technology is quite legitimately viewed as directly optimizing space-time locations of computation  - ensuring every event (computation) is within the event-horizon of it's causal predecessors (prior computations).

 

http://www.fpl2012.org/Presentations/Keynote_Steve_Teig.pdf

 

Ignore the bits about FPGAs and reconfigurable computing...skip ahead to around slide 24.

 

If you decide this is cool I can dredge up more specifics, diagrams, slides, computational models, etc.

[Mike in SA]

This will be WAY off the beaten path, but the connection to relativity is much more than just marketing - the technology is quite legitimately viewed as directly optimizing space-time locations of computation  - ensuring every event (computation) is within the event-horizon of it's causal predecessors (prior computations).

 

http://www.fpl2012.org/Presentations/Keynote_Steve_Teig.pdf

 

Ignore the bits about FPGAs and reconfigurable computing...skip ahead to around slide 24.

 

If you decide this is cool I can dredge up more specifics, diagrams, slides, computational models, etc.

 

Not sure I follow the link to cosmology studies?  I see the discussion of Minkowski space, but it's highly simplified...  Am I missing the point?

[Mike in SA]

Bump - any takers among our physicists?

[desertflower]

I had to take differential equations when I was thinking about majoring in physics.  But since you are limiting mechanics.......not sure I understand your train of thought. 

 

This looks like a good path at any rate. 

[Mike in SA]

I had to take differential equations when I was thinking about majoring in physics. But since you are limiting mechanics.......not sure I understand your train of thought.

 

This looks like a good path at any rate.

Thanks for the feedback.

 

Differential equations is embedded to the extent that is necessary, within calculus and differential geometry. We figure that will keep a formal diffeq course relevant once he hits college while focusing on the track more likely to be limited during undergraduate studies.

 

We tried to follow MIT prerequisite guidelines where suggested. They don't require a whole lot of classical mechanics, which is a good thing in our minds. Classical thinking limits multidimensional creativity...

[8filltheheart]

I just looked at their physics 3 course and the first half is mechanics:

 

Physics lll: Mechanical vibrations and waves; simple harmonic motion, superposition, forced vibrations and resonance, coupled oscillations, and normal modes; vibrations of continuous systems; reflection and refraction; phase and group velocity. Optics; wave solutions to Maxwell's equations; polarization; Snell's Law, interference, Huygens's principle, Fraunhofer diffraction, and gratings.

 

http://student.mit.edu/catalog/m8a.html

 

Also, what is AP physics BC? There is physics C mechanics and physics C E&M. (The equivalent of basic physics 1 and 2.) Fwiw, it seems like you really don't want any input, only affirmation, so I am not sure any one is going to be much help.

[lewelma]

Bump - any takers among our physicists?

 

Why don't you PM Regentrude.  She doesn't frequent this board very often and probably hasn't seen it.

 

Ruth in NZ

 

ETA:  I see that Regentrude and I were posting at the same time.  :001_smile:

Edited April 18, 2016 by lewelma

[regentrude]

My background is in condensed mater theory and not in cosmology, so I am not an expert, but I am puzzled by the "limiting mechanics" approach.

One reason why it is useful to study mechanics is that it is easier to introduce abstract concepts and methods by analyzing systems students can see and experience directly and thus have some expectation of their behavior.

I am not quite sure it makes sense to skip the foundational physics - mechanics, electrodynamics, quantum and statistics - to jump into astrophysics. What astrophysics can one do without those basic tools? (Or are you talking about basic astronomy, planetary motion etc when you use this term?)

 

I am not smart enough to understand the argument that "Classical thinking limits multidimensional creativity."

How can one understand relativity without a thorough grounding in classical physics?

 

 

Edited April 18, 2016 by regentrude

[Mike in SA]

I just looked at their physics 3 course and the first half is mechanics:

 

Physics lll: Mechanical vibrations and waves; simple harmonic motion, superposition, forced vibrations and resonance, coupled oscillations, and normal modes; vibrations of continuous systems; reflection and refraction; phase and group velocity. Optics; wave solutions to Maxwell's equations; polarization; Snell's Law, interference, Huygens's principle, Fraunhofer diffraction, and gratings.

 

http://student.mit.edu/catalog/m8a.html

 

Also, what is AP physics BC? There is physics C mechanics and physics C E&M. (The equivalent of basic physics 1 and 2.) Fwiw, it seems like you really don't want any input, only affirmation, so I am not sure any one is going to be much help.

 

Sorry, C.  :)  Actually, I really would like input.  DS wants to get to the meat of cosmology as quickly as possible, and he actually gets the ideas, so we're trying to help to the extent that it is feasible.

 

 

 

My background is in condensed mater theory and not in cosmology, so I am not an expert, but I am puzzled by the "limiting mechanics" approach.

One reason why it is useful to study mechanics is that it is easier to introduce abstract concepts and methods by analyzing systems students can see and experience directly and thus have some expectation of their behavior.

I am not quite sure it makes sense to skip the foundational physics - mechanics, electrodynamics, quantum and statistics - to jump into astrophysics. What astrophysics can one do without those basic tools? (Or are you talking about basic astronomy, planetary motion etc when you use this term?)

 

I am not smart enough to understand the argument that "Classical thinking limits multidimensional creativity."

How can one understand relativity without a thorough grounding in classical physics?

 

Thanks for the feedback!  Yes, the astrophysics text we are looking at is very introductory - it might be considered a solid astronomy course.  It actually does delve into gravitational lensing, etc, so it requires a decent working knowledge of physics, as you mentioned.  DS does have a lot of that already (to the extent required by the text, which isn't that much).  Our idea was to introduce enough of the experimental techniques used to understand discussions of recent findings, such as LIGO & gravitational waves, while reinforcing some of the basic physics with just a smattering of relevant math.  The key math should be covered by the first year math course - again, only just enough to understand, and not really apply yet.  It's a bit surprising, but it's actually a very effective text.

 

We understand that some mechanics is required.  But, we have also seen people struggle with the abstractions of spacetime when their thinking gets confined to Newtonian mechanics.  DS has absolutely NO issues with the abstractions, so we wanted to limit classical mechanics to the minimum level needed to really start working on his topic of choice.  Also, our thinking goes, he'll get plenty of classical physics in college, and he can set to that with a rather unique perspective.  Does it make any sense at all?  Yes?  No?

 

He also wants to dig deeply into quantum mechanics, but we get the feeling that is going require more than we'll be able to give him.  We tried stretching things out as far as possible, but at the end of that ^ run, it's getting downright silly.  That's about when we guess his language arts skills will catch up sufficiently to plop him in a college setting.  For QM, he'll probably have to make do with the basics for now.  Then again, if you have any suggestions that make the content accessible, we'd be all ears!  He is convinced that there are dimensions beyond our perception, and wants to develop well-supported theories about relationships between dark matter, dark energy, entropy, and quantum entanglement.

 

I do have a physics degree, but I would never consider myself a qualified physics instructor at the levels he is looking for.  Both DW and I can, however, teach math all the way up to what is listed.  DS has had some abstract algebra already, so we know what we're working with there.  It's really the physics path to develop a cosmologist that we aren't confident in.

[regentrude]

He also wants to dig deeply into quantum mechanics, but we get the feeling that is going require more than we'll be able to give him.  We tried stretching things out as far as possible, but at the end of that ^ run, it's getting downright silly.  That's about when we guess his language arts skills will catch up sufficiently to plop him in a college setting.  For QM, he'll probably have to make do with the basics for now.  Then again, if you have any suggestions that make the content accessible, we'd be all ears! 

 

I don't know how to make quantum mechanics "accessible". I cannot imagine doing it before theoretical mechanics - where the student becomes familiar with the Hamiltonian, for example, and develops a feeling for what it is and does, before encountering it in the very abstract Schroedinger equation. Of course one can learn to simply manipulate the math, but developing an abstract understanding without the concrete understanding obtained in mechanics? I guess my brain does not do that. YMMV.

Not to mention that I don't see how you can do quantum mechanics without differential equations.

 

So basically, to me, an approach that skips the foundations and jumps ahead to a field that uses those techniques does not make sense.

 

Oh, btw, I had a question about your math listings. What do you consider "Calculus 3" since you list it separately from vector calculus? Normally, calc 3 is vector calc... but since you guys are mathematicians, you must see a difference.

Edited April 19, 2016 by regentrude

[Mike in SA]

I don't know how to make quantum mechanics "accessible". I cannot imagine doing it before theoretical mechanics - where the student becomes familiar with the Hamiltonian, for example, and develops a feeling for what it is and does, before encountering it in the very abstract Schroedinger equation. Of course one can learn to simply manipulate the math, but developing an abstract understanding without the concrete understanding obtained in mechanics? I guess my brain does not do that. YMMV.

Not to mention that I don't see how you can do quantum mechanics without differential equations.

 

So basically, to me, an approach that skips the foundations and jumps ahead to a field that uses those techniques does not make sense.

 

Oh, btw, I had a question about your math listings. What do you consider "Calculus 3" since you list it separately from vector calculus? Normally, calc 3 is vector calc... but since you guys are mathematicians, you must see a difference.

 

:)  For vector calculus, there's still a lot to learn after cal 3.  We'd be covering topics in real & complex analysis, vector fields, curls, and tensors, since our differential geometry texts are introductory.  Different universities may call it "advanced calculus" or "vector and tensor analysis," and some do not even offer it. 

 

We picked the math sequence very deliberately to take a non-standard route, so that the "typical" route will still be fresh when required.  It took a bit of work to make sure the texts flowed coherently, but to be completely frank, it's the route many early 20th-century mathematicians took, and provides an outstanding foundation in pure math.  It's a bit of a shame that geometry is becoming a lost art, because it is so imminently applicable.  Developing calculus from a geometric perspective may seem odd, but it is very effective (it is NOT easy!).

 

The Hamiltonian is an interesting one, and sort of why we didn't think QM would be in reach.  We actually cover that in chemistry (another MIT-OCW offering).  Atkins introduces it in a very understandable way, but without formal partial diffeq, usage is obviously a wee bit limited.  A conceptual understanding will be there, but that won't satisfy DS.  Maybe we'll find a way to get him up to the level that the founders of the theory had, but I think we have our hands full already...

[EmilyGF]

What are your CS plans? Can't neglect those for research.

 

Emily

[Mike in SA]

What are your CS plans? Can't neglect those for research.

 

Emily

C++ for game programmers, java8, and lisp. Once again, MIT has the best course of the three (lisp / scheme). It's tough, but really teaches it right.

[lewelma]

I'm glad to hear that you think geometry is useful.  My ds does quite a lot of geometry for the IMO, and I was concerned he was kind of wasting his time.  He is currently working on geometric inequalities, which are apparently very hard. 

[Mike in SA]

I'm glad to hear that you think geometry is useful. My ds does quite a lot of geometry for the IMO, and I was concerned he was kind of wasting his time. He is currently working on geometric inequalities, which are apparently very hard.

Try Kiselev's books. They cover it well, but they are hard and provide no solutions.

 

They aren't as hard once you have a couple of approaches in your toolkit.

[shawthorne44]

I have a B.S. in Physics and an Engineering Masters.   I read Physics books for fun.  As a child my parents had to cancel a skating birthday party of mine for a good reason but they felt bad so they asked if I had a desired extra birthday present.   I named a Physics textbook I'd been eyeing at the local University.  They said Yes before they realized the cost, it covered the first three semesters of Physics.   BwaaaHahahahaha.   So, I'm a Physics geek.   In my entire life I have had a real understanding of General relativity for about a half hour time span.  That was in Physics graduate school after an evening of prolonged drinking, then I went home slept for a few hours, woke up to pee, and while on the toilet I genuinely understood General Relatively.  It was really cool, I just sat there because I didn't want to break the spell.   I didn't want to go back to sleep, but I was tired.   My point is that true understanding of it is sort of like Buddist enlightenment.  You can study your way to a sort of understanding.  You can understand the formulas.  You can lay the groundwork.  But the enlightenment might not come.   

That said, your track looks interesting.  

[lewelma]

Off topic here but:

 

I'm definitely interested in anything that would help.  This is the problem he solved last night:

 

An ant starts on the boundary of a circular disk with radius one meter and walks in a straight line.  Every now and then it turns left by 60 degrees or right by 60 degrees, alternating each time.  When the ant reaches the boundary of the disc again, it decides to stop for a rest.  What is the maximum distance that the ant could have travelled?

 

Would Kiselev help with this kind of thing? This is problem 6 out of 22, and they only get harder.

[Mike in SA]

I have a B.S. in Physics and an Engineering Masters. I read Physics books for fun. As a child my parents had to cancel a skating birthday party of mine for a good reason but they felt bad so they asked if I had a desired extra birthday present. I named a Physics textbook I'd been eyeing at the local University. They said Yes before they realized the cost, it covered the first three semesters of Physics. BwaaaHahahahaha. So, I'm a Physics geek. In my entire life I have had a real understanding of General relativity for about a half hour time span. That was in Physics graduate school after an evening of prolonged drinking, then I went home slept for a few hours, woke up to pee, and while on the toilet I genuinely understood General Relatively. It was really cool, I just sat there because I didn't want to break the spell. I didn't want to go back to sleep, but I was tired. My point is that true understanding of it is sort of like Buddist enlightenment. You can study your way to a sort of understanding. You can understand the formulas. You can lay the groundwork. But the enlightenment might not come.

 

That said, your track looks interesting.

Can't express how much I love this story.

[SeaConquest]

Listening in... Sacha and I have been watching The Elegant Universe, and now he is off trying to mod Minecraft to include strings. As a humanities person, I am seriously intimidated thinking about what he will be like in a few years. Thank G-d for MOOCs, and this board.

[Mike in SA]

Off topic here but:

 

I'm definitely interested in anything that would help. This is the problem he solved last night:

 

An ant starts on the boundary of a circular disk with radius one meter and walks in a straight line. Every now and then it turns left by 60 degrees or right by 60 degrees, alternating each time. When the ant reaches the boundary of the disc again, it decides to stop for a rest. What is the maximum distance that the ant could have travelled?

 

Would Kiselev help with this kind of thing? This is problem 6 out of 22, and they only get harder.

(edited after looking it up)

 

Yes it would. It covers things like mensuration (auto correct almost got me on that one), divisibility, maxima and minima. This one will require a little synthesis, but all good problems do!

 

It's a relatively inexpensive little book (well, 2 books, but you're looking for the first one - Planimetry). No harm adding it to the library if you can get it reasonably cheap.

 

Here's what you are looking for: http://www.cut-the-knot.org/books/Reviews/KiselevsGeometry.shtml

 

Stereometry is great, too, but for solid geometry, vector arithmetic, and basic non-Euclidean geometry.

 

http://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&text=A.+P.+Kiselev&search-alias=books&field-author=A.+P.+Kiselev&sort=relevancerank

 

Edited April 20, 2016 by Mike in SA

[Mike in SA]

Off topic here but:

 

I'm definitely interested in anything that would help. This is the problem he solved last night:

 

An ant starts on the boundary of a circular disk with radius one meter and walks in a straight line. Every now and then it turns left by 60 degrees or right by 60 degrees, alternating each time. When the ant reaches the boundary of the disc again, it decides to stop for a rest. What is the maximum distance that the ant could have travelled?

 

Would Kiselev help with this kind of thing? This is problem 6 out of 22, and they only get harder.

I'm curious. Did he figure this one out? I believe it can be proven almost completely via construction (faster with a little trig).

[lewelma]

Yes, he did.  His instructor misread the problem and thought that the path could never cross, so found the question incredibly non-trivial.  But ds said that if the path can cross that the furthest distance the ant can walk is (4*sqr3)/3 meters.  

 

I see phrases in his proof like "construct equilateral triangle" and "apply ptolemys inequality."  I think I see some pythagoras but no trig.  Beats me what he did though.  

Edited April 20, 2016 by lewelma

[shawthorne44]

Can't express how much I love this story.

I had no idea people would like my toilet story so much.  Either you guys are geeks too, or I tell it better in print.  Not many people know because I get the "you have two heads" look normally.