Spivak's Calculus

[Janice in NJ]

Would anyone like to discuss this book with me?

 

Rather than discuss the content (which I really like), I would like to discuss its placement in the curriculum sequence and the implications.

 

I have seen it listed (along with Apostol and others like it) as Honors Calculus at select schools for strong math students who start college with the BC exam under their belt. 

 

Among other things, it really is an intro to analysis text. It's a beautiful warm up to Baby Rudin. 

 

What happens to kids who never discover it? Did you? Did you find it and devour it on your own, or were you luck enough to find yourself in a class where it was used? Did you know how lucky you were, or did you assume your situation was "normal." Or maybe you never really liked the book. 

 

For me, it was one of those books that left me wondering, "Where has THIS been all my life? Sheesh! If someone had put this in front of me â€‹before this or before this or this, how might things have been different?" I find the book to be full of immense ideas. Each one seems small at first, but if you let them roll around in you for a while, they swell and grow. It's as if they have a life of their own. I just don't find that with Larson or Stewart or others like it (doorstop texts). 

 

I guess I would love to hear about your experiences. At this point, I feel that students who understand and can see the beauty in this text are at a distinct advantage when it comes to upper level math as undergrads and beyond - in terms of content and pedagogue. But I'm not sure I could have read it on my own before I had the other experiences that would have been better if the text had informed them. Which, of course, doesn't make any sense. 

 

Discussions about Apostol/Courant/Hardy and others are welcome as well. 

 

Baby Rudin? (Another ball of fun, I know!) I haven't mastered the text yet, but I am determined to keep at it. It makes me cry; I actually have cried. I find it so beautiful and so satisfying. Hard going, yes.  Such SLOW going. But when an idea really settles in, it produces such strong emotions in me. The ideas are so beautiful; I just end up weeping. 

 

Would love to talk more about this.

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

[Arcadia]

 

Rather than discuss the content (which I really like), I would like to discuss its placement in the curriculum sequence and the implications.

 

My husband and I are not from the US education system so no idea on sequence or college implications

 

Link is to the free archive.org version for anyone who want to read

https://archive.org/details/Calculus_643

 

My hubby finds the archive version too wordy though and he prefers applied math a lot more than theoretical math.  My kids are like me and would just make do with any book they find as long as they don't fall asleep on the book.  No mathematicians in our extended family though.

[Emerald Stoker]

Thank you for mentioning this book, Janice! (Also I see the first review that comes up on Amazon is by mathwonk!) It looks as though it is wonderful; perhaps I might get brave enough to try reading it some day!

 

I should just hide quietly in the corner and listen...and I mostly will, since I don't know anything. But--mostly by way of a bump for you--I did want to mention a couple of other books I have been finding interesting lately (I never took calculus--not offered in small high schools in my province back in the--cough,cough--dark ages of my teens). So I am trying to learn it on my own now that I am old and grey, and have been enjoying two books I found at abebooks: Israel Herstein and Reuben Sandler's Introduction to the Calculus, and Barry Mitchell's Calculus without Analytic Geometry. I found them recommended on the stack exchange website, and am finding them very clear (and Mitchell, I must say, is absolutely hilarious! I laugh out loud several times per chapter.) Also, and I find this very interesting as a math layperson, both books are very short (right around 300 pages each). The problems are fun to do, and so far I understand what's going on! So maybe when I am done those two books, I will, with some trepidation, take a look at the books you mention here!

 

Thanks, Janice!

[quark]

Bumping.

 

Won't be able to contribute but happy to lurk and absorb the awesomeness. :001_wub:

[Julie of KY]

I'm eyeing this book now as it looks fantastic. I've been helping my son through AP calculus and I think he'd love this book.

[Arcadia]

I'm eyeing this book now as it looks fantastic. I've been helping my son through AP calculus and I think he'd love this book.

I also found a 1965 version for Courant's Introduction to Calculus and Analysis Vol I on a french website.

 

There is also a Courant Differential and Integral Calculus on archive.org

 

Vol I

https://archive.org/details/ost-math-courant-differentialintegralcalculusvoli

 

Vol II

https://archive.org/details/DifferentialIntegralCalculusVol2

 

You would probably have to PM Mathwonk for a comparison.

[Kathy in Richmond]

Checking in as an Apostol lover here. Can't comment on Spivak's text since I never bought it. I like Apostol so much that I never saw the need!

 

I went to U Rochester in the 70's as a chemical engineering student. I was first generation to go to college in my family & from a very small rural town, meaning I had no knowledge of what to look for & no real advisors. It was a shot in the dark based on good financial aid and location.

 

For my first year, I had AP credits for calculus & was encouraged to try the honors calculus sequence. I thought about skipping ahead to Calc 2, but I was intrigued by the department's promise that we'd learn a new way of looking at math if we took honors instead, and that it would not be a repeat of the AP class.

 

So...freshman year went well & we learned to write proofs, but the textbook did nothing to draw me in, though...it was a text written by one of the UR faculty, and it was fairly lousy. I did well enough that they did ask me to TA the honors course as a sophomore, though, and that year they switched to Apostol. I was *given* the 2-volume set. I still had no idea how lucky I was! It was AMAZING.

 

TA'ing that year taught me more mathematics than any of my regular courses. It was such a treat! I still remember an afternoon spent poring over the theorems concerning reduction of matrices to diagonal forms. A thing of beauty!  My book still flips open to that chapter, so I must have spent lots of time there, LOL. Having to teach Apostol to other students in recitation really motivated me to learn it.

 

The result: I added math as a 2nd major. Later on, I abandoned engineering entirely & went to grad school in math. So I guess Apostol was one of many motivating factors for that decision.

 

Where do books like Apostol & Spivak fit in the curriculum?

 

I think they're best used as an freshman intro course for prospective math majors. I'm not sure about Spivak, but Apostol can replace linear algebra, multivariable calc, and a first course in differential equations. It includes all that material and more, from a mathematician's point of view, (unlike their typical counterparts taught from an engineering point of view). Like Janice said, it's an intro to real analysis, too, and will help students succeed in rigorous upper level math major courses later on.

 

I used Apostol at home with my son for his last couple of years after he finished BC calculus. He mainly self-taught, but I was able to check his work. Without an answer key, though, that's going to be rough for most kids. So if interested, I'd look for colleges that use one of these books in their freshman honors classes. DS went on to use Baby Rudin in his first year of college and was well-prepared.

 

 

Edited February 20, 2016 by Kathy in Richmond

[lewelma]

Small hijack here.

 

Kathy, how would you recommend my ds attack one of these 2 books?  His multivariate class in July will be using Anton, which is definitely practical.  I don't see how he would get through the class if he tries to use both.  Would you pair it with an analysis class? Or should he just try to do it on his own over a period of years when he has time?  I know he would love one of them for all the reasons Janice describes in her experience

 

And why no answers?  There seems to be some online answers, but is there no solutions manual?

 

Ruth in NZ

[Kathy in Richmond]

Small hijack here.

 

Kathy, how would you recommend my ds attack one of these 2 books?  His multivariate class in July will be using Anton, which is definitely practical.  I don't see how he would get through the class if he tries to use both.  Would you pair it with an analysis class? Or should he just try to do it on his own over a period of years when he has time?  I know he would love one of them for all the reasons Janice describes in her experience

 

And why no answers?  There seems to be some online answers, but is there no solutions manual?

 

Ruth in NZ

 

Hi Ruth,

 

If I'm remembering correctly, your son is using AoPS to teach himself calculus, right? If so, I'd recommend he go directly to volume 2 of Apostol. It covers linear algebra, differential equations, and multivariable calc in that order. I like that because without linear algebra under your belt, you can only do a superficial treatment of the other two topics. An intro to real analysis will be woven into the text (for ex, the multivariable section starts with a treatments of open sets, limits, and continuity in higher dimensions).

 

(I can't speak for Spivak, so hopefully someone else with knowledge of that text will speak up)

 

A kid like yours could self-teach this. The explanations are very good, and while there's no solution manual that I know about, there are short answers to most of the exercises in the back of the book. They're not like AoPS solutions, but good enough to check whether you're on the right track.  And he can always ask on the AoPS forums or shoot me an email if there's a question..

 

When should he tackle this? He'd probably find it interesting & useful to follow along in Apostol during his multivariable class (just match topics). Other than that, I'd just suggest working on his own over the next few years. It's handy to finish before tackling a rigorous real analysis class (on the level of baby Rudin).

 

Why no solutions manuals? Maybe because it hasn't even been revised since the 1969 edition! It's a classic, but not a book that ever was used widely, so probably no big $$ involved for publishers.

 

[SeaConquest]

Kathy,

 

I thought the standard sequence was to do Multivariate Calculus before Linear Alegebra and Differential Equations. Can you elaborate on why you should reverse the sequence? Are there schools who use the approach you've recommended?

 

As a humanities person with a STEM kid, I learn so much from these threads. :)

[Kathy in Richmond]

Kathy,

 

I thought the standard sequence was to do Multivariate Calculus before Linear Alegebra and Differential Equations. Can you elaborate on why you should reverse the sequence? Are there schools who use the approach you've recommended?

 

As a humanities person with a STEM kid, I learn so much from these threads. :)

 

Monique, you're absolutely right that the usual sequence is to take multivariable calculus after regular calc and before linear algebra. You certainly can teach kids to calculate partial derivatives & multiple integrals & even crank through Green's and Stokes theorems at that point. If the goal is to be able to apply those tools in engineering or physics, then there's some sense to that sequence. It gets you to what you want to do faster.

 

But to really understand what's going on mathematically in multiple dimensions, we need to understand how functions behave on those spaces. And that entails understanding when they're continuous, when they're differentiable, & when their inverses exist. That requires a bit of real analysis and linear algebra.

 

For example, look up "Jacobian determinants"...Determinants belong to linear algebra, but the Jacobian is used when you do a change of variables (spherical to rectangular, etc) in a multiple integral. A typical book like Stewart's calculus, for instance, gives formulas for how to compute the Jacobian at that point.

 

Sure, I can teach you how to calculate the determinant in 3 dimensions, but when I teach you what a determinant *is* mathematically, it doesn't seem so much like a bag of tricks pulled out of the blue to be memorized & then promptly forgotten.

 

The Jacobian determinant arises naturally when you study which functions on multidimensional spaces have inverses. And that inverse function is what comes into play in the change of variable formula in multiple integration. Books like Apostol go into all of this in a mathematically correct way.

 

These subjects (multivariable calc, real analysis, linear alg) are SO intertwined, and I think the best approach for a student seriously interested in mathematics is to teach them together as the Apostol texts do.

 

Now that's not the only book that takes that approach. I used Marsden & Tromba's Vector Calculus after AP calc with dd. It introduces the needed linear algebra and real analysis as it goes, & does a super job of it, not just skimming or giving formulas.

 

At Stanford, dd took Honors Multivariable calculus during her freshman year, which used Simon's Introduction to Multivariable Mathematics in the first quarter. If you look at the table of contents, it's divided into 4 parts: (1) linear algebra, (2) analysis in Rⁿ, (3) more linear algebra, and (4) more real analysis...! NOT "partial derivatives" and "multiple integrals," though they're all covered in there.  [btw, I don't recommend this book for self-study since it's pretty dense. It's just lecture notes verbatim without a lot of context.]

 

Diff Eq is much the same as multivariable calc...Knowing linear algebra is important when you get to a study of eigenvalues & eigenvectors, for example.

 

Is all lost if you do a typical multivariable math first? No, I'd never be that pessimistic! You can pick it up later in a good real analysis course and it will make more sense then. But with a capable & motivated student, I'd rather use the integrated approach from the start.

hope that helps!

[Lilaclady]

That is great to know Kathy. My dd will be doing calculus next year with AOPS. Where else can she take linear algebra after doing calc BC ?

[Arcadia]

My dd will be doing calculus next year with AOPS. Where else can she take linear algebra after doing calc BC ?

Stanford OHS (singe course) and CTY JHU are popular here for online. Dual enroll for those whose kids prefer B&M classroom.

 

https://ohs.stanford.edu/academics/courses/linear-algebra

 

http://cty.jhu.edu/ctyonline/courses/mathematics/linear_algebra.html

[Kathy in Richmond]

Hmmm...good question Lilaclady!  My kids studied with me at home, one using Apostol and the other using Linear Algebra Done Right after Marsden's Vector Calculus.

 

I bet there's some good OCW out there or maybe a MOOC? Let me see...

 

I'd recommend the MIT OCW Linear Algebra by Gilbert Strang. It's on OCW Scholar, so it has videos and more helps than the typical OCW course. Strang is a fantastic teacher!

 

 

Edited February 21, 2016 by Kathy in Richmond

[Janice in NJ]

A HUGE "LIKE!" to everything Kathy said. A school like Princeton treats LA and Multi as "companion courses" for math majors. In a perfect world, it would be great if you could take each course with the other as a pre-req. Which you can't. I suspect that's why Kathy understands all of this so well though. :001_smile: They say that as a grad student, you should try to land a Linear Algebra TA position as soon as possible. The course just has that coalescing quality. I've heard that you find yourself making connections left and right.

 

I recall the time on these boards that I complained about Jane Eyre. This was a long time ago. I was reading the book, but I hated it. So I spoke up. A wonderful gal - I'm sorry but I can't remember your name even though the things you said left a huge impact on me that has produced a ripple effect in so many directions; this gal wrote her Master's Thesis on the book. She chimed in, rescued me, and informed my home school, my kids, and my future reads. Big change! When I started, I didn't have enough in me to read Jane. But I just starting reading it anyway. And if I hadn't just started, I would have never have learned what I learned. But of course, what I learned would have made starting the book from the beginning that much better. You can roll the chicken/egg thing around inside of yourself forever and never really know for sure what to do. You just can't read a book for the second time the first time.

 

I honestly think they just push kids through the Calc I, II, Multi sequence before they given them LA because of the college schedule and/or because so many kids will fall away from the math major into other disciplines that don't require LA, so why introduce it too early. However, the subject informs SO much of mathematics. I've heard it said that you can "never have enough LA." And no, an undergrad class is not going to get through the subject. It's a bit like Shakespeare. If you're serious about mathematics, you should get started as soon as you can. It's a fun playground that offers a lot to think about. Have you ever read a Shakespeare play and then seen/heard a reference to the play within the next six months and asked yourself, "How many more references am I completely missing because I haven't read the rest of his plays?"

 

But of course, you can't put your life on hold so you can figure out how to live your life in the most efficient way. In the end, as Kathy said, no one is going to be ruined by going through things the "wrong" way. If the college the student attends puts things in a different order, a student could gain MUCH by pulling Apostol from the library, reading on their own, and approaching their Mult Calc prof during office hours. I suspect they would LOVE to help that kind of student see the connections. Who knows, the student might find a mentor. A lot can happen when things are done well the wrong way too.  :001_smile:

 

Obviously Kathy knows WAY more about this than I do. I do not have a Master's Degree in math. I took my 1st LA course a few years ago, but I think it is a very satisfying subject. (My many-moons-ago BS was in electrical engineering; they was no room in the schedule for LA. No one even suggested it. Although I would have LOVED it. Sigh.) I find it so very satisfying. Addicting. Beautiful. 

 

Kathy - I have Apostol. Haven't worked through it from cover to cover. But you are the 2nd wise person who has suggested it. I would be foolish to argue.  :001_smile: Thank you! 

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

[Janice in NJ]

Another BIG THUMBS UP for Gilbert Strang's course. Agreeing - fantastic teacher!!

 

Janice

Edited February 21, 2016 by Janice in NJ

[SeaConquest]

Kathy, thank you for taking the time to explain. I am tucking all of this wonderful information away for future studies, including my own -- I never loved math until I had an amazing Calc professor in ugrad. I considered double majoring in math, and now wish I had done so. I look forward to exploring these areas of mathematics, so thank you for your generosity in sharing.

Edited February 21, 2016 by SeaConquest

[Kathy in Richmond]

Janice, I just love your passion for learning!! So inspiring! There's nothing like taking advantage of these years when homeschooling winds down to expand our horizons. And it's often in unexpected directions....Off to my piano lessons and music theory books...then to the gym to work on weights...haha so different from the younger version of me. It's NEVER too late!

[Janice in NJ]

Hey Kathy,

 

Depending on where you are with your music theory... I have a COMPLETE set of a college-level Norton music theory course. TE's and all. It's all there. I don't need it and have thought about tossing it, but it seems like such a waste. I would be happy to give it to you. (DS used it for a bit and then headed into another direction for theory)

It's a 1st edition set of this one:  http://books.wwnorton.com/books/webad.aspx?id=4294990317 

 

I played piano through sophomore year of college before heading in other directions. At the time, I would have died if I couldn't play every day. And then of course, I eventually stopped but didn't die. DS played so we own a beautiful grand piano (still can't believe we squeezed a full-size grand into this dinky house, but we did). It is the nicest instrument I have ever had regular contact with, but all I do is dust it. Sometimes the irony of it is remarkable to me. It is one of the things I SWEAR I am going to get back to. Although I suspect that I will probably obsess over it if I do. 

 

So glad you are playing. Sharing your joy!

 

Peace,

Janice

 

 

Edited February 21, 2016 by Janice in NJ

[Kathy in Richmond]

Wow, Janice, what a kind offer! Are you sure? I'd gladly pay for postage :-D Sounds like a very thorough set!

 

I'm truly a beginner at piano, but used to play clarinet so I know treble clef and a bit about chords. I'm currently plunking around in my kids' old piano books, mostly level 3 & 4, and their old theory books. I just bought the Schirmer Library "First Lessons in Bach," and am having lots of fun with that, too. The mathematician inside me just adores Bach!

 

thanks again! :001_smile:

 

 

[Lilaclady]

Thank you so much Kathy and also Janice for starting this post. Really great info.

[quark]

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Edited February 21, 2016 by quark

[quark]

That is great to know Kathy. My dd will be doing calculus next year with AOPS. Where else can she take linear algebra after doing calc BC ?

 

My son moved on to DE. Apart from Arcadia's OHS suggestion which might involve enrollment in the online high school (I'm not sure, please check), there is also OHSx which I think enables you to take the one course at a time online for just uni-level math and physics. I'm not really sure what the difference is between OHS and OHSx but I have it bookmarked just in case DS suddenly decides to not take further math through DE.

 

http://ohsx.stanford.edu/university-math

 

[quark]

.

Edited February 24, 2016 by quark

[Arcadia]

My son moved on to DE. Apart from Arcadia's OHS suggestion which might involve enrollment in the online high school (I'm not sure, please check), there is also OHSx which I think enables you to take the one course at a time online for just uni-level math and physics. I'm not really sure what the difference is between OHS and OHSx but I have it bookmarked just in case DS suddenly decides to not take further math through DE.

 

http://ohsx.stanford.edu/university-math

 

I don't know what is the difference either. OHS has a single course option which was what I am looking at as backup for languages but OHSx has a lower tuition.

 

My guess is OHS is for kids high school and under, while OHSx is for everyone because the x tag usually means the extension arm of an education provider.

 

ETA:

Someone uploaded Apostol's book on the internet so if someone wants to browse before buying a used copy, google is your friend.

Not linking because don't know if book is no longer copyrighted

Edited February 21, 2016 by Arcadia

[quark]

My guess is OHS is for kids high school and under, while OHSx is for everyone because the x tag usually means the extension arm of an education provider.

 

Good guess! I think that sounds like what it is!

 

I just wanted to add, silly though it may sound, that I wish I could invite all of you over for tea today or someday soon for a real life math chat. I won't be much help but I can keep everyone fed and comfortable while DS soaks up all this wisdom. We can have home-made banana fritters with a dash of cayenne pepper if that will tempt anyone to actually make this happen. :laugh:

[Brad S]

I'd recommend the MIT OCW Linear Algebra by Gilbert Strang. It's on OCW Scholar, so it has videos and more helps than the typical OCW course. Strang is a fantastic teacher!

 

Gilbert Strang's book, Linear Algebra and Its Applications, is also terrific.  Although I used an earlier edition many moons ago, I basically learned virtually everything in the course from the book as the teacher was dreadful IMO.  The book was delightful as well as very clear!!

[Kathy in Richmond]

Kathy, DS's university DE course (an honors section) combines Linear Algebra with Differential Equations. He is also currently DE-ing multivariable at a different college. Would this be a good integrated introduction in your opinion? We didn't plan this. It just happened because of a serendipitous combination of interest and his schedule working out for both courses in one semester. He is enjoying the math very much!

 

The unis' is a customized Lay textbook for Lin Alg/Diff Eq. Multivariable is with Stewart. Where would you suggest inserting Apostol or which volume of Apostol? We have the free pdfs Arcadia linked to a while back (thanks Arcadia!).

 

His private tutor introduced eigenvalues/ eigenvectors to him as part of a group theory course so he has had quite a nice introduction to the various connections within the different math areas.

 

May I also ask what you would suggest as a possible next step after this semester (non-contest based)? Self study is a possibility or he also has a choice of more honors sections to concurrently enroll in. I think he would enjoy the group sections vs studying alone at home without solutions to check.

 

 Your son's integrated honors linear algebra with diff eqs sounds like a winner, quark. I've heard good things about the Lay book, but I never used it myself. If he wants to dig deeper in multivariable calc, then I'd suggest volume 2 of Apostol, chapters 8 through 12. I'd start with Ch 8 since there's some nice introductory analysis at the beginning before it gets into partial derivatives. After that, it should be fairly easy to match up topics.

 

In fact the whole of vol 2 would mesh nicely with his current math lineup. He's really soaking up the math this term!

 

What next? Real analysis would be the next logical step, but he could also branch off into different directions at this point (higher level algebra, number theory, etc). The only caution is that I would do real analysis before pde's or complex analysis.

 

If he's enjoying dual enrollment at the uni, could he continue there? The OHSx classes sound like a new name for what dd took through EPGY. They are good, meaty courses, but there's not a lot of human interaction if it's run the same way. DD had an assigned tutor, and she occasionally communicated by email or phone with him, but most days it was working independently.

 

Good guess! I think that sounds like what it is!

 

I just wanted to add, silly though it may sound, that I wish I could invite all of you over for tea today or someday soon for a real life math chat. I won't be much help but I can keep everyone fed and comfortable while DS soaks up all this wisdom. We can have home-made banana fritters with a dash of cayenne pepper if that will tempt anyone to actually make this happen. :laugh:

I was just out for dinner with dh. While I was sippng a nice cup of ginger mint tea at the end of the meal, I was thinking the same thing! Your banana fritters sound heavenly. :)

[quark]

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Edited February 24, 2016 by quark

[quark]

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Edited February 22, 2016 by quark

[raptor_dad]

Kathy,

 

I thought the standard sequence was to do Multivariate Calculus before Linear Alegebra and Differential Equations. Can you elaborate on why you should reverse the sequence? Are there schools who use the approach you've recommended?

 

 

 

Our local university's honors high school math calculus component for 8th graders through high school takes the same approach and their description[1]  and Kathy's sound almost verbatim:

 

 

Our Calculus I course actually looks quite similar to any AP/IB or college Calculus course in terms of topics covered. (The mathematical culture and writing expectations are very different; see below.) Our Linear Algebra and Multivariable Calculus courses are at a different level, however. Most Linear Algebra (or Linear Algebra and Differential Equations) courses are very heavy on matrix algebra and other calculations, whereas ours takes a geometric, theoretical approach. Once we have covered Linear Algebra, we can do Multivariable Calculus in full generality in Calculus III; the vast majority of Multivariable Calculus courses taught at colleges or high schools use the last third of a large Calculus text, which does not assume any Linear Algebra and therefore cannot cover all of the relevant definitions and ideas. We use a Vector Calculus book which assumes knowledge of matrices and linear transformations and can therefore “tell the whole story.†There is nothing inherently wrong with using a standard Calculus book for Multivariable–in fact, that’s why my own Multivariable class was like as a student–but anybody who is interested in mathematics, physics or certain types of engineering will eventually have to learn the more general version, so there is an advantage to doing it that way the first time.

 

For those with a mathematical background, the following list gives a short summary of the curriculum through the six-semester UMTYMP Calculus sequence. If this is outside of your comfort zone and would make your eyes glaze over, please feel free to skip over the list!

 

Calculus I Fall.  Limits, derivatives, applications and geometric interpretation of derivatives. Extended Mean Value Theorem.

Calculus I Spring. Riemann integral, integration techniques and applications, rigorous treatment of sequences and series with ε definitions. 

 

Calculus II Fall. Differential equations. 3D coordinate systems. Vectors and products. Set theory, logic and methods of proof. More with ε − δ definitions.

Calculus II Spring. Theoretical linear algebra. 

 

Calculus III Fall. Differential geometry of curves and surfaces in R 3 . Multivariable differentiation and optimization using matrices. Limit definition of multivariable derivative. Taylor’s Theorem in higher dimensions.

Calculus III Spring. Topology of R n, vector fields, classical vector analysis (Stokes Theorem, etc.).

 

 

[1] http://mathcep.umn.edu/docs/why-UMTYMP-2013.pdf has a little more detail comparing this sequence and their honors undergrad sequence to the standard science and engineering calculus course.

[lewelma]

Kathy, you may have just convinced ds.   :001_smile:  I read him your post, and he was very interested in this alternative order that Mike has also suggested. Biggest problem is that ds is just chomping at the bit for multivariate calc.  I'm not sure why, but seems to be some sort of right of passage in his eyes.  But luckily for us, linear algebra is taught in July 2016, Analysis in Feb 2017, and Multivariate in July in 2017.  Linear algebra uses Anton's Elementary linear algebra (which given his calc book, I'm assuming in more practical than theoretical), and Analysis uses Judson's Abstract Algebra. Are those books going to get the job done?  Should he read Apostol concurrently during all 3, or wait until multivariate?

 

Can you tell my ds what he would learn in linear algebra that he has not already covered with AoPS precalc and calc?

 

Also, is linear algebra more useful to the IMO than multivariate calc?  Or is its content also excluded?

 

Thanks!

[Kathy in Richmond]

Kathy, you may have just convinced ds.   :001_smile:  I read him your post, and he was very interested in this alternative order that Mike has also suggested. Biggest problem is that ds is just chomping at the bit for multivariate calc.  I'm not sure why, but seems to be some sort of right of passage in his eyes.  But luckily for us, linear algebra is taught in July 2016, Analysis in Feb 2017, and Multivariate in July in 2017.  Linear algebra uses Anton's Elementary linear algebra (which given his calc book, I'm assuming in more practical than theoretical), and Analysis uses Judson's Abstract Algebra. Are those books going to get the job done?  Should he read Apostol concurrently during all 3, or wait until multivariate?

 

Can you tell my ds what he would learn in linear algebra that he has not already covered with AoPS precalc and calc?

 

Also, is linear algebra more useful to the IMO than multivariate calc?  Or is its content also excluded?

 

Thanks!

 

It does seem that *everyone* does multivariate calc as the next course after calc here in the US, too..(me included back in my student days).

 

I love the schedule you have laid out better, though. How convenient & nice of your local uni to cooperate! :001_smile:

 

I've never seen either the Anton linear algebra text nor the Judson abstract algebra. I read some comments online, and Judson sounds very promising. But I'd probably have something like Apostol or Strang on the side to supplement the Anton.

If you do go with Apostol, then yes, he could definitely start reading at any time since vol 2 begins with linear algebra and integrates in the real analysis before tackling multivariable calc.

 

AoPS precalculus has a terrific intro to linear alg, but there's still lots more good stuff to come! For instance: vector spaces, subspaces, direct sums, bases, change of bases, null spaces or kernels, rank, trace, inner product spaces, characteristic polynomials, self-adjoint operators, the Spectral Theorem & Jordan canonical form,....that's just what I can come up with tonight... Linear algebra is a deep & satisfying subject & I keep learning more each time I tackle it.

 

As for IMO, it doesn't cover anything beyond precalculus topic-wise, but it just requires amazing amounts of depth & the ability to make connections and think deeply.  At least for myself, I use more of my deep thinking muscles on linear alg & real anal than I do on multivariable calc.

 

Edited February 23, 2016 by Kathy in Richmond

[lewelma]

Got it!  I'll read this to ds tonight and contact the head of department at the local uni later this week. Thanks heaps.

 

Ruth in NZ

[raptor_dad]

A while back you mentioned a camp alum in CS maybe at Harvard piquing his interest with tales of lucrative jobs. Linear algebra is a great subject to explore CS applications in.... Google PageRank, computer graphics, econ/ecology/linguistics with Markov chains, lots of fun stuff. You  might want to take some time to explore that in addition to the pure math.