# Differential Equations, Linear Algebra and Multivariable Calculus

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I’m trying to figure out math for my dd’s senior year.  I see Khan Academy has instruction for all three of these on their website, but does anyone know if they cover all the main topics?  Linear algebra and differential equations both look like they are extremely short.

My dd has got calculus down pretty well at this point, but it’s not her favorite subject.  I was reading that linear algebra involved more proofs and abstract thinking vs the visualization needed for calculus.  Since she really liked doing proofs in geometry, would this type of math likely be more up her alley than calculus?  Also, what is differential equations like?

Another thought — is there any point in her considering taking statistics II at the cc if she does well on the AP exam for statistics?

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I think she just finds calculus kind of dull.  I’m not sure why.  I don’t believe she has done any proofs in calculus, though.  Unless proofs are introduced in Multivariable Calculus?  She used Derek Owens for Calculus AB and then Khan Academy to review and cover the BC material.  It’s not that she hates it.  It’s just at the bottom of her list as far as subjects go.

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MIT has a great OpenCourseWare multivariable calculus course. Lectures, problem sets, and exams with solutions are all available. I went through it with my senior this year. It was enjoyable.

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6 minutes ago, square_25 said:

You'll need linear algebra before multivariable calculus, by the way.

Really?

Strange.

Can't say I've ever seen that assertion before.

If Calc doesn't excite her, Diff Eq is not likely to, either. I would try linear algebra from a MOOC (trying to find a good prof) as she might like that. Dh & I would recommend some matrix theory. 😉

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If she likes proofs, a Discrete Math or an Analysis course might suit her.  Hmm.  She can't take Analysis without Multivariable Calc though, I wouldn't think.

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4 minutes ago, square_25 said:

I mean, I guess you could take them out of order, but I don't see why you would want to be introduced to vectors and matrices in an already difficult context.

I think this depends on how you studied it (or, in some cases, in retrospect, how you wish you'd studied it). I took Calc 1, 2 & 3, then Matrix Theory. (I never took a proof-based "linear algebra" class. I'm a Mechanical Engineer. Wasn't required.) Dd#1 is planning to be a math major. None of the three "final picks" among her colleges recommended or required linear algebra before Calc 3, but some have Calc 2 as a prereq for Linear Alg (others have Calc 3 as a prereq). I did see a Calc 4 (Advanced something something) with Linear Alg as a prereq  (or coreq?) at one. (Some have Matrix Theory as its own class & others don't.) I'm sure some schools do it differently, though. I've just never heard that assertion.

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20 minutes ago, square_25 said:

Well, it's all in terms of vectors and matrices. You ought to know what those are and how to multiply them before you can do multivariable calculus in a serious way. I mean, I guess you could take them out of order, but I don't see why you would want to be introduced to vectors and matrices in an already difficult context.

What did her Precalc class cover? My dd's class has been doing matrices for almost 2 months. I took Trig in high school back in the last century so I'd never seen matrices when we started Calc our senior year.

Edited by chiguirre
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Okay, you all are talking over my head with the math, lol.

36 minutes ago, square_25 said:

I'd start with Discrete Math before analysis, anyway :-). I don't think you need multivariable calculus for analysis, but it's been a while... I could be forgetting something.

Thanks! So, any ideas where I can find a discrete math class?

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1 hour ago, Caroline said:

MIT has a great OpenCourseWare multivariable calculus course. Lectures, problem sets, and exams with solutions are all available. I went through it with my senior this year. It was enjoyable.

Edited by Mom0012
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1 hour ago, square_25 said:

You'll need linear algebra before multivariable calculus, by the way.

I've never seen this either.

1 hour ago, RootAnn said:

(I never took a proof-based "linear algebra" class. I'm a Mechanical Engineer. Wasn't required.) Dd#1 is planning to be a math major. None of the three "final picks" among her colleges recommended or required linear algebra before Calc 3, but some have Calc 2 as a prereq for Linear Alg (others have Calc 3 as a prereq). I did see a Calc 4 (Advanced something something) with Linear Alg as a prereq  (or coreq?) at one. (Some have Matrix Theory as its own class & others don't.) I'm sure some schools do it differently, though. I've just never heard that assertion.

Interesting.  My three sons are all engineers and all had to take linear algebra but DH, also an engineer, never took it.  So, maybe it's a newer requirement?  Here, Calc 2 is a prereq for Linear Algebra and Differential Equations.  DH always says Diff Eq is Calc 4 but I don't know.  Ds3 took Diff Eq without Calc 3 and said some Calc 3 knowledge was assumed and he had to figure it out.  He took Linear Algebra and was fine.  Dd will take linear algebra next year with only calc 2 background and I'm hoping she'll be ok.

Edited by Kassia
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6 minutes ago, square_25 said:

I’m really confused by this sequence. You use no calculus in linear algebra whatsoever but you use vectors and matrices in multivariable calculus. I’ve taught all of these! What am I missing here?

At Georgia Tech, linear algebra used to be covered in the first part of calculus 2. Now students take linear algebra after calculus 1, then multivariable after that. But really, it was just a name change of the class. Calculus 2 was mostly linear algebra.

The MIT course I referred to above has a vectors and matrices unit as the first unit. Then it goes into partial derivatives, double integrals and triple integrals.

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I think there is a naming thing, but I checked & this book is used by one of DD's top two college choices for their Linear Alg class which has a Calc 2 prereq. Note the overview description:

"This book is for sophomore-level or junior/senior-level first courses in linear algebra and assumes calculus as a prerequisite."

Adding that their Calc sequence uses 6th edition Stewart Calculus.

And this book is used in their second Linear Algebra course in the sequence.

Edited by RootAnn
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Sorry, BTW, Lisa. Didn't mean to run off on a tangent in your thread.

@square_25 I added another book link in my previous post. That particular college lists the texts in the course descriptions, but you have to look up textbooks separately at the others. I think there is a naming & coverage difference at different schools.

So, what are we looking for now -- a Discrete Math course?

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4 minutes ago, RootAnn said:

Sorry, BTW, Lisa. Didn't mean to run off on a tangent in your thread.

@square_25 I added another book link in my previous post. That particular college lists the texts in the course descriptions, but you have to look up textbooks separately at the others. I think there is a naming & coverage difference at different schools.

So, what are we looking for now -- a Discrete Math course?

I’m just glad everyone is enjoying the discussion.😊

Honestly, I don’t know what I’m looking for, but maybe discrete math.

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4 minutes ago, square_25 said:

Yeah, sorry!!

I know AoPS has probability and combinatorics classes. Would that work?

I will take a look.  Honestly, this may just be a get it done year for us with math since my dd isn’t super interested and would like to carry on with 4 languages and a slew of other things.

I guess that’s why I was originally wondering about the Khan Academy materials.

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What is her goal here? Is she looking to do math or engineering in college? Is she just an advanced math student who needs something rigorous for senior year?  Am I understanding correctly that she has completed math through Calc BC but has also done AP Stats?

If she is considering engineering, I think doing more calculus at the CC is probably going to serve her better than Stats 2.

Some school districts consider computer science a subject in the math category. If you are just looking to fill the math hole, and she doesn't have specific goals, that might be an option.

Editing to add that I missed your last post, and now see that you are looking to "get 'er done" as far as math! If that's the case, I think the stats 2 would be fine, or comp sci if she has an interest. The AP Comp Sci through PA Homeschoolers was really quite good.

Edited by GoodGrief1
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re: calc 3 and linear algebra. It is quite uncommon to require linear algebra as a prereq for multivariable calculus; it is generally taught on an as-needed basis. Calculus is usually required as a prerequisite for linear, not because the linear algebra content itself requires calculus, but because a certain level of mathematical maturity is required. Calculus examples are used in some of the popular linear algebra textbooks (Anton, for one) but they are asterisked and could be omitted. There are other courses (modern algebra, for example) where technically they could be done after precalculus (there are modern algebra texts that are structured for intro to proofs) but it would be rare indeed for a student to have that level of maturity and not had calculus.

re: op, if she thinks calculus is easy and boring I would not do diffeq or mvc, I think it would be more of the same. I think that a discrete math course would be a wonderful idea. I had taken calc 1-3, linear (computationally oriented), diffeq, and stats, all of which were easy and boring. I took discrete, which at my college was intro to proofs, but it was also the last class required for my math minor, and halfway through the semester I changed my major to math. I think I'd consider the following:

The aops discrete math classes -- the intro level might not be very challenging to someone who has already had calculus, but the topics are fun and it certainly will fill the bill of a senior math class. If nt/combinatorics are completed I'd say 1cr discrete math. For each one I'd say a semester's credit would be totally reasonable.

The stats II -- AP stats often doesn't transfer in very well, and especially if she took stats at the CC. Stats II should transfer as something.

A CC liberal arts math class -- again, it will be a bunch of fun topics that won't be especially challenging, but will be generally chosen to be of either high interest or high applicability. It'd probably end up transferring and counting for a general elective.

Doing math for liberal arts on your own with Lippman's math and society (free) and/or jacobs MHE, choose your own units based on interest.

Many of these would be great one semester courses, so you could mix and match. I've headed towards the easier and of more interest to the humanities side because you mentioned her desire to git'r'done and study languages.

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5 hours ago, GoodGrief1 said:

What is her goal here? Is she looking to do math or engineering in college? Is she just an advanced math student who needs something rigorous for senior year?  Am I understanding correctly that she has completed math through Calc BC but has also done AP Stats?

If she is considering engineering, I think doing more calculus at the CC is probably going to serve her better than Stats 2.

Some school districts consider computer science a subject in the math category. If you are just looking to fill the math hole, and she doesn't have specific goals, that might be an option.

Editing to add that I missed your last post, and now see that you are looking to "get 'er done" as far as math! If that's the case, I think the stats 2 would be fine, or comp sci if she has an interest. The AP Comp Sci through PA Homeschoolers was really quite good.

Hmm, hadn’t even thought about computer science.  I think pretty much anything would excite her more than calculus.  And she liked stats, so stats II could work then, too.

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2 hours ago, kiana said:

re: calc 3 and linear algebra. It is quite uncommon to require linear algebra as a prereq for multivariable calculus; it is generally taught on an as-needed basis. Calculus is usually required as a prerequisite for linear, not because the linear algebra content itself requires calculus, but because a certain level of mathematical maturity is required. Calculus examples are used in some of the popular linear algebra textbooks (Anton, for one) but they are asterisked and could be omitted. There are other courses (modern algebra, for example) where technically they could be done after precalculus (there are modern algebra texts that are structured for intro to proofs) but it would be rare indeed for a student to have that level of maturity and not had calculus.

re: op, if she thinks calculus is easy and boring I would not do diffeq or mvc, I think it would be more of the same. I think that a discrete math course would be a wonderful idea. I had taken calc 1-3, linear (computationally oriented), diffeq, and stats, all of which were easy and boring. I took discrete, which at my college was intro to proofs, but it was also the last class required for my math minor, and halfway through the semester I changed my major to math. I think I'd consider the following:

The aops discrete math classes -- the intro level might not be very challenging to someone who has already had calculus, but the topics are fun and it certainly will fill the bill of a senior math class. If nt/combinatorics are completed I'd say 1cr discrete math. For each one I'd say a semester's credit would be totally reasonable.

The stats II -- AP stats often doesn't transfer in very well, and especially if she took stats at the CC. Stats II should transfer as something.

A CC liberal arts math class -- again, it will be a bunch of fun topics that won't be especially challenging, but will be generally chosen to be of either high interest or high applicability. It'd probably end up transferring and counting for a general elective.

Doing math for liberal arts on your own with Lippman's math and society (free) and/or jacobs MHE, choose your own units based on interest.

Many of these would be great one semester courses, so you could mix and match. I've headed towards the easier and of more interest to the humanities side because you mentioned her desire to git'r'done and study languages.

Thanks, Kiana.  I had noticed that the stats II class wouldn’t transfer to the university closest to us, so it made me wonder if the class was worth taking.  Thanks for the other suggestions.  I will look more closely at each of them and see if there is something there that sparks an interest for her.

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8 hours ago, square_25 said:

I’ll take a look at what they have :-). I think making math more fun for a year might be nice, though... but not if you don’t think she’ll enjoy it.

Don’t get me wrong.  It would be great if I could find something she’d enjoy.  I’d love that.  I just don’t want to get her into something really intense and then have it turn out that she doesn’t enjoy it.  She had been exploring the idea of actuary work, and had been thinking about prepping for the first actuary exam for her math this year — probability — but then I think she got so sick of calculus, she started to think she might not like that kind of work.  Prepping for that exam is probably still the thing she is most interested in doing.

i guess if she did that, I could call it a probability course, culminating in the actuary exam, vs “actuary exam prep”.  Hmmm.

Edited by Mom0012
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10 minutes ago, square_25 said:

Just to reiterate: if she likes proofs and her calculus class had none, the issue isn’t that she doesn’t like math, it’s that she doesn’t like rote calculations. I can’t tell you whether she’d enjoy actuarial work, but a basic class like Intro to Probability at AoPS would clarify what she enjoys and doesn’t. And it wouldn’t be too onerous, either. It’s really a gentle, fun, proof-based intro.

I will take a look at AOPS’ probability class.  Thank you.

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16 minutes ago, square_25 said:

It’s relevant to actuarial work, too, since it’s probability :-). I’ve taught it, it’s a fun class :-).

So, it would be the Intro to Counting and Probability class that I should look at?

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13 hours ago, Mom0012 said:

Honestly, this may just be a get it done year for us with math since my dd isn’t super interested and would like to carry on with 4 languages and a slew of other things.

See if she is interested in something like an independent study on math and languages since your daughter likes proofs.  Kathy in Richmond’s daughter majored in math and classics at Stanford.

DS13 did Introduction to Counting & Probability and Introduction to Number Theory courses. He wasn’t interested enough to do the intermediate courses and he is more of an applied math person all along.

DS14 did the books for those and did the course for Intermediate Number Theory and he enjoys classes where he needs to learn. He wasn’t using the course for high school credit.

My library actually has copies of the AoPS books and solution manuals. So you might want to check there first. I bought the books for DS14 and then he decides whether he wants to just self study or do the online course as well.

DS14 makes his own math choices, I just budget and pay. He finds linear algebra the hardest so he picked to do CTY Linear Algebra as a non-credit foundation, then SPCS Multivariable Differential Calculus followed by SPCS Multivariable Integral Calculus. He would likely do SPCS Linear Algebra in Fall semester as we are concentrating on humanities (his weak area) in summer.

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See﻿﻿﻿ if sh﻿e﻿ is interes﻿ted in something like an in﻿dependent study ﻿on math and languages since your daughter likes p﻿ro﻿o﻿f﻿﻿s﻿.﻿﻿﻿﻿﻿  ﻿Kathy in Richmond’s daughter majored in math ﻿and classics at Stanford.﻿﻿﻿﻿﻿﻿﻿﻿﻿﻿

DS13 did Introduction to Counting & Probability and Introduction to Number Theory courses. He wasn’t interested enough to do the intermediate courses and he is more of an applied math person all along.

DS14 did the books for those and did the course for Intermediate Number Theory and he enjoys classes where he needs to learn. He wasn’t using the course for high school credit.

My library actually has copies of the AoPS books and solution manuals. So you might want to check there first. I bought the books for DS14 and then he decides whether he wants to just self study or do the online course as well.

DS14 makes his own math choices, I just budget and pay. He finds linear algebra the hardest so he picked to do CTY Linear Algebra as a non-credit foundation, then SPCS Multivariable Differential Calculus followed by SPCS Multivariable Integral Calculus. He would likely do SPCS Linear Algebra in Fall semester as we are concentrating on humanities (his weak area) in summer.

Arcadia, what would an independent study on math and languages look like?

Are you saying the C&P classes are not applied math?  Or that they are, but that your son still didn’t care for them?  I am wondering because when my dd was looking at becoming an actuary, it was recommended that she get a degree in applied math vs actuarial studies in order to excel in the field and I wasn’t sure what was meant by that.

I will see if our library has the AOPS books available.  Thank you!

Oh, and what is “SPCS”?

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2 minutes ago, Mom0012 said:

Arcadia, what would an independent study on math and languages look like?

Are you saying the C&P classes are not applied math?  Or that they are, but that your son still didn’t care for them?

Oh, and what is “SPCS”?

I am good with creating and doing independent studies but bad at quantifying what would be worth a high school credit. You could start a thread on that because math and languages have lots of interlap actually in the logic sense and probably in more areas that I am unaware of.

The AoPS C&P courses are more theoretical than typical high school C&P topics. My son was thinking of designing sports cars and fighter airplanes as a career then so definitely prefer a get it done Math than AoPS style of C&P. You can take a look at Except 1 and Except 2 on this link https://artofproblemsolving.com/store/item/intro-counting

SPCS is Stanford Precollegiate Studies https://ulo.stanford.edu/mathematics

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15 hours ago, RootAnn said:

I think there is a naming thing, but I checked & this book is used by one of DD's top two college choices for their Linear Alg class which has a Calc 2 prereq. Note the overview description:

"This book is for sophomore-level or junior/senior-level first courses in linear algebra and assumes calculus as a prerequisite."

Adding that their Calc sequence uses 6th edition Stewart Calculus.

And this book is used in their second Linear Algebra course in the sequence.

That had been my experience as well.   Cal 1, 2 and 3 were taken then you did other stuff (assuming you were taking lots of math).  I would have liked to take Linear Algebra earlier though.   Cal 2 kicked me around, but Linear Algebra was breath of fresh air.

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5 hours ago, Mom0012 said:

So, it would be the Intro to Counting and Probability class that I should look at?

I think that the first C&P class will be too basic for someone who has had Calc I and II and a semester of stat. Trinqueta is a good student and likes math, but she's not super advanced, and she took that class in 8th grade. I think your dd would find a better fit with the Intermediate class although she might need to read through the Intro book on her own if she hasn't covered all the material.

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1 hour ago, Mom0012 said:

Are you saying the C&P classes are not applied math?  Or that they are, but that your son still didn’t care for them?  I am wondering because when my dd was looking at becoming an actuary, it was recommended that she get a degree in applied math vs actuarial studies in order to excel in the field and I wasn’t sure what was meant by that.

There are different kinds of applied math. Someone who's looking at engineering would be looking more at the calculus side. Someone who's looking at actuarial science would need the probability side (and the calculus side, but).

University-major applied math is still pretty theoretical. I went back and looked at the requirements for one school I attended. They required calc 1-3, linear, diffeq, modeling, advanced calculus 1-2 (this is a theory class, calculus with proofs), a survey of probability and statistics, and three electives. Their electives all included significant quantities of theory. Examples were differential equations with proofs, complex analysis (calculus with proofs and complex numbers), numerical analysis (again, proofs). They also needed to pursue an approved minor. Someone who was looking at actuarial science, for example, might choose something like economics or finance. Someone else might choose physics because it's more in line with their interest.

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1 hour ago, square_25 said:

It's true that it's a fairly low level class :-). But I think it'd be better to start there and see if you like the material rather than starting with more advanced stuff.

I agree. I'd skim through it and if it piqued interest, go for the intermediate -- if not, choose another topic.

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Thanks again!  I showed her all the options and it sounds like she’s interested in either taking a discrete math class or statistics II at the cc.

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• 7 months later...

Updating -

She took an online discrete math class through the cc and liked it. Her professor commented that she was really great at proofs. So, she really loved proofs when she did geometry and she liked them in discrete math. Maybe she would like calculus better if it was a proof-based class.

Depending on what college she winds up at and how the credits transfer, she may need to take calculus again, so that’s something I’ll point out to her in case that is an option.

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Another vote for introductory counting and probability (if she hasn't taken it yet), intro to number theory, and/or intro to programming in python.  All are offered at AoPS, or the math classes you can do self-study with the textbooks.

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On 12/4/2019 at 1:32 PM, Mom0012 said:

Updating -

She took an online discrete math class through the cc and liked it.

In case anyone is still following this part of the thread, both of my dds' universities require multivariable calculus before linear algebra. Both are engineering schools (but neither of mine are engineers) and one is very strong in STEM overall.

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