Relearning Calculus

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saltycaper
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Re: Relearning Calculus

Post by saltycaper » Fri Feb 02, 2018 11:23 pm

patrick013 wrote:
Fri Feb 02, 2018 8:53 pm
Yeah, see I'm not a math major. Whatever the practical calculus in the GMAT
math refresher includes is all I can admit to. But it does help my overall
math sometimes. Ain't it so ? Fer the stuff I wee need.

Thanks for the response.

I'm only a couple courses away from an MS Finance I'm told but who needs all
that hedging. Options on index futures with regression......have mercy on
that.
The GMAT does not assume any knowledge of calculus, so it is unlikely to be covered in any review guide--certainly not in any detail. There may be a problem you stumble across where calculus could help in some way, but those are rare in my experience.

Congrats on nearing the completion of your degree.
Quod vitae sectabor iter?

jalbert
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Re: Relearning Calculus

Post by jalbert » Fri Feb 02, 2018 11:32 pm

If you want a good theoretical treatment, the series by Tom Apostol is good and available for free as a PDF:

https://www.pdfdrive.net/tom-m-apostol- ... 40211.html
Index fund investor since 1987.

nimo956
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Re: Relearning Calculus

Post by nimo956 » Fri Feb 02, 2018 11:53 pm

Work through the undergrad text Abbott's Understanding Analysis.
50% VTI / 50% VXUS

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patrick013
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Re: Relearning Calculus

Post by patrick013 » Fri Feb 02, 2018 11:57 pm

saltycaper wrote:
Fri Feb 02, 2018 11:23 pm

The GMAT does not assume any knowledge of calculus, so it is unlikely to be covered in any review guide--certainly not in any detail. There may be a problem you stumble across where calculus could help in some way, but those are rare in my experience.

Congrats on nearing the completion of your degree.
You're right, the GMAT math book covers many topics tho not
specifically calculus. The word problems they have don't qualify
as calculus I guess. If you took 4 years of math in high school
you could pass. Still a neat reference.

My grad degree has enough credits to pursue Mgmt Sci which is
mostly linear stuff or MS Fin which is mostly stat stuff but who
needs it ? Going to stand still and use the macro economic stuff
for profit. Not pursuing any further business degrees.
age in bonds, buy-and-hold, 10 year business cycle

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Clever_Username
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Re: Relearning Calculus

Post by Clever_Username » Sat Feb 03, 2018 12:24 am

triceratop wrote:
Fri Feb 02, 2018 3:43 pm
Clever_Username wrote:
Fri Feb 02, 2018 2:30 pm
Houe wrote:
Fri Feb 02, 2018 12:28 pm
Take a calculus class at a local college. In my opinion you will get a lot more out of it than just going through books on your own. I'm an electrical engineer and decided to take a class on digital signal processing post graduate. I enjoyed it a lot more than my college years. Being able to concentrate on a single class (or two) and take something you really enjoy made for an wonderful experience.
Are there really calculus classes at a local college that would cover what OP wants to learn though? In my experience, most calculus classes (and other maths, sadly) are mechanical computation and an occasional small problem solving fitting narrowly into a framework we've already seen. That isn't what OP is after.

I hope that didn't come out as rude; I'd be extremely excited if you came back with "yes, these classes exist and are common."
It depends on what your locale is. There is an excellent school in my city in southern california which teaches calculus out of Apostol.

But yes if by local college a "community/city college" is meant, the OP should look elsewhere and definitely not take such a class.
That sounds amazing and I wish I took that class. I even took all my Calculus classes in various locations in Southern California.
"What was true then is true now. Have a plan. Stick to it." -- XXXX, _Layer Cake_

jarhead1
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Re: Relearning Calculus

Post by jarhead1 » Sat Feb 03, 2018 12:30 pm

I teach calculus and if you are learning on your own I recommend a Ron Larson textbook, plenty on amazon used for a song. there is accompanying website with videos and solutions to odd problems. Note that most folks who struggle with calculus are weak in algebra skills and trigonometry.

MindBogler
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Re: Relearning Calculus

Post by MindBogler » Sat Feb 03, 2018 1:17 pm

CFR wrote:
Fri Feb 02, 2018 1:49 pm
To understand why/how, then study Physics, Semiconductor Physics, Statics, Dynamics, Engineering in general.

CFR
I wholeheartedly agree. Studying Physics made calculus both tangible and useful.

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Re: Relearning Calculus

Post by jeffarvon » Sat Feb 03, 2018 1:19 pm

Specifically, at Coursera, I took and got a lot out of the Calculus 1 class developed by a couple of professors from Ohio State.

I see the nex class starting Feb 19 https://www.coursera.org/learn/calculus1
"Enough is as good as a feast" - Mary Poppins

ccieemeritus
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Re: Relearning Calculus

Post by ccieemeritus » Sun Feb 04, 2018 4:09 am

The “why” of calculus?

The most important question in calculus is to calculate the optimal dimensions of a soda can. What ratio of height to radius will give the maximum volume to surface area ratio (maximize soda while minimizing aluminum cost)?

One way to solve the problem is to set the radius to 1 and calculate the volume/area as the height increases. Differentiate to find the maximum.

The other way to solve the problem is to go to the refrigerator with a ruler and measure a Pepsi can. :-)

In the next class (physics) you’ll discover 4 ways to determine the height of a building with a barometer.

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Re: Relearning Calculus

Post by wrongfunds » Sun Feb 04, 2018 9:22 am

Epsilon Delta wrote:
Fri Feb 02, 2018 6:53 pm
Real analysis is beautiful. There isn't enough beauty in the world, which should be enough reason to study it. But I'm not sure it will tell you what you want to know or make you a better engineer. There are places between real analysis and rote application of calculus that might be a better fit.

I did real analysis as an entry level course with no prerequisites. Real analysis starts from scratch, logically there are no prerequisites, although almost everybody knew calculus up to at least second order differential equations (simple harmonic oscillator).

If your interested I'd do a two part test.

1) Learn the proof that the square root of 2 is not rational. This should take about ten minutes.

2) Learn the proof of the intermediate value theorem. This also requires learning a definition of real numbers (and why sqrt(2) is real). This is probably two to ten hours.

If you're still excited maybe real analysis is for you. While you said you don't want a lecturer I think it's best to write out proofs and have somebody grade or correct them, either as class works or exams. Sometimes you need somebody to tell you "you need to be more explicit in step 2." or a little less explicit in step 2.7.1.8.3.
This is great approach. Nothing to do with Calculus, but add Pythagoras theorem to the list too.

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Re: Relearning Calculus

Post by dbr » Sun Feb 04, 2018 9:29 am

wrongfunds wrote:
Sun Feb 04, 2018 9:22 am
Epsilon Delta wrote:
Fri Feb 02, 2018 6:53 pm
Real analysis is beautiful. There isn't enough beauty in the world, which should be enough reason to study it. But I'm not sure it will tell you what you want to know or make you a better engineer. There are places between real analysis and rote application of calculus that might be a better fit.

I did real analysis as an entry level course with no prerequisites. Real analysis starts from scratch, logically there are no prerequisites, although almost everybody knew calculus up to at least second order differential equations (simple harmonic oscillator).

If your interested I'd do a two part test.

1) Learn the proof that the square root of 2 is not rational. This should take about ten minutes.

2) Learn the proof of the intermediate value theorem. This also requires learning a definition of real numbers (and why sqrt(2) is real). This is probably two to ten hours.

If you're still excited maybe real analysis is for you. While you said you don't want a lecturer I think it's best to write out proofs and have somebody grade or correct them, either as class works or exams. Sometimes you need somebody to tell you "you need to be more explicit in step 2." or a little less explicit in step 2.7.1.8.3.
This is great approach. Nothing to do with Calculus, but add Pythagoras theorem to the list too.
I really like the concept of learning the definition of real numbers. That is the point at which seeing what is behind calculus and a lot of other things comes to the front.

If real analysis is not the idea of what relearning calculus is all about, then I would consider looking into differential equations as that is where most of the practical examples in science and engineering originate. The reason for this is that many laws of physics are formulated as differential equations. A first example would be Newton's law F=ma. A second example would be Maxwell's equations.

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Re: Relearning Calculus

Post by obgraham » Sun Feb 04, 2018 11:55 am

OP's original question tweeked my interest.

Back in my first year University days, some 55 years ago, I struggled with Calculus 101. This having scored rather high math scores on the ACT and SAT, so it wasn't for lack of basic math knowledge. I just never caught on to the concepts of calculus. By midterm I was carrying a D -- disaster for one aiming for med school. So I sat down and memorized the book, the theorems, and the problems. Aced the final, thus earning a A- for the course, without learning a single concept! Never took another math class.

Now looking back, I can really say that not once in those 55 years, including a successful professional career, did I come across something that I could not solve because I knew zero of calculus.

However, I have for many years wondered about that. Most of us have some knowledge block. I think we accommodate by learning something else instead, or by changing our fields of endeavor.

I've great respect for OP, trying to learn calculus now. I've often wondered the same. But I think I'll pass!

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Re: Relearning Calculus

Post by dbr » Sun Feb 04, 2018 12:05 pm

obgraham wrote:
Sun Feb 04, 2018 11:55 am
OP's original question tweeked my interest.

Back in my first year University days, some 55 years ago, I struggled with Calculus 101. This having scored rather high math scores on the ACT and SAT, so it wasn't for lack of basic math knowledge. I just never caught on to the concepts of calculus. By midterm I was carrying a D -- disaster for one aiming for med school. So I sat down and memorized the book, the theorems, and the problems. Aced the final, thus earning a A- for the course, without learning a single concept! Never took another math class.

Now looking back, I can really say that not once in those 55 years, including a successful professional career, did I come across something that I could not solve because I knew zero of calculus.

However, I have for many years wondered about that. Most of us have some knowledge block. I think we accommodate by learning something else instead, or by changing our fields of endeavor.

I've great respect for OP, trying to learn calculus now. I've often wondered the same. But I think I'll pass!
The need to know and understand various topics in mathematics is vastly over-rated as it is patently obvious that most of the population is successful doing whatever they are doing with almost no mathematical skills and knowledge at all. The people who do need to know and use mathematics know who they are. You are correct that the major cost of flaking out in math is closed doors to some opportunities. That is mostly in the gate-keeping function of prerequisites to programs that provide the qualifying certification. Even most of a career in academic physics or in science and engineering generally is not the daily exercise of advanced math compared to figuring out how to convince someone somewhere to pay for something. There are also situations where people would not be flim-flammed or make bad decisions if a little basic mathematical thinking were in the tool chest to apply to the problem. We see that on this forum from time to time.

The question in the OP is really interesting. One notes the amount of emphasis on "go learn real analysis" which is basically correct, but is probably a topic to which not more than 0.001% of the population has ever been exposed.

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Epsilon Delta
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Re: Relearning Calculus

Post by Epsilon Delta » Sun Feb 04, 2018 1:04 pm

obgraham wrote:
Sun Feb 04, 2018 11:55 am
Most of us have some knowledge block. I think we accommodate by learning something else instead, or by changing our fields of endeavor.
The ideal solution is to know what you don't know and ask somebody who might. Often you don't even need to ask how to do it, just ask them to do it. It's depressing to work in a organizations that try to ensure that everybody has exactly the same skills.

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Re: Relearning Calculus

Post by rgs92 » Sun Feb 04, 2018 1:22 pm

Yeah, that stuff with integration by parts and trigonometric substitutions and all the theory stuff in advanced calculus can be a real bear. I can relate. I moved over from pure math to comp. sci. because of that, which was much easier to me. It's better for your career prospects anyway, so maybe you can do what I did.

The beginning few courses in calculus with understandable problems and even moderately complex proofs was OK with me in the first year or so, but later on I struggled, but I still managed to get good enough grades to get an undergrad pure math degree. I took lots of comp sci and liberal arts courses to pump up my overall GPA to something nice and high to balance out the mediocre math grades.

Good luck and my sympathies.

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Re: Relearning Calculus

Post by FullYellowJacket » Sun Feb 04, 2018 1:35 pm

My reasonings for wanting to tackle this are more personal than professional. I'm a Materials Engineer, so actually needing to know advanced math beyond statistics and manipulating rate equations is questionable. What I think this is going on is I am having an early mid-life crisis, where I regret not using all the time and resources I had in my college years (and earlier) to really learn about the world of math, science, and humanities. I've already made up for the major slacking off I did while earning my BS in MSE with getting my masters and really applying myself and excelling at it (all A's and some papers i am very proud of). I really worked hard at learning Math, up until middle high school when an apathy set in towards learning in general. This apathy lasted until the end of undergrad. Note to parents of young children: don't ever tell you kids how smart they are or that they don't even need to study to get A's; they might take it to heart.

So basically, I am trying to relearn (and take seriously) everything that I should have learned from 11th grade towards the end of undergrad. And since this is all out of personal fulfillment, I'm willing to dive even deeper than my described goal if the itch is there.

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Re: Relearning Calculus

Post by Pajamas » Sun Feb 04, 2018 1:50 pm

FullYellowJacket wrote:
Sun Feb 04, 2018 1:35 pm
My reasonings for wanting to tackle this are more personal than professional. I'm a Materials Engineer, so actually needing to know advanced math beyond statistics and manipulating rate equations is questionable. What I think this is going on is I am having an early mid-life crisis, where I regret not using all the time and resources I had in my college years (and earlier) to really learn about the world of math, science, and humanities. I've already made up for the major slacking off I did while earning my BS in MSE with getting my masters and really applying myself and excelling at it (all A's and some papers i am very proud of). I really worked hard at learning Math, up until middle high school when an apathy set in towards learning in general. This apathy lasted until the end of undergrad. Note to parents of young children: don't ever tell you kids how smart they are or that they don't even need to study to get A's; they might take it to heart.

So basically, I am trying to relearn (and take seriously) everything that I should have learned from 11th grade towards the end of undergrad. And since this is all out of personal fulfillment, I'm willing to dive even deeper than my described goal if the itch is there.
Is there anything that you could tackle learning that you would actually use and that would actually help you in your work or otherwise in life?

I ask that because I took calculus in high school and again in college and have literally never applied it after graduating college for the first time. Other things that I studied in school have proved much more useful in life, even specifically in the area of math, such as advanced geometry.

If it is an exercise for the brain or just to prove to yourself that you can do it, what about tackling something like Latin or a modern language such as Spanish that would actually be useful? Or something that overlaps materials engineering that would allow you to expand your use of your current knowledge?

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Epsilon Delta
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Re: Relearning Calculus

Post by Epsilon Delta » Sun Feb 04, 2018 2:22 pm

FullYellowJacket wrote:
Sun Feb 04, 2018 1:35 pm
Note to parents of young children: don't ever tell you kids how smart they are or that they don't even need to study to get A's; they might take it to heart.
Kids know how hard they are working and what grades they are getting. Not acknowledging reality leads kids to deduce you are a liar or an idiot. Not a good way to influence them. And yes this is directed at the faculty and staff of ****** ****** School.

If kids are not studying and still getting A's the solution is not to study harder to (still) get A's; It's to study something that is harder and earn a B or even a C. Straight A's are for lazy people.

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Re: Relearning Calculus

Post by bertilak » Sun Feb 04, 2018 4:47 pm

FullYellowJacket wrote:
Fri Feb 02, 2018 11:21 am
Hi everyone! I've decided to go on a personal endeavor to relearn calculus from the ground up. I've realized in the past few years that I am a product of the 2000s USA education system, and I only know how to solve Calculus problems. The issue is I don't understand the why behind a lot of it. This resulted in passing Calculus 1-3 with trouble.

Does anyone have a good recommendation for Calculus books/text books that will help me truly understand calculus (Single Variable through Differential Equations)?

So far I am starting with Spivak's Calculus.
I took, and passed, Calculus in college but felt I didn't really understand it. I think I went through three courses.

In 1910 Silavnus P. Thompson published a book called Calculus Made Easy. It covers Differential and Integral calculus. There was an update in 1945. I read it about 1970.

I did find it "worked" for me. It made a distinction that is often not made: the difference between the "idea" of what's being done and the "tricks of the trade" that help you actually get something done. He gives clear derivations of the "tricks." Thus you a) feel justified in using those tricks, b) find them easier to remember, and c) recognize when to use them. That last is important!
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Re: Relearning Calculus

Post by obgraham » Sun Feb 04, 2018 5:11 pm

study something that is harder and earn a B or even a C. Straight A's are for lazy people.
There indeed is a wise comment. It certainly applied to me.

Unfortunately, in some fields, the almighty GPA still reigns supreme. Like applying to medical school. So courses in which a student is not getting a top grade are avoided or dropped, and some other "easy A" class selected instead.

Even in advanced fields of study there are plenty of folks with superb ability to ace the exams, but no common sense or ability to apply that knowledge creatively. Conversely, there are plenty of C students who have gone on to great success.

investorpeter
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Re: Relearning Calculus

Post by investorpeter » Sun Feb 04, 2018 5:15 pm

I would love to restart my mathematics education just for pure enjoyment. I only got as far as linear algebra and differential equations before getting side tracked onto other "more practical" endeavors. But I remember the first time I finally understood the meaning of integral calculus and it was an epiphany. By combining infinitesimally small discrete units into a whole, it allowed one to grasp infinity. That kind of blew my mind.

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Re: Relearning Calculus

Post by Herekittykitty » Sun Feb 04, 2018 7:33 pm

investorpeter wrote:
Sun Feb 04, 2018 5:15 pm
I would love to restart my mathematics education just for pure enjoyment. I only got as far as linear algebra and differential equations before getting side tracked onto other "more practical" endeavors. But I remember the first time I finally understood the meaning of integral calculus and it was an epiphany. By combining infinitesimally small discrete units into a whole, it allowed one to grasp infinity. That kind of blew my mind.
Me too. I started college later than most, wanted to be a doctor. I took Calculus I because I didn't want to take Statistics. Then Physics nearly clobbered me, and it was what stood between me and getting through to medical school. In the course of getting nearly clobbered by beginning Physics and turning it around to doing really well in it, I did all the things I suggested in my post earlier in this thread (except I didn't use any DVD's as there weren't any then.) I was really lucky to have a Physics professor who made us derive equations for every problem, not just memorize stuff, who told me I was going to flunk because my math wasn't good enough, and who was going to flunk me if I didn't start understanding Physics and Math (he ended up being my advisor although I also kept my pre-med advisor).

I still remember after years have passed, the moment I really grasped what Calculus was - Wow!!!!! And where Physics fit it - Wow!!!!!!! The more I took of Math and of Physics, the more I loved them. My mind and how I think were changed forever by Math and Physics.

Anyway, I did go on to become a doctor, and I have never regretted it, because I could just go in one direction - Medicine (I really like working with patients and figuring medical things out) or Math/Physics. Math and Physics are to me like the person you used to love but you loved someone else too who was better suited to you, and you could just marry one, so you make your choice and enjoy your life.

But I do have to say, I dream of starting over in Math and Physics from the beginning and learning them all over again when I retire. I could not get competitive in either again (actually I was never destined for greatness in either, maybe masters degree level but I think I would have hit my limit in those areas - yep, joke intended - at that point). And I just might do it.

:)

Edited because I keep typing too fast and making typing mistakes.
I don't know anything.

dbr
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Re: Relearning Calculus

Post by dbr » Sun Feb 04, 2018 7:43 pm

There is never anything that stops a person from delving again and more into anything that is interesting. Many people do this all the time in any variety of areas that are of interest to them. I recently enjoyed delving into some mathematical concepts of average of a set of numbers that was new to me but was part of a thread. I enjoyed that because the concepts were generalization of ideas and generalization is always one of the objects and rewards of much of mathematics.

bawr
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Re: Relearning Calculus

Post by bawr » Thu Feb 08, 2018 2:57 pm

The two volume Apostol calculus book mentioned by others is quite good.

Another, more advanced book is Principles of Mathematical Analysis by Walter Rudin. It has been used as a text for UC Berkeley's introductory analysis course for undergraduates (Math 104) for a long time.

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Re: Relearning Calculus

Post by camden » Thu Feb 08, 2018 6:08 pm

I applaud the OP for being ambitious enough to want to deeply understand calculus, and stand in awe of those who do, a group which probably includes many who have answered above. No wish to offend anyone, but as for me.....

I hated, with a passion, having to learn calculus in the first place. Now that I am much older, and my time left on the planet much shorter, I cannot conceive of spending a millisecond of that time with that stuff. 😆

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Re: Relearning Calculus

Post by URSnshn » Thu Feb 08, 2018 7:26 pm

What a cool post! I love this OP! If you've the interest go for it! I came to mathematics in an odd way. I didn't even have all I needed when I realized in Physics that I was missing a major piece and had to take more math to actually understand what I was doing in Physics.

When you follow your thirst for understanding and knowledge and/or even a simple curiosity - you never know where you'll end up. Some of these excursions can lead to wonderful experiences and a new path or the deepening of something. I hope you'll post about your journey now again here.

I've also enjoyed the many resources this thread includes!

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Re: Relearning Calculus

Post by CppCoder » Thu Feb 08, 2018 8:11 pm

Epsilon Delta wrote:
Sun Feb 04, 2018 2:22 pm
If kids are not studying and still getting A's the solution is not to study harder to (still) get A's; It's to study something that is harder and earn a B or even a C. Straight A's are for lazy people.
??? For some of us, they didn't make a class we couldn't get an A in. I agreed with your statement until the end. I think the strategy should be to study something harder and get A's in that too, or maybe to study something interesting without bothering with a letter grade. Grades are for others to assess or judge you. Knowing that people use GPA as a (stupid) hurdle, just get the A's so as not to ruin future opportunities. Once you reach a certain point, though, the concept of being graded on learning becomes infantile.

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Re: Relearning Calculus

Post by GraduateStudent » Thu Feb 08, 2018 9:20 pm

FullYellowJacket wrote:
Fri Feb 02, 2018 12:23 pm
Epsilon Delta wrote:
Fri Feb 02, 2018 12:15 pm
This depends on what you want to learn and what you mean by "from the ground up". The foundations go well below ground level and get deeper every decade. You don't need to go all the way down to have a good understanding.

In my experience a "calculus" course is a mathematical methods course. It teaches you how to solve problems using calculus. The text book is often mathematical methods for physicists or some permutation. You can substitute for "physics" if you have a subject area preference.

If you want the foundations it's typically in a "real analysis" course. This should teach you about limits, proofs, Riemann and Lebesgue integrals. The text book will be called something like A first introduction to the foundations of basic real analysis. The more qualifiers claiming simplicity the harder it will be.

Some people will tell you to read "Principia". They are nuts. We've learned a lot since Newton. And any way Deists against Dotage.
I think this is on the right track. I should clarify that I can "Do" Calculus, i.e. find the integral then solve for C. I want to have some deeper understanding of it.

https://ocw.mit.edu/courses/mathematics ... fall-2010/
Looks like it has promise.
I like the book written by John Hubbard. It sounds like it'll be right up your alley.
Life after grad school is great.

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Re: Relearning Calculus

Post by grabiner » Thu Feb 08, 2018 9:31 pm

FullYellowJacket wrote:
Fri Feb 02, 2018 11:21 am
The issue is I don't understand the why behind a lot of it. This resulted in passing Calculus 1-3 with trouble.

So far I am starting with Spivak's Calculus.
Spivak's book is a great book for understanding calculus from the viewpoint of a mathematician; it isn't as good for using calculus as a computational tool. Therefore, if you can follow the book (which requires you to be comfortable with proofs), it may meet your own needs.

The book covers single-variable calculus, but at the level of rigor of an advanced course. Everything has to be proved, and those proofs are well-motivated. If you go on to real analysis, you will see many of the same concepts, applied to different areas; for example, real analysis has a more general definition of the integral.

I am a mathematician, and this is my favorite math textbook. A copy sits on my desk at work, and I still use it for reference, usually to look up an exercise to remember how I learned something.
Wiki David Grabiner

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Re: Relearning Calculus

Post by qwertyjazz » Thu Feb 08, 2018 9:37 pm

Rudin Principles of Mathematical Analysis
Spivak is a great book to learn theory
Have fun
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Re: Relearning Calculus

Post by wintermute » Sat Feb 10, 2018 11:38 pm

MathBFF, patrickJMT, Kahn; all on YouTube.

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Re: Relearning Calculus

Post by Dominic » Sat Feb 10, 2018 11:53 pm

GettingThere wrote:
Fri Feb 02, 2018 11:56 am
I find that there are good resources on-line that are not official courses. For example, the YouTube channel 3Blue1Brown (https://www.youtube.com/channel/UCYO_ja ... V4b17AJtAw) has a great series of videos introducing Calculus - "Essence of Calculus."
3Blue1Brown is outstanding. I highly recommend his videos. They should give you a really strong intuition for the fundamentals.

I'd also like to throw Paul's Online Math Notes into the mix. This isn't a formal textbook, but complete lecture notes with worked examples for calc I-III and diff eqs. I found this site indispensable when I was learning calculus.

Best of luck in your studies OP!

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