What are the best books to study Multivariable Calculus? Something like the AOPS series. Thanks

[Guest derdepperte]

The art of problem solving series only cover to single variable calculus, and I need a resource to continue the road of calculus-dom. I think books are the best format to impart wisdom, so I would like to find one. Any suggestion? Is there any book that's the equivalent of AOPS for higher mathematics? Thank you.

[regentrude]

I asked a similar question a while ago, here is the link to that thread:

http://forums.welltrainedmind.com/topic/455093-multivariable-calculus/?hl=%2Bmultivariable+%2Bcalculus

[Kathy in Richmond]

Here are a few more old threads for you:

 

Multivariable Calculus

Multivariable Calc Question

Multivariable Calc

 

I don't know of any textbook choices similar to AoPS in style, though :sad:

[Mike in SA]

MIT-OCW all the way.  It'll be hard to find a closer fit unless you develop the curriculum yourself.

[1053]

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[Kathy in Richmond]

We just chose our path based on what best suited the student & didn't worry at all about earning credit. Ds and dd both took multivariable, diff eqns, and linear alg at home with me.

 

You don't need official grades in every single course for college acceptance; our mommy grades in those courses were backed up with other testing, math mentor LORs, and math competition results. As far as credit goes, my son did end up earning some credit by successfully completing end of course challenge exams once he arrived at college. My daughter chose to repeat those three courses at the honors level at her university (and they were MUCH tougher, far more abstract beasts there!)

[Mike in SA]

We just chose our path based on what best suited the student & didn't worry at all about earning credit. Ds and dd both took multivariable, diff eqns, and linear alg at home with me.

 

You don't need official grades in every single course for college acceptance; our mommy grades in those courses were backed up with other testing, math mentor LORs, and math competition results. As far as credit goes, my son did end up earning some credit by successfully completing end of course challenge exams once he arrived at college. My daughter chose to repeat those three courses at the honors level at her university (and they were MUCH tougher, far more abstract beasts there!)

 

100% agree.  We grade DS according to state standards for his age.  Why does anyone need to know he actually used MIT for middle school life science?  He wants depth, and we are happy to support him.

 

College credit, btw, can limit your options going forward.  Not a problem for most state schools, but some of the elites don't want to see more than X credits in your past, and you can't simply choose not to report the credits attempted.

 

Most kids are better off repeating in a more rigorous section at the university.  If one is not available, just about all universities make accommodations for classes not needed.

[SeaConquest]

100% agree.  We grade DS according to state standards for his age.  Why does anyone need to know he actually used MIT for middle school life science?  He wants depth, and we are happy to support him.

 

College credit, btw, can limit your options going forward.  Not a problem for most state schools, but some of the elites don't want to see more than X credits in your past, and you can't simply choose not to report the credits attempted.

 

Most kids are better off repeating in a more rigorous section at the university.  If one is not available, just about all universities make accommodations for classes not needed.

 

I have never heard this. Do you have a source? I started undergrad with around 60 units of credit from DE/CC and credits earned from a military language school, and it was never an issue. All of the credit was elective credit, but it did place me into higher level Russian courses immediately.

[EndOfOrdinary]

I have never heard this. Do you have a source? I started undergrad with around 60 units of credit from DE/CC and credits earned from a military language school, and it was never an issue. All of the credit was elective credit, but it did place me into higher level Russian courses immediately.

All of my math credits done during summers at CalTech and my CC credits transferred as well. It got me through a fairly large chunk of my credits for my math certification.

 

I know that Community College credits will not transfer to elite schools if you are dual enrollment, but I had not heard this about four year university credits. Those have been asked for on all the applications I have looked at, and considered transfer credits. Associates and CC credits are to be declared, but often do not garner credit, they just place you into higher level classes. You are considered a freshmen and not transfer.

[Mike in SA]

It isn't an issue with transfer of credits -- it has more to do with the allocation of available spots and freshman status.  Coming in as a transfer is very different.  I mentioned it because of how quickly we look at DE as an option.  It's ok to an extent, but eventually you stop being a freshman.

[Butler]

I have never heard of "too many" transfer credits being a problem. Either the receiving school gives credit and the student starts as a freshman with advanced standing (which can come in handy when registering since the freshman window usually opens last) or the receiving school gives no credit but the student can take classes at a more advanced level (if desired) as a freshman. There are benefits to either scenario. A student starting with many credits can either graduate sooner than the usual four years and save $$ or he/she can keep to the four year schedule and take many more advanced/graduate level courses and beef up the resume or grad school application.

 

What schools have you found that disqualify a highschool dual enrollment student as a freshman?

[1053]

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[Kathy in Richmond]

From the MIT admissions site:
 

"I have been attending college during high school. Should I apply as a transfer?
The transfer process is intended for students who have finished high school and completed at least one year of college. If you are still in high school, you are considered a freshman applicant regardless of how many classes you may have taken at the university level."

 

It is my understanding that as long as your student takes college classes before their high school graduation date, then they will be considered as freshmen in the admissions process. In fact, we've met kids there who entered with LOTS of dual enrollment credits.

 

They may, if they bring in enough credits (transfer credit from dual enrollment is evaluated on a case by case basis), be eligible for sophomore standing at mid-year. But even the students who choose early sophomore standing usually stay around for the full four years at MIT.

 

1053, maybe you were referring to this question about true transfer students, ie, those who took college courses for credit after they graduated from high school but before applying to MIT: (again, from the admissions website)

 

"I have completed more than 2.5 years of college. Am I still eligible to apply?
Unfortunately, no. We do not admit students who will complete more than 2.5 years of college by the time they enter MIT. We encourage you to complete your undergraduate degree at your current institution and apply to MIT for graduate school."

[Mike in SA]

It does happen, but depends on the school and the interpretation.  Some only care if the credit was earned from an actual university (meaning DEs may be ignored), but some consider any credit to be a credit, and once you hit a certain level, you are no longer a freshman.  I've had a Harvard recruiter mention this to me once.

 

I do not know of many schools who deny admissions based upon credit like MIT -- but it can affect the application pool you enter under (so selectivity is higher), and sometimes the scholarships you are eligible for can decrease.  Transferability is another subject altogether.

 

Please keep in mind that 95%+ of schools really don't care -- as far as I am aware, this only applies to a select few at the elite level.  For most cases, the savings from earning credits before college way outweigh the value of the scholarship foregone.

 

Sorry - I didn't mean to start a debate!

[8filltheheart]

My kids take almost all of their DE credits at 4 yr universities (and it is still considered de). The only issue we have ever encountered is whether or not they would be accepted as credit or simply used for advanced placement but still having the same number of required credit hours to graduate. They have never been viewed negatively as too many hrs.

 

As far as homeschooling or at a university, we have opted for the DE route for the classes that are beyond my abilities. For us that means math and science. Lit/comp/history I can manage at home. I'm very glad we went the DE route bc my kids have thrived in the college classrooms and it was a nice way to transition to classes outside of home before leaving as a full-time student.

 

Eta: Mike I think you are confusing credits which make you a transfer student with DE credits. They are not the same thing. If you take any credits after graduating from high school you are automatically considered a transfer student. I have only ever heard of 1 school considering you a transfer student if crossed a certain credit hr threshold via DE and that was NOT a top school. Top schools are used to kids havIng lots of AP, IB, and DE credit hrs. They just don't normally give you credit hrs for them.

[Mike in SA]

Eta: Mike I think you are confusing credits which make you a transfer student with DE credits. They are not the same thing. If you take any credits after graduating from high school you are automatically considered a transfer student. I have only ever heard of 1 school considering you a transfer student if crossed a certain credit hr threshold via DE and that was NOT a top school. Top schools are used to kids havIng lots of AP, IB, and DE credit hrs. They just don't normally give you credit hrs for them.

 

No, not confusing them -- if you have a school that you are interested in, check on their policies on treatment.  That's all I meant to suggest.  Again, it really is nothing to the vast majority of schools.  I really wish I hadn't said anything...  Oh, well!

[8filltheheart]

It does happen, but depends on the school and the interpretation. Some only care if the credit was earned from an actual university (meaning DEs may be ignored), but some consider any credit to be a credit, and once you hit a certain level, you are no longer a freshman. I've had a Harvard recruiter mention this to me once.

 

I do not know of many schools who deny admissions based upon credit like MIT -- but it can affect the application pool you enter under (so selectivity is higher), and sometimes the scholarships you are eligible for can decrease. Transferability is another subject altogether.

 

Please keep in mind that 95%+ of schools really don't care -- as far as I am aware, this only applies to a select few at the elite level. For most cases, the savings from earning credits before college way outweigh the value of the scholarship foregone.

 

Sorry - I didn't mean to start a debate!

No, not confusing them -- if you have a school that you are interested in, check on their policies on treatment. That's all I meant to suggest. Again, it really is nothing to the vast majority of schools. I really wish I hadn't said anything... Oh, well!

I think you misunderstood Kathy's post. MIT does not deny admission to freshman due to number of DE hrs. That quote was for transfer students.

 

ETA: I also wanted to point out that tippy top schools do not really offer merit scholarships. Everything offered is in terms of financial aid. Parents are expected to pay their EFC based on the NPC of the school. The lower ranked schools, the ones with the most scholarship $$ are also the school's most willing to give credit for DE credits. So, say your student has completed 60-70 credit hours. At Harvard they are accepted as a freshman (Harvard does not give credit for any college courses used toward a high school diploma) and still need 4 yrs to graduate and the cost is based upon income and assets. (Over-simplification, but generally true.). Student accepted into state U as a ***freshman*** and eligible for all freshman scholarships and may end up with significant credit hrs toward graduation and can graduate in 2 to 2 1/2 yrs. After their first semester they achieve advanced standing. (Our Ds was admitted as a freshman, but in Jan he will be a jr.)

[1053]

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[Butler]

Grad school acceptance (especially to the top schools in their fields) is dependent on much more than grades and courses. Even if the above hypothetical student ended up at local mid level university he/she wouldn't necessarily want to graduate quickly. He/she would need time to form relationships with professors who will supply the required and very important letters of recommendation. He/she would also need time to get involved in research. As far as courses go, taking longer to graduate would give him/her more time to take advanced/graduate courses which (at least for math, but I'm pretty sure it's the same in other fields) are often expected. (Evidence of completion of pre-requisite courses is not usually necessary to start at an advanced level. Placement into higher level courses can be obtained through other means than a transcript)

[kiana]

What Butler says. Math is one of the fields where it is the easiest to convince them to give you advanced standing if not advanced credit. It is most common to have a cumulative final where a student who can pass the final clearly does not need to take the course and can substitute (for example) Advanced Differential Equations as credit instead of Differential Equations. Also, racing through a degree at a mid-tier university is unlikely to lead to a top-tier graduate school -- the student would be far better off doing something like Freshman and Sophomore years, complete all gen eds + all advanced math classes the school offers, Junior Year, go for study abroad someplace like Budapest Semesters in Mathematics or the Moscow equivalent (I know people who did both and it was a brilliant experience, very rigorous classes), Senior Year, apply for graduate schools, do independent studies and/or research, and pick up any missing gen eds.

[1053]

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[kiana]

I actually thought it was quite common for high ability students who end up at the top universities for graduate study, to have gone through average universities for undergraduate study.

 

It happens, but it's also common for them to be rejected.

 

However, I would like to point out that what I said was "racing through a degree" and not just "doing a degree". The idea of doing more advanced material sooner is a fine one -- the idea of finishing in 2 years because of advanced standing and then going to a good graduate school is less so. The extra time should be used to do things like research, advanced independent study electives, start on grad-school courses (graduate courses taken at a second-tier university would be kind of like DE classes at undergrad -- the first-tier graduate school is likely to require them to be repeated but they will strengthen the application significantly).

[8filltheheart]

I agree with both Butler and Kiana, but it really depends on objectives. If a student simply wants to complete a BS in engineering, for example, than co-oping and finishing early is a far different scenario than a student who wants to go to grad school. Just b/c our ds will be technically be a jr in less than 2 weeks does not mean he wants to graduate early. As a matter of fact, the research honors program he is in takes 4 yrs to complete. He plans on taking 4 yrs to graduate. He will take grad level courses and participate in research. Right now he is thinking of participating in 2 research projects......one for math and one for physics. He loves them both. He plans on a phD in physics, but the math dept has been encouraging him to get involved in math research and it really appeals to him. Since he entered with so many credits, he really does have the luxury of exploring and spending time doing both.

[kiana]

Yes yes yes. If someone just wants to get into the world of work then finishing early or using freed-up time to do internships is a GREAT idea. For example, someone who wants to be a high school math teacher and enters with 30 credits of math and 30 credits of gen eds through DE/AP/CLEP would have no reasons other than intellectual fulfillment to spend extra time taking courses.