Courses Need help in choosing math classes for a HS student

[Homelilly]

Hi all! I need help choosing math classes for a boy (who will be in the 11th grade, so two more years of math), who has taken AP Calculus BC and Intro to Linear Algebra, Proofs. I am thinking about taking MVC and DE. What other options might be good? He is doing AP Physics C both and preparing for F=ma/USAPhO.

His future path is physics, applied math, or electrical Engineering. He is not a fan of programming, though he is finishing the second year of a university-level CS course (he really hates Haskell and is stuck on the tests).

Should he go the classical way—MVC/DE—or take something like Mathematical Statistics (he had AP Statistics on his own; the exam was 5) or Discrete Math?
He wants to spend a lot of time on math olympiads, too.

Thanks a lot for the advice! (Parents had math at school 30+ years ago and are completely useless).

 

[jedishrfu]

The big four for college freshmen heading for the natural sciences are:
- Calculus 1,2,3
- Linear Algebra
- Differential Equations
- Statistics

The website mathispower4u.com lists free video shorts on topics in these courses that could keep your student engaged for quite a while. Their format is simple: the presenter shows a problem and then walks through the solution.

When I was an undergrad in the 1970s, you learned Statistics in the context of a Statistical Physics/Thermodynamics course. The other three math groups were taught by the math department.

Higher-level math, such as Vector Analysis, was included in EM Theory and Partial Differential Equations in a Boundary Values course. The math department offered Advanced Calculus, Group Theory, and Real Analysis as pure math courses, providing valuable support for physics CM and QM courses.

From there, the students' interest drove what they learned next. In my case, it was Tensor Analysis and General Relativity, of which I've retained very little other than that it was an amazing trip using Wheeler's Gavitation book in preprint form.

 

[ohwilleke]

Parents had math at school 30+ years ago and are completely useless.

Give it five years or so, and your opinion of your parents will improve greatly.

One virtue of being sufficiently advanced at a young age is that you don't have to rush to cram in the usual intermediate university math courses like differential equations, discrete math, and calculus based probability and statistics.

Multi-variable calculus is normally taken before differential equations or real analysis or complex analysis. If you have multi-variable calculus, linear algebra, and calculus based probability and statistics under your belt, discrete math would probably be pretty trivial at that point.

You could study applied analysis (a big part of which is tensor calculus), abstract algebra, number theory, fractals and chaos, complex analysis, operations research, or topology, for example. Sometimes, in one of the analysis subjects, you'd study Hilbert spaces (although where this is taught varies somewhat as they have so many varied applications).

The more abstract topics, like number theory, topology, and abstract algebra, can be particularly useful in a Math Olympiad. So can studying "gee-whiz" arithmetic parlor tricks for doing calculations with large numbers quickly, or memorizing numbers with many digits.

Math adjacent subjects that hover on the math-physics boundary are also a direction you could explore. For example, the mathematics of spinors and twistors, "amplitudeology", non-commutative geometry, and "complexity".

 

[Muu9]

While discrete math might be useful for math competitions, specifically studying for math competitions will be a more efficient use of time.

You can see the math required for physics Olympiad here: http://ipho.org/PDF/2015-12-06 Syllabus of IPhO.pdf (basically it's only math up to single variable calculus)

How did he feel about the proofs class? Is he taking the courses at a community college or four year university? What are the prerequisites for mathematical statistics?

 

[Homelilly]

The big four for college freshmen heading for the natural sciences are:
- Calculus 1,2,3
- Linear Algebra
- Differential Equations
- Statistics

The website mathispower4u.com lists free video shorts on topics in these courses that could keep your student engaged for quite a while. Their format is simple: the presenter shows a problem and then walks through the solution.

When I was an undergrad in the 1970s, you learned Statistics in the context of a Statistical Physics/Thermodynamics course. The other three math groups were taught by the math department.

Higher-level math, such as Vector Analysis, was included in EM Theory and Partial Differential Equations in a Boundary Values course. The math department offered Advanced Calculus, Group Theory, and Real Analysis as pure math courses, providing valuable support for physics CM and QM courses.

From there, the students' interest drove what they learned next. In my case, it was Tensor Analysis and General Relativity, of which I've retained very little other than that it was an amazing trip using Wheeler's Gavitation book in preprint form.

Thanks for the suggestions! I will discuss what's possible for him in Dual Enrollment. He will be fine with the big four next year. He wants game theory, I am not sure where to put it into the curriculum.

 

[Homelilly]

Give it five years or so, and your opinion of your parents will improve greatly.

One virtue of being sufficiently advanced at a young age is that you don't have to rush to cram in the usual intermediate university math courses like differential equations, discrete math, and calculus based probability and statistics.

Multi-variable calculus is normally taken before differential equations or real analysis or complex analysis. If you have multi-variable calculus, linear algebra, and calculus based probability and statistics under your belt, discrete math would probably be pretty trivial at that point.

You could study applied analysis (a big part of which is tensor calculus), abstract algebra, number theory, fractals and chaos, complex analysis, operations research, or topology, for example. Sometimes, in one of the analysis subjects, you'd study Hilbert spaces (although where this is taught varies somewhat as they have so many varied applications).

The more abstract topics, like number theory, topology, and abstract algebra, can be particularly useful in a Math Olympiad. So can studying "gee-whiz" arithmetic parlor tricks for doing calculations with large numbers quickly, or memorizing numbers with many digits.

Math adjacent subjects that hover on the math-physics boundary are also a direction you could explore. For example, the mathematics of spinors and twistors, "amplitudeology", non-commutative geometry, and "complexity".

Thanks a lot! That's very valuable for me—I am the homeschooling mom making the curriculum for him. Unfortunately, we as parents can only shake our heads while looking at what he is doing in math/physics and cannot help. That's why I am seeking advice here.
He will take MVC next fall and DE in spring. He has also taken some Intro courses to NT and is doing self-paced NT Olympiad books (also algebra, geometry, etc.).
Since he likes physics and is preparing for the USAPhO, I must also determine which math courses can help him.


Thanks a lot for sharing your opinion and options! I will think over what to do next.

 

[Muu9]

If he wants to do game theory, I say let him do game theory. Does he want to self study or take a college course? The former would be more useful for math Olympiads, where game theory questions occasionally show up. I recommend the book Lessons in Play by Michael Albert

 

[Homelilly]

If he wants to do game theory, I say let him do game theory. Does he want to self study or take a college course? The former would be more useful for math Olympiads, where game theory questions occasionally show up

Could you suggest some good introductory textbook? I have Fudenberg and Tirole, but he said it's complicated for him.

 

[Homelilly]

While discrete math might be useful for math competitions, specifically studying for math competitions will be a more efficient use of time.

You can see the math required for physics Olympiad here: http://ipho.org/PDF/2015-12-06 Syllabus of IPhO.pdf (basically it's only math up to single variable calculus)

How did he feel about the proofs class? Is he taking the courses at a community college or four year university? What are the prerequisites for mathematical statistics?

We are homeschooling, so he takes classes with online schools like AoPS. The community college is not good around us. I will try to find a class at a four-year university, but I cannot understand if he needs to spend more time on Olympiad math (his choice) or take the school path with MVC and DE (I want to suggest counselors for college).
He is the first one, and I am afraid to make a mistake. He wants to do math olympiad for the whole year, and I am not sure it will be good for college if he lacks the classes others are doing at that time.


He is taking a proof class, loves it a lot, and is doing pretty well. In the summer, he did Zeitz art of problem solving on his own.


His algebra and Geometry are good (was in the middle at HMMT in November), combinatorics is not so good - could not find a good online class with good explanations, and this kid, who was making uf crazy demanding what is "there", what is "table", what is "is" need only explaining why and what, hates memorizing.

 

[Homelilly]

If he wants to do game theory, I say let him do game theory. Does he want to self study or take a college course? The former would be more useful for math Olympiads, where game theory questions occasionally show up. I recommend the book Lessons in Play by Michael Albert

Thanks for the recommendation! I ordered it. Self-study is hard—when he needs help, we cannot do anything, so I will look into a university nearby.

 

[Muu9]

We are homeschooling, so he takes classes with online schools like AoPS. The community college is not good around us. I will try to find a class at a four-year university, but I cannot understand if he needs to spend more time on Olympiad math (his choice) or take the school path with MVC and DE (I want to suggest counselors for college).
He is the first one, and I am afraid to make a mistake. He wants to do math olympiad for the whole year, and I am not sure it will be good for college if he lacks the classes others are doing at that time.


He is taking a proof class, loves it a lot, and is doing pretty well. In the summer, he did Zeitz art of problem solving on his own.


His algebra and Geometry are good (was in the middle at HMMT in November), combinatorics is not so good - could not find a good online class with good explanations, and this kid, who was making uf crazy demanding what is "there", what is "table", what is "is" need only explaining why and what, hates memorizing.

For school he will likely need ay least one math credit per year, so he could take one math course each year in the fall or spring to meet the requirement. (A one semester college DD course counts as a one year HS credit). Does the community college offer an independent study course? What has he tried so far in regards to combinatorics? Which math books has he enjoyed learning from?

 

[jedishrfu]

Have you checked for colleges in your state that offer remote learning?

I know in Texas Texas Tech offers many courses for high-school students.

https://www.depts.ttu.edu/k12/online/online-high-school/

 

[Muu9]

WTAMU offers dual enrollment for $150

If money is not an issue, then check out UIUC Netmath or UCSD extended studies

 

[mathwonk]

You and he are the best judges of what he is enjoying, and that is key. Since you ask for advice however, I will mention one of my principles for advanced kids in math. Namely I suggest that such kids, will benefit more from materials designed for honors students, rather than using the same materials used for average students but just using them earlier. So I suggest having a look at Calculus by Apostol, or Differential and Integral Calculus by Courant. The insights in these books go much deeper than standard BC calc prep books.

 

[Homelilly]

For school he will likely need ay least one math credit per year, so he could take one math course each year in the fall or spring to meet the requirement. (A one semester college DD course counts as a one year HS credit). Does the community college offer an independent study course? What has he tried so far in regards to combinatorics? Which math books has he enjoyed learning from?

For school he will likely need ay least one math credit per year, so he could take one math course each year in the fall or spring to meet the requirement. (A one semester college DD course counts as a one year HS credit). Does the community college offer an independent study course? What has he tried so far in regards to combinatorics? Which math books has he enjoyed learning from?

He will take MVC, and since his calculus and Linear algebra and proofs are all in overleaf, it's enough job. I have awesome math combinatorics, and thinking about AoPS intermediate combinatorics, but he was taking intro with AoPS and he was stuck that he could not understand the problems. He loves reading - something like Alex Bellos in math, Michael Lewis or James Stewart (in non fiction). I am thinking of taking awesome math combinatorics books for summer.

 

[Homelilly]

WTAMU offers dual enrollment for $150

If money is not an issue, then check out UIUC Netmath or UCSD extended studies

Thanks! will look at them - have never heard of them before.

 

[Homelilly]

You and he are the best judges of what he is enjoying, and that is key. Since you ask for advice however, I will mention one of my principles for advanced kids in math. Namely I suggest that such kids, will benefit more from materials designed for honors students, rather than using the same materials used for average students but just using them earlier. So I suggest having a look at Calculus by Apostol, or Differential and Integral Calculus by Courant. The insights in these books go much deeper than standard BC calc prep books.

Thanks! That's a great decision - in the same way, I gave him toppler in physics, but he chose HRK and loves it greatly. Then, he will not use the textbook by school but use one of the suggested. If I take volume 2 of calculus by Apostol, will it be ok for MVC? He is using Thomas right now for calculus BC with calculus by Patrick from AoPS. Is Courant for MVC or DE?

 

[Homelilly]

Have you checked for colleges in your state that offer remote learning?

I know in Texas Texas Tech offers many courses for high-school students.

https://www.depts.ttu.edu/k12/online/online-high-school/

I would like to take him to university.

 

[jedishrfu]

To visit or to go there?

To regularly attend then maybe a local community college would work.

To visit maybe you could contact the college for a campus tour and specify what you're interested in.

When I was in high school and had a physics project of finding the Lagrange points between two celestial bodies. My math teacher connected me with a math professor at a local college.

I visited the college, had a nice talk with the professor about my problem and got to see some of the cool things they did. Basically, he was the reason I attended that college. A very nice professor, who taught Linear Algebra and Advanced Calculus and maintained the college's historic Olivier models.

 

[mathwonk]

There are two volumes of Apostol, and the best preparation for vol 2 Apostol is vol 1 Apostol. I.e. books like Thomas do not prepare one as well for Apostol vol. 2. There are also two volumes of Courant, and I recommend both of them together as well. Courant vol. 1 has a brief intro (about 27 pages) to some diff eq, and vol. 2 has a chapter (about 80 pages) on diff eq. Apostol also has a 50 page intro in vol 1, and vol.2 is subtitled something like Calculus of several variables with an introduction to differential equations and probability. Vol.2 starts with about 140 pages of linear algebra followed by 100 pages of diff eq. Both volumes of Apostol (but not Courant) treat linear algebra. In general the volume 1's are single variable and the volume 2's are multi variable.

These books are more serious and slower going than modern books, so it is advised to have your son preview and approve them before spending a lot for them. Just as a guess, I think it likely that having studied from Thomas, is adequate preparation for vol.1 of one of these books, but not vol. 2. Both Courant and Apostol assume one is prepared to do proofs along with calculus, right from the beginning. Of course it is possible he will sail through Apostol vol. 2 from the start, but I think he would miss something not having read vol. 1.

When you say "Thomas", I am not sure exactly what that means. I.e. Thomas died long ago, and the recent books with his name attached that I have seen are not very good in my opinion. I.e. recommended editions written by Thomas himself include the 2nd and 4th, but I would avoid any after about the 9th.

I might add that a book I dislike may be useful for learning some things, but in less depth. E.g. one might learn the chain rule or the fundamental theorem of calculus from a mediocre book, but one is less likely to learn why they are true, or exactly when they can be used, if you value that sort of insight.

If a good college library is available, browsing the stacks in the math section can be very illuminating
Hope this is useful. Again, be your own judge. My opinions reflect my own bias as a pure mathematician.

 

[Muu9]

He will take MVC, and since his calculus and Linear algebra and proofs are all in overleaf, it's enough job. I have awesome math combinatorics, and thinking about AoPS intermediate combinatorics, but he was taking intro with AoPS and he was stuck that he could not understand the problems. He loves reading - something like Alex Bellos in math, Michael Lewis or James Stewart (in non fiction). I am thinking of taking awesome math combinatorics books for summer.

Have you tried the AoPS intro to combinatorics book? If he can't understand that, the AwesomeMath book will not be helpful as it's just a collection of challenging combinatorics problems. Alternatively, there is a textbook on combinatorics by the people who run AwesomeMath titled "A Path to Combinatorics for Undergraduates: Counting Strategies"

 

[jedishrfu]

This may be totally off-base here but I'd thought I'd give it a try.

It could be a seed for your son for what math topics to pursue.

The video is a history on the physics principle of least action presented by Veritaseum on Youtube:



And now back to our regularly scheduled thread.

 

[mathwonk]

In my opinion, a well regarded introductory book on combinatorics is Mathematics of Choice, by Ivan Niven.

 

[mathwonk]

By the way have you read any of the thread Should I become a mathematician? in this forum. You might browse posts 1-40 or at least posts 36-40 there for a very opinionated take on several topics and a long list of sometimes quite advanced books.
https://www.physicsforums.com/threads/should-i-become-a-mathematician.122924/page-2

There is also zapper's thread for future physicists, which I took as a model.

But be aware those posts contained my views from some 20 years ago, and are far more adamant and confident than my older ones now. In particular my somewhat negative views of AP courses were formed by trying to teach students in college who skipped beginning college calc based on their AP high school prep. In general, high school students who had beginning calc only in AP classes were not qualified for my second year college calc classes. But so many such students existed that we had to water down the second year curriculum to accommodate them. This does not necessarily apply to really strong students like apparently your son. The AOPS series of books are also mostly quite good, but were just getting started back then, and I did not know about them. The books I recommend are still a bit higher level though.

Anyway, my advice for young students is that it matters much more for them to pursue topics they really enjoy than topics thought to be required for a certain professional future. good luck!

 

[Homelilly]

There are two volumes of Apostol, and the best preparation for vol 2 Apostol is vol 1 Apostol. I.e. books like Thomas do not prepare one as well for Apostol vol. 2. There are also two volumes of Courant, and I recommend both of them together as well. Courant vol. 1 has a brief intro (about 27 pages) to some diff eq, and vol. 2 has a chapter (about 80 pages) on diff eq. Apostol also has a 50 page intro in vol 1, and vol.2 is subtitled something like Calculus of several variables with an introduction to differential equations and probability. Vol.2 starts with about 140 pages of linear algebra followed by 100 pages of diff eq. Both volumes of Apostol (but not Courant) treat linear algebra. In general the volume 1's are single variable and the volume 2's are multi variable.

These books are more serious and slower going than modern books, so it is advised to have your son preview and approve them before spending a lot for them. Just as a guess, I think it likely that having studied from Thomas, is adequate preparation for vol.1 of one of these books, but not vol. 2. Both Courant and Apostol assume one is prepared to do proofs along with calculus, right from the beginning. Of course it is possible he will sail through Apostol vol. 2 from the start, but I think he would miss something not having read vol. 1.

When you say "Thomas", I am not sure exactly what that means. I.e. Thomas died long ago, and the recent books with his name attached that I have seen are not very good in my opinion. I.e. recommended editions written by Thomas himself include the 2nd and 4th, but I would avoid any after about the 9th.

I might add that a book I dislike may be useful for learning some things, but in less depth. E.g. one might learn the chain rule or the fundamental theorem of calculus from a mediocre book, but one is less likely to learn why they are true, or exactly when they can be used, if you value that sort of insight.

If a good college library is available, browsing the stacks in the math section can be very illuminating
Hope this is useful. Again, be your own judge. My opinions reflect my own bias as a pure mathematician.

Thanks a lot for your help and advice! It's very needed. I changed "Thomas" 15th ed to Apostol Volume 1, he will finish it this year, I hope. The issue was that the assignments are from Thomas, so I was reluctant to do other books and the assignments from Thomas for the class, plus he was doing all them in overleaf. Last fall, Linear Algebra in Overleaf took much time.

Will it be ok to take Apostol Volume 2 for MVC next fall?

As for Courant, is it for MVC or DE? When is it better to be used? After Apostol?
I apologize for so many questions and appreciate a lot the help - as I said, I had precalculus 30 years ago, and it's hard to understand what he needs. If I look only at what he likes, he would do only olympiad, but he needs math classes to graduate.

 

[Homelilly]

By the way have you read any of the thread Should I become a mathematician? in this forum. You might browse posts 1-40 or at least posts 36-40 there for a very opinionated take on several topics and a long list of sometimes quite advanced books.
https://www.physicsforums.com/threads/should-i-become-a-mathematician.122924/page-2

There is also zapper's thread for future physicists, which I took as a model.

But be aware those posts contained my views from some 20 years ago, and are far more adamant and confident than my older ones now. In particular my somewhat negative views of AP courses were formed by trying to teach students in college who skipped beginning college calc based on their AP high school prep. In general, high school students who had beginning calc only in AP classes were not qualified for my second year college calc classes. But so many such students existed that we had to water down the second year curriculum to accommodate them. This does not necessarily apply to really strong students like apparently your son. The AOPS series of books are also mostly quite good, but were just getting started back then, and I did not know about them. The books I recommend are still a bit higher level though.

Anyway, my advice for young students is that it matters much more for them to pursue topics they really enjoy than topics thought to be required for a certain professional future. good luck!

Thanks for the recommendation - I will read it.
I am not a big fan of AP and understand that they are very simplified and made to promote "Oh, you have done the first year of university level and pay less" propaganda. That's why I added more books, and he is sailing through the exam. As a homeschooler, he needs proof of the classes.
The AOPS books are good, and he has completed all the intro and intermediate levels (except intermediate counting).
The issue I found is that if he takes classes at community college or online, the books are usually not good or deep enough (and he hates wasting time memorizing, though he can spend three days (at least four hours) on one problem). However, the assignments should be done, and it's hard to do a good book on his own, plus the class.
He does not know what he wants to do yet - right now it's just a lot of math and physics. We cannot help in any of these areas.

 

[mathwonk]

You don't need both Courant and Apostol. Both Courant vol.1 and Apostol vol. 1 are mainly one variable calculus. Volume 1 of Courant, and also volume 1 of Apostol both include introductions to diff eq. Both Courant vol.2 and Apostol vol. 2 cover multivariable calculus, and both vol.2 of Courant and vol.2 of Apostol have longer treatments of diff eq.

Apostol is a little more modern than Courant, in particular Apostol, but not Courant, treats and uses linear algebra in discussing multivariable calc, so all you need is both volumes of Apostol. Vol.1 Apostol will treat one variable calc very thoroughly, and also will introduce diff eq. and linear algebra. The first section of vol.2 of Apostol will repeat the linear algebra, (some of which you can thus skip), and then apply it to a treatment of diff eq. Then in the 2nd section it will treat multivariable calc. The 3rd and last section is more specialized, on probability and numerical analysis, and is less standard, and is not included in most courses, so could be skipped, or postponed.

As to his preferences, the more he knows, the better he will likely do on contests. As my son remarked when I asked him if the junior high contests he took tested the algebra I had taught him: "they don't specifically test it, but if you know algebra, you can use it".

I would also be sensitive to the student's reaction to these books. I.e. if he prefers Thomas, he should probably use Thomas, perhaps supplemented by Apostol. And it always helps to have more than one book. Thomas may make some things look easier, or offer more useful computational practice. The best book is the one that speaks to and engages and enlightens him.

 

[Homelilly]

You don't need both Courant and Apostol. Both Courant vol.1 and Apostol vol. 1 are mainly one variable calculus. Volume 1 of Courant, and also volume 1 of Apostol both include introductions to diff eq. Both Courant vol.2 and Apostol vol. 2 cover multivariable calculus, and both vol.2 of Courant and vol.2 of Apostol have longer treatments of diff eq.

Apostol is a little more modern than Courant, in particular Apostol, but not Courant, treats and uses linear algebra in discussing multivariable calc, so all you need is both volumes of Apostol. Vol.1 Apostol will treat one variable calc very thoroughly, and also will introduce diff eq. and linear algebra. The first section of vol.2 of Apostol will repeat the linear algebra, (some of which you can thus skip), and then apply it to a treatment of diff eq. Then in the 2nd section it will treat multivariable calc. The 3rd and last section is more specialized, on probability and numerical analysis, and is less standard, and is not included in most courses, so could be skipped, or postponed.

As to his preferences, the more he knows, the better he will likely do on contests. As my son remarked when I asked him if the junior high contests he took tested the algebra I had taught him: "they don't specifically test it, but if you know algebra, you can use it".

I would also be sensitive to the student's reaction to these books. I.e. if he prefers Thomas, he should probably use Thomas, perhaps supplemented by Apostol. And it always helps to have more than one book. Thomas may make some things look easier, or offer more useful computational practice. The best book is the one that speaks to and engages and enlightens him.

Thanks! That's very detailed response.
He took linear algebra with Penney textbook. The teacher was amazing, and the grading was pretty tough, so I am sure he knows the intro to linear algebra.
Assuming he does not need an introduction to linear Algebra and will finish Apostol volume 1 by August, the AoPS book Calculus will be done in two weeks, and proofs by Simon Euler Circle, which is better to take for the fall—Courant volume 2 or Apostol Volume 2?
Thanks a lot, and I apologize for asking so many questions.

 

[mathwonk]

After vol.1 Apostol, he should use vol.2 Apostol. In particular, multivariable differential calculus is conceptually much more natural using linear algebra, which Courant does not do.

By the way, here is a free, short (about 125 pages), book on linear algebra that I wrote for fun.
https://www.math.uga.edu/sites/default/files/laprimexp.pdf

It is an expansion of a much shorter version (15 pages) available on the same website, which I wrote just as an exercise to see how short I could make a book on linear algebra, leaving many easier facts as exercises.

By the way, the book by Richard Penney looks quite good, but there are many ways to present the subject, and even though his book is almost 500 pages, and covers many, many things mine does not, he still does not quite cover all the same bases. E.g. he presents the Jordan form of a matrix, but not the rational canonical form. One difference between them I like to point out, is that one cannot usually actually compute the Jordan form in practice, since it requires factoring the characteristic polynomial, and there is no practical way to factor polynomials. There is however an algorithm for computing the rational canonical form by hand, so one can actually find this form in practice. My little book covers both topics, and even proves the (less important) uniqueness of the reduced echelon form of a matrix, which Penney omits. I also at least sketch a more general form of the spectral theorem than he does, for "normal" operators. He also seems to omit any applications of linear algebra to differential equations, which my book includes. I think this is curious, although common, since differentiation is arguably the most important example of a linear operator. Another topic my book discusses and uses, is the basic concept of "quotient space", which Penney also omits. So it never hurts to have more than one book around.

Another glance at Penney reveals that he also treats numerical methods. In the computer age, this is another response to the fact that polynomials are hard to factor. i.e. since one cannot always compute
"eigenvalues" exactly by factoring, one can use numerical methods to at least compute them approximately. I do not treat this (out of ignorance).

Another look at Penney shows he is working with vector spaces over "scalars" but never says what the scalars are. Well he says scalars means "numbers" but never says what numbers are. Apparently they are usually real numbers and sometimes complex numbers. I work with vector spaces over any "field" of scalars, and I at least do summarize what that means.

 

[Muu9]

Thanks for the recommendation - I will read it.
I am not a big fan of AP and understand that they are very simplified and made to promote "Oh, you have done the first year of university level and pay less" propaganda. That's why I added more books, and he is sailing through the exam. As a homeschooler, he needs proof of the classes.
The AOPS books are good, and he has completed all the intro and intermediate levels (except intermediate counting).
The issue I found is that if he takes classes at community college or online, the books are usually not good or deep enough (and he hates wasting time memorizing, though he can spend three days (at least four hours) on one problem). However, the assignments should be done, and it's hard to do a good book on his own, plus the class.
He does not know what he wants to do yet - right now it's just a lot of math and physics. We cannot help in any of these areas.

For next year, check out https://loveofmath.com/course?courseid=multivariable for MVC - it's run by a former AoPS teacher. What's his physics background?

 

[Homelilly]

For next year, check out https://loveofmath.com/course?courseid=multivariable for MVC - it's run by a former AoPS teacher. What's his physics background?

I love Heather with all my mom's heart - she is doing an incredible job! Our two kids have a lot of fun learning with with her - and the kid is taking MVC there next year, and other two are having Geometry and Algebra. He is doing HRK now and taking AP Physics C in May.

 

[Homelilly]

After vol.1 Apostol, he should use vol.2 Apostol. In particular, multivariable differential calculus is conceptually much more natural using linear algebra, which Courant does not do.

By the way, here is a free, short (about 125 pages), book on linear algebra that I wrote for fun.
https://www.math.uga.edu/sites/default/files/laprimexp.pdf

It is an expansion of a much shorter version (15 pages) available on the same website, which I wrote just as an exercise to see how short I could make a book on linear algebra, leaving many easier facts as exercises.

By the way, the book by Richard Penney looks quite good, but there are many ways to present the subject, and even though his book is almost 500 pages, and covers many, many things mine does not, he still does not quite cover all the same bases. E.g. he presents the Jordan form of a matrix, but not the rational canonical form. One difference between them I like to point out, is that one cannot usually actually compute the Jordan form in practice, since it requires factoring the characteristic polynomial, and there is no practical way to factor polynomials. There is however an algorithm for computing the rational canonical form by hand, so one can actually find this form in practice. My little book covers both topics, and even proves the (less important) uniqueness of the reduced echelon form of a matrix, which Penney omits. I also at least sketch a more general form of the spectral theorem than he does, for "normal" operators. He also seems to omit any applications of linear algebra to differential equations, which my book includes. I think this is curious, although common, since differentiation is arguably the most important example of a linear operator. So it never hurts to have more than one book around.

Another glance at Penney reveals that he also treats numerical methods. In the computer age, this is another response to the fact that polynomials are hard to factor. i.e. since one cannot always compute
"eigenvalues" exactly by factoring, one can use numerical methods to at least compute them approximately. I do not treat this (out of ignorance).

Thanks a lot for sharing! That's very kind of you! I will print it and he will go through it.

 

[mathwonk]

If he finds my book impenetrable, I will be happy to answer questions on it. If he has the appetite, here is another short (67 pages) free linear algebra book I wrote. This one includes a complete treatment of determinants. These are based on lectures from a course I taught.
https://www.math.uga.edu/sites/default/files/inline-files/4050sum08.pdf

Warning: It is possible these short "books" of mine may not be so easy to read, even though he has completed the course of linear algebra from Penney.

I am interested in his reaction to them however, if he does look at them. It is quite possible they are too terse, or too abstract, but I hope they may be useful at least for a different perspective. And perhaps the fact they are brief may make them more useful for review, highlighting and recalling the main ideas.

As long as I am at it, here are the notes from my course on rigorous proof - based geometry, from Euclid, taught in the first year (2011) of epsilon camp, a 2 week summer experience for brilliant kids aged roughly 10-12. These notes start from scratch and go far enough to deduce the volume of a 4 dimensional ball by techniques Archimedes would have understood. They also discuss how Euclid's and Archimedes' ideas pave the way for, and relate to, those of Newton in calculus.
https://www.math.uga.edu/sites/default/files/inline-files/10.pdf