Relearning Calculus

 Posts: 316
 Joined: Thu Apr 03, 2014 8:21 am
Relearning Calculus
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 13 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.
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.

 Posts: 53
 Joined: Fri Jan 27, 2017 9:50 am
Re: Relearning Calculus
Coursera? Lynda?

 Posts: 10
 Joined: Wed Oct 14, 2015 12:01 pm
Re: Relearning Calculus
I used Howard Anton while in school, now I recommend Calculus by Anton, Bivens, and Davis.
I find that there are good resources online 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."
I use Khan Academy (https://www.khanacademy.org/) to aid in my homeschooling efforts; my sons are currently studying Prealgebra and Geometry, but I expect to use their Calculus material when the time comes.
Also, if you are looking for a college level course, MIT has many of their classes online for free: https://ocw.mit.edu/index.htm
Good Luck!
I find that there are good resources online 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."
I use Khan Academy (https://www.khanacademy.org/) to aid in my homeschooling efforts; my sons are currently studying Prealgebra and Geometry, but I expect to use their Calculus material when the time comes.
Also, if you are looking for a college level course, MIT has many of their classes online for free: https://ocw.mit.edu/index.htm
Good Luck!

 Posts: 651
 Joined: Mon Jul 31, 2017 10:43 am
Re: Relearning Calculus
*cringe*
Re: Relearning Calculus
https://www.khanacademy.org/ is excellent and free

 Posts: 316
 Joined: Thu Apr 03, 2014 8:21 am
Re: Relearning Calculus
Anything that involves a lecturer is probably a nonstarter for me, unfortunately. I'll keep the youtube channel in mind.

 Posts: 316
 Joined: Thu Apr 03, 2014 8:21 am
Re: Relearning Calculus
Why so? From what I have read it's one of the best, but perhaps a little dense.
 Epsilon Delta
 Posts: 7423
 Joined: Thu Apr 28, 2011 7:00 pm
Re: Relearning Calculus
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.
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.

 Posts: 316
 Joined: Thu Apr 03, 2014 8:21 am
Re: Relearning Calculus
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.Epsilon Delta wrote: ↑Fri Feb 02, 2018 12:15 pmThis 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.
https://ocw.mit.edu/courses/mathematics ... fall2010/
Looks like it has promise.
Re: Relearning Calculus
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.
 triceratop
 Moderator
 Posts: 5257
 Joined: Tue Aug 04, 2015 8:20 pm
 Location: la la land
Re: Relearning Calculus
I would also go for Spivak given the OP's stated preferences.
"To play the stock market is to play musical chairs under the chord progression of a bidask spread."
Re: Relearning Calculus
MIT Open Courseware ...
https://ocw.mit.edu/courses/findbytop ... t=calculus
EdX has many calculus courses available...
From MIT:
https://www.edx.org/course/calculus1a ... 18011x0
https://www.edx.org/course/calculus1b ... 18012x0
https://www.edx.org/course/calculus1c ... 18013x0
From Harvard:
https://www.edx.org/course/calculusapp ... calcapl1x
https://www.edx.org/course/precalculusasuxmat170x
https://www.edx.org/course/aprcalculus ... calapbcx0
https://ocw.mit.edu/courses/findbytop ... t=calculus
EdX has many calculus courses available...
From MIT:
https://www.edx.org/course/calculus1a ... 18011x0
https://www.edx.org/course/calculus1b ... 18012x0
https://www.edx.org/course/calculus1c ... 18013x0
From Harvard:
https://www.edx.org/course/calculusapp ... calcapl1x
https://www.edx.org/course/precalculusasuxmat170x
https://www.edx.org/course/aprcalculus ... calapbcx0
Re: Relearning Calculus
Spivak's is an excellent book. However, I am concerned with the statement that "this resulted in passing Calculus 13 with trouble." My experience is that students reading Spivak find Calculus 13 easy. But of course, this was in the 2000s and maybe you were not as mature as you are now. In any case, if you are understanding Spivak's, keep reading.
 saltycaper
 Posts: 2650
 Joined: Thu Apr 24, 2014 8:47 pm
 Location: The Tower
Re: Relearning Calculus
^Did you use Spivak then or something else, like Stewart?
Quod vitae sectabor iter?
Re: Relearning Calculus
Agree with Khan Academy. Absolutely a great resource and its free. Nothing on this site is overwhelming and is broken up in videos of around 10 minutes followed by some exercises.
 FreeAtLast
 Posts: 630
 Joined: Tue Nov 04, 2014 9:08 pm
Re: Relearning Calculus
How about Salas and Hille's text? It provides as clear an explanation of basic Calculus problems as I have ever seen. If you want to learn further about fundamental concepts of analysis: A Course of Pure Mathematics by G.H.Hardy, a very famous and wellrespected British mathematician.
Edit: An excellent, easytoread text on Elementary Differential Equations is by Boyce and DiPrima. Free copies in PDF format are available on the Internet (Google it!).
Edit: An excellent, easytoread text on Elementary Differential Equations is by Boyce and DiPrima. Free copies in PDF format are available on the Internet (Google it!).
Illegitimi non carborundum.

 Posts: 651
 Joined: Mon Jul 31, 2017 10:43 am
Re: Relearning Calculus
FullYellowJacket wrote: ↑Fri Feb 02, 2018 12:05 pmWhy so? From what I have read it's one of the best, but perhaps a little dense.
Perhaps. It's been quite a while since I've taken the course, but our analysis class used this book and I found it rather opaque. From memory it was "lemma lemma lemma theorem proof lemma proof theorem proof lemma proof" and not much in the way of descriptive text or explanation. About 10% of the class excelled, 90% were completely lost. Might have had just as much to do with the instructor as the book. *shrugs* Like I said, it's been a while and I didn't like the course in general, so...triceratop wrote: ↑Fri Feb 02, 2018 12:31 pmI would also go for Spivak given the OP's stated preferences.
That said, if the book works for you, then great. I agree that it is highly regarded, so if the style fits you then you'll probably learn a lot.
Re: Relearning Calculus
To understand why/how, then study Physics, Semiconductor Physics, Statics, Dynamics, Engineering in general.FullYellowJacket wrote: ↑Fri Feb 02, 2018 11:21 amHi 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 13 with trouble.
CFR
Re: Relearning Calculus
If the goal is to *understand* Calculus as well as be able to "crank the machinery" then I'd recommend Apostol's Calculus text.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 13 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)?
No matter which textbook you select, I recommend that you also pick up a copy of "Prof. E McSquared's Calculus Primer" by Swann. Yes, it is a cartoon. The differential calculus that it teaches is real. The book is great at what it tries to do, it just isn't complete. But it doesn't pretend to be complete.
Re: Relearning Calculus
I highly recommend betterexplained.com if you want to understand math on an intuitive level so you "get" calculus. They have a calculus class.FullYellowJacket wrote: ↑Fri Feb 02, 2018 11:21 amHi 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 13 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.
Most math classes and teachers are so rigid and formal it makes it hard to grasp the core idea of what a thing is, what it does, and why you would want to use it. Better explained goes in the opposite direction by starting with rough approximations and analogies and then works toward formality.
As a side note, i would also recommend Street Fighting Mathematics and an essay A Mathematician's Lament by Paul Lockhart.
Re: Relearning Calculus
Are you looking for a better conceptual understanding or for a more theoretical development with rigorous proofs?
If you are looking for a conceptual approach, then Stewart's Concepts and Contexts series is very good  it favors graphical/conceptual "plausibility" arguments over rigorous proofs, but may help give you a better informal understanding of "why" some things work the way that they do. If you are looking for a more traditional book, then I like Stewart's regular book.
openstax.org has an open source free book that might be worth looking at  it's free in pdf or reasonably priced from Amazon if you wanted a hard copy. You can look over the free pdf to see if it's something you are interested in.
As already mentioned, Khan Academy and MIT have good open courseware.
If you are looking for a conceptual approach, then Stewart's Concepts and Contexts series is very good  it favors graphical/conceptual "plausibility" arguments over rigorous proofs, but may help give you a better informal understanding of "why" some things work the way that they do. If you are looking for a more traditional book, then I like Stewart's regular book.
openstax.org has an open source free book that might be worth looking at  it's free in pdf or reasonably priced from Amazon if you wanted a hard copy. You can look over the free pdf to see if it's something you are interested in.
As already mentioned, Khan Academy and MIT have good open courseware.
Re: Relearning Calculus
That is way overkill.CFR wrote: ↑Fri Feb 02, 2018 1:49 pmTo understand why/how, then study Physics, Semiconductor Physics, Statics, Dynamics, Engineering in general.FullYellowJacket wrote: ↑Fri Feb 02, 2018 11:21 amHi 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 13 with trouble.
CFR
You don't need to take it to that level to get the why/how of calculus.
Here is a much simpler example to introduce someone to calculus. Give them two problems.
One they know how to solve easily:
1> If you give your child an allowance of $1/week and they don't spend any of it, how much money will they have on their 18th birthday?
One they don't know how to solve as easily:
2> If you give your child an allowance of $1/week for each year old they are (e.g. $1/week until their first birthday, then $2/week, then $3/week) and they don't spend any of it, how much money will they have on their 18th birthday?
Using this simple example, you can teach what an integral is, what it does, and why you would want to use one.
 Clever_Username
 Posts: 1040
 Joined: Sun Jul 15, 2012 12:24 am
 Location: Southern California
Re: Relearning Calculus
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.Houe wrote: ↑Fri Feb 02, 2018 12:28 pmTake 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.
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."
"What was true then is true now. Have a plan. Stick to it."  XXXX, _Layer Cake_

 Posts: 97
 Joined: Tue Jul 05, 2011 6:56 pm
Re: Relearning Calculus
Derive the basic concepts yourself:
Evaluate an expression to a limit, ie. x>0 or x>infinity
If an expression can not be evaluated at its limit, the derivative/integral is undefined at that point.
Integrals are the sum of an infinite number of small rectangles or trapezoids underneath a curve (+ a constant to account for initial conditions)
derivatives are the slope of two adjacent points on a curve as you approach 0 distance between the points.
Beyond that its techniques, applications and tables.
Evaluate an expression to a limit, ie. x>0 or x>infinity
If an expression can not be evaluated at its limit, the derivative/integral is undefined at that point.
Integrals are the sum of an infinite number of small rectangles or trapezoids underneath a curve (+ a constant to account for initial conditions)
derivatives are the slope of two adjacent points on a curve as you approach 0 distance between the points.
Beyond that its techniques, applications and tables.
 Epsilon Delta
 Posts: 7423
 Joined: Thu Apr 28, 2011 7:00 pm
Re: Relearning Calculus
Do not learn calculus from an engineer.
Fortunately the things engineers have to deal with are relatively benign. If they weren't the entire built world would vanish in an uncountable number of infinite discontinuities.
To be fair you should not learn engineering from a mathematician. They spend far too much time worrying about an uncountable number of infinite discontinuities.
Fortunately the things engineers have to deal with are relatively benign. If they weren't the entire built world would vanish in an uncountable number of infinite discontinuities.
To be fair you should not learn engineering from a mathematician. They spend far too much time worrying about an uncountable number of infinite discontinuities.
Re: Relearning Calculus
These statements are true, but not really helpful to *understanding* calc.engineer1969 wrote: ↑Fri Feb 02, 2018 2:51 pmDerive the basic concepts yourself:
Evaluate an expression to a limit, ie. x>0 or x>infinity
If an expression can not be evaluated at its limit, the derivative/integral is undefined at that point.
Integrals are the sum of an infinite number of small rectangles or trapezoids underneath a curve (+ a constant to account for initial conditions)
derivatives are the slope of two adjacent points on a curve as you approach 0 distance between the points.
Beyond that its techniques, applications and tables.
It doesn't explain why integrals are cool. It doesn't explain why you would want to use one. what capability do I get using them that i don't get using other approaches....etc.
Re: Relearning Calculus
Sorry to be pedantic. If a limit of a function exists at x=a and is equal to f(a), then the function is continuous at x=a. Continuity is a necessary condition for differentiability, but is not a sufficient condition; for example a continuous function is not differentiable at a sharp corner, cusp, or vertical tangent. On the other hand, continuity does imply that the function is integrable. The fundamental theorem requires continuity, but discontinuous functions can have integrals; in fact, any bounded function with a countable number of discontinuities still has a Riemann integral.engineer1969 wrote: ↑Fri Feb 02, 2018 2:51 pmDerive the basic concepts yourself:
Evaluate an expression to a limit, ie. x>0 or x>infinity
If an expression can not be evaluated at its limit, the derivative/integral is undefined at that point.
Integrals are the sum of an infinite number of small rectangles or trapezoids underneath a curve (+ a constant to account for initial conditions)
derivatives are the slope of two adjacent points on a curve as you approach 0 distance between the points.
Beyond that its techniques, applications and tables.

 Posts: 316
 Joined: Thu Apr 03, 2014 8:21 am
Re: Relearning Calculus
I am an engineer so you can see my predicament. This whole exercise was prompted because I had to calculate flux during diffusion at an interface.Epsilon Delta wrote: ↑Fri Feb 02, 2018 3:17 pmDo not learn calculus from an engineer.
Fortunately the things engineers have to deal with are relatively benign. If they weren't the entire built world would vanish in an uncountable number of infinite discontinuities.
To be fair you should not learn engineering from a mathematician. They spend far too much time worrying about an uncountable number of infinite discontinuities.
Even though I could calculate the flux, I'm still not able to convince myself that I'm sure of what flux is. I figured this stemmed from my lack of Understanding calculus, even if I could solve the problems.
Re: Relearning Calculus
Here are four books that cover calculus from different, entertaining perspectives. These are easy, truly fun reads that explain the fundamental techniques and concepts in useful ways. I'm not suggesting these as a means to really learning calculus, but rather as enjoyable readings that will help you think about calculus in new ways.
1. Calculus Made Easy, Silvanus Thompson
https://www.amazon.com/CalculusMadeEa ... 004TGKJNY/
2. A Tour of the Calculus, David Berlinski
https://www.amazon.com/TourCalculusDa ... 679747885/
3. How to Ace Calculus: The Streetwise Guide
https://www.amazon.com/HowAceCalculus ... B0151UPVF2
4. How to Ace the Rest of Calculus
https://www.amazon.com/HowAceRestCal ... 016IA0A6Y/
1. Calculus Made Easy, Silvanus Thompson
https://www.amazon.com/CalculusMadeEa ... 004TGKJNY/
2. A Tour of the Calculus, David Berlinski
https://www.amazon.com/TourCalculusDa ... 679747885/
3. How to Ace Calculus: The Streetwise Guide
https://www.amazon.com/HowAceCalculus ... B0151UPVF2
4. How to Ace the Rest of Calculus
https://www.amazon.com/HowAceRestCal ... 016IA0A6Y/
 triceratop
 Moderator
 Posts: 5257
 Joined: Tue Aug 04, 2015 8:20 pm
 Location: la la land
Re: Relearning Calculus
It depends on what your locale is. There is an excellent school in my city in southern california which teaches calculus out of Apostol.Clever_Username wrote: ↑Fri Feb 02, 2018 2:30 pmAre 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.Houe wrote: ↑Fri Feb 02, 2018 12:28 pmTake 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.
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."
But yes if by local college a "community/city college" is meant, the OP should look elsewhere and definitely not take such a class.
"To play the stock market is to play musical chairs under the chord progression of a bidask spread."

 Posts: 155
 Joined: Sun Aug 21, 2016 1:51 pm
Re: Relearning Calculus
if you are interested in the physical meaning of flux, that is more like what some physics courses teach. It takes some vector calculus, which is beyond the firstyear calculus required of many students, so many physics programs take some time out to teach that bit of math inside the physics course. But usually the motivating subject is electromagnetics, so the examples are of electric and magnetic flux, and how to go between the integral and differential forms of Maxwell's equations. Ramo, Whinnery, and Van Duzer, "Fields and Waves in Communication Electronics", once the alltime most popular engineering book, was excellent. But you probably want something which uses flux in a diffusion application. Maybe Schaum's Outline of Heat Transfer if it has to do with heat transfer and diffusion. Or look for a heat transfer text which has a math section which goes over the required vector calculus. Or if it is some other application, like Process Engineering, just look for a text with a section or appendix which covers the "advanced calculus" required for the subject. Often the math isn't a general requirement, so the application field will try to include it as background material. Then they can use physical examples that are helpful, from the actual application you are interested in, that make sense.
Apostol is favored by math majors mostly because of its mathematical rigor. I wasn't that mathematical, and preferred engineering approaches. I am sure Apostol covered most of those subjects well, but I can hardly remember any of it. Sokolnikoff & Redheffer is a good applied math text if you are looking for an applied math book.
Apostol is favored by math majors mostly because of its mathematical rigor. I wasn't that mathematical, and preferred engineering approaches. I am sure Apostol covered most of those subjects well, but I can hardly remember any of it. Sokolnikoff & Redheffer is a good applied math text if you are looking for an applied math book.
Re: Relearning Calculus
How deep of an understanding do you need? It sounds like you might want an undergraduate Real Analysis course. You start really basic, recreating all aspects of math (literally addition and subtraction), and culminate with definitions of derivatives and integrals.
Real Analysis is not basic at allmany math majors struggle with it. But you will get as behindthescenes as you can get. Typical calculus sequence courses handwave a bunch of stuff and it's easy to think you've gained understanding when you haven't completely.
Real Analysis is not basic at allmany math majors struggle with it. But you will get as behindthescenes as you can get. Typical calculus sequence courses handwave a bunch of stuff and it's easy to think you've gained understanding when you haven't completely.

 Posts: 316
 Joined: Thu Apr 03, 2014 8:21 am
Re: Relearning Calculus
Interestingly MIT's OCW for Real Analysis lists Multivariable Calculus and Differential Equations as prereqs. Maybe this is something to look at if I still have an itch to scratch when I am done.rocket354 wrote: ↑Fri Feb 02, 2018 5:30 pmHow deep of an understanding do you need? It sounds like you might want an undergraduate Real Analysis course. You start really basic, recreating all aspects of math (literally addition and subtraction), and culminate with definitions of derivatives and integrals.
Real Analysis is not basic at allmany math majors struggle with it. But you will get as behindthescenes as you can get. Typical calculus sequence courses handwave a bunch of stuff and it's easy to think you've gained understanding when you haven't completely.

 Posts: 316
 Joined: Thu Apr 03, 2014 8:21 am
Re: Relearning Calculus
Relearning Physics is another conversation for another day... I'll post that thread when I've gone on my mathematics journey.ReadyOrNot wrote: ↑Fri Feb 02, 2018 4:56 pmif you are interested in the physical meaning of flux, that is more like what some physics courses teach. It takes some vector calculus, which is beyond the firstyear calculus required of many students, so many physics programs take some time out to teach that bit of math inside the physics course. But usually the motivating subject is electromagnetics, so the examples are of electric and magnetic flux, and how to go between the integral and differential forms of Maxwell's equations. Ramo, Whinnery, and Van Duzer, "Fields and Waves in Communication Electronics", once the alltime most popular engineering book, was excellent. But you probably want something which uses flux in a diffusion application. Maybe Schaum's Outline of Heat Transfer if it has to do with heat transfer and diffusion. Or look for a heat transfer text which has a math section which goes over the required vector calculus. Or if it is some other application, like Process Engineering, just look for a text with a section or appendix which covers the "advanced calculus" required for the subject. Often the math isn't a general requirement, so the application field will try to include it as background material. Then they can use physical examples that are helpful, from the actual application you are interested in, that make sense.
Apostol is favored by math majors mostly because of its mathematical rigor. I wasn't that mathematical, and preferred engineering approaches. I am sure Apostol covered most of those subjects well, but I can hardly remember any of it. Sokolnikoff & Redheffer is a good applied math text if you are looking for an applied math book.
Re: Relearning Calculus
I'm so far away from an actual calculus course I can hardly suggest sources, but in agreement with other posts I think there are two directions to go:
1. Foundations of mathematical theory, aka Real Analysis. The nitty gritty is probably a first year graduate course for PhD candidates in math, but there are certainly undergraduate courses and books in analysis. This can be followed by Calculus of Functions of a Complex Variable or divert into Differential Geometry. This is not light weight stuff by any account but is the answer to what underlies calculus.
2. Books and courses using applications of calculus. There were some good examples listed before referencing such topics as dynamics (motion of bodies) from the basic calculus of one dimensional motion to advanced texts such as Goldstein or texts casting mechanics in the language of differential geometry. Electrodynamics can be studied from simpler levels up through Maxwell's equations such as Jackson. Diffusion phenomena, heat transfer were mentioned. Another area is aerodynamics and hydrodynamics. Quantum mechanics is also an application of calculus as in Schroedinger equation but tends to divert into linear algebra as well. While general relativity is essentially geometric the formulation of the properties of noneuclidean spaces involves differential equations and differential geometry. Possibly the general subject that underlies most of physics and engineering is differential equations.
1. Foundations of mathematical theory, aka Real Analysis. The nitty gritty is probably a first year graduate course for PhD candidates in math, but there are certainly undergraduate courses and books in analysis. This can be followed by Calculus of Functions of a Complex Variable or divert into Differential Geometry. This is not light weight stuff by any account but is the answer to what underlies calculus.
2. Books and courses using applications of calculus. There were some good examples listed before referencing such topics as dynamics (motion of bodies) from the basic calculus of one dimensional motion to advanced texts such as Goldstein or texts casting mechanics in the language of differential geometry. Electrodynamics can be studied from simpler levels up through Maxwell's equations such as Jackson. Diffusion phenomena, heat transfer were mentioned. Another area is aerodynamics and hydrodynamics. Quantum mechanics is also an application of calculus as in Schroedinger equation but tends to divert into linear algebra as well. While general relativity is essentially geometric the formulation of the properties of noneuclidean spaces involves differential equations and differential geometry. Possibly the general subject that underlies most of physics and engineering is differential equations.
 Epsilon Delta
 Posts: 7423
 Joined: Thu Apr 28, 2011 7:00 pm
Re: Relearning Calculus
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.
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.
Re: Relearning Calculus
I guess you really love it given your username.
Re: Relearning Calculus
My daughter agrees.
...I might be just beginning 
I might be near the end. Enya 

C'est la vie

 Posts: 877
 Joined: Sat Mar 04, 2017 5:28 pm
 Location: Western Washington
Re: Relearning Calculus
I'm not sure exactly how deep you're going, but where I got my degree, real analysis was a 400 level course, and we learned limits, Riemann sums, and basic proofs in a 200 level course freshsman year intended for science and engineering majors. According to the OP, MIT's open courseware follows the same pattern. The book we used was this one, which from brief discussions with coworkers seems to be very popular at many colleges, so it should be easy to find used:Epsilon Delta wrote: ↑Fri Feb 02, 2018 12:15 pmThis 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.
https://www.amazon.com/Calculus7thJam ... 0538497815
Perhaps more importantly, the common usage of Stewart's book suggests to me the progression from Riemann sums to lay the basic foundation of what calculus is so you actually understand what you're doing when you use it, then on to the numerous shortcut methods as you move on to applying it is the norm, as far as I know, for those tasked with teaching its practical use.
However, I will point out that I didn't find Stewart easy to read on my own when I had reasons to miss lectures. The lectures were really helpful to me.
That was very different from business calculus, which was a 100 level course, and from the limited times I helped business majors with their homework, sounds more like what you're describe.
 patrick013
 Posts: 2253
 Joined: Mon Jul 13, 2015 7:49 pm
Re: Relearning Calculus
I'd go to a major bookstore and get and old GMAT math refresher.
Has lasted me thru many forgetful things. A step up would be
an engineering math guide and I think their math is a step up from
business math. What can't you do then ?
Has lasted me thru many forgetful things. A step up would be
an engineering math guide and I think their math is a step up from
business math. What can't you do then ?
age in bonds, buyandhold, 10 year business cycle
 triceratop
 Moderator
 Posts: 5257
 Joined: Tue Aug 04, 2015 8:20 pm
 Location: la la land
Re: Relearning Calculus
You cannot do proofbased calculus.patrick013 wrote: ↑Fri Feb 02, 2018 8:25 pmI'd go to a major bookstore and get and old GMAT math refresher.
Has lasted me thru many forgetful things. A step up would be
an engineering math guide and I think their math is a step up from
business math. What can't you do then ?
"To play the stock market is to play musical chairs under the chord progression of a bidask spread."
 patrick013
 Posts: 2253
 Joined: Mon Jul 13, 2015 7:49 pm
Re: Relearning Calculus
OK correct me then. What exactly then is proofbased calculustriceratop wrote: ↑Fri Feb 02, 2018 8:27 pmYou cannot do proofbased calculus.patrick013 wrote: ↑Fri Feb 02, 2018 8:25 pmI'd go to a major bookstore and get and old GMAT math refresher.
Has lasted me thru many forgetful things. A step up would be
an engineering math guide and I think their math is a step up from
business math. What can't you do then ?
as opposed to general math problem......etc. ?
age in bonds, buyandhold, 10 year business cycle
 triceratop
 Moderator
 Posts: 5257
 Joined: Tue Aug 04, 2015 8:20 pm
 Location: la la land
Re: Relearning Calculus
A calculus problem might be: optimize this multivariable function on a given domain.patrick013 wrote: ↑Fri Feb 02, 2018 8:30 pmOK correct me then. What exactly then is proofbased calculustriceratop wrote: ↑Fri Feb 02, 2018 8:27 pmYou cannot do proofbased calculus.patrick013 wrote: ↑Fri Feb 02, 2018 8:25 pmI'd go to a major bookstore and get and old GMAT math refresher.
Has lasted me thru many forgetful things. A step up would be
an engineering math guide and I think their math is a step up from
business math. What can't you do then ?
as opposed to general math problem......etc. ?
Or another one might be: compute the fourier transform of a Gaussian. (this requires computing a nontrivial integral)
A proofbased calculus problem might be: If a function f(z) is integrable, what does its Fourier transform converge to as z > \infinity.
One of these has a formulaic answer in the sense that there are a series of steps one can carry out to arrive at the result; the other is more abstract and foundational. For one thing, maybe the fourier transform converges to nothing at all? You have to prove each piece of the argument, in general terms without relying on the specific form of the function but instead on its properties as a class.
"To play the stock market is to play musical chairs under the chord progression of a bidask spread."
 patrick013
 Posts: 2253
 Joined: Mon Jul 13, 2015 7:49 pm
Re: Relearning Calculus
Yeah, see I'm not a math major. Whatever the practical calculus in the GMATtriceratop wrote: ↑Fri Feb 02, 2018 8:38 pmOne of these has a formulaic answer in the sense that there are a series of steps one can carry out to arrive at the result; the other is more abstract and foundational. For one thing, maybe the fourier transform converges to nothing at all? You have to prove each piece of the argument, in general terms without relying on specifics of the function other than its properties.
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.
age in bonds, buyandhold, 10 year business cycle

 Posts: 570
 Joined: Wed Apr 16, 2014 8:11 pm
 Location: Flyover Country
Re: Relearning Calculus
If you struggled with Calculus 13 and want to relearn Calculus from the ground up, I don't think I would start with a Calculus book I would start with Algebra and make sure I had Algebra down cold. Then I would learn Trigonometry cold. Then possibly Geometry or Precalculus, or you may be okay with just Algebra and Trig. Only then (after Algebra and Trigonometry at minimum) would I move on to Calculus I.
I do not have specific books in mind as I took/learned math long ago. But I would likely use two books with one being a basic textbook and one being a book of problems with solutions for Algebra, and similarly up the line.
My guess is that your struggle with Calculus I3 could have been due to weaknesses in the basics, and if so, doing as I suggest could get you started in Calculus with a more solid foundation.
Another thought  in addition, not instead, of the above is to invest in a set of DVD's from The Great Courses on relevant mathematical subjects  for example the following (I haven't seen that one myself but I have a lot of The Great Courses DVD's and love them.)
"Mathematics Describing the Real World: Precalculus and Trigonometry"
https://www.thegreatcourses.com/courses ... metry.html
I would do something like that before Calculus !.
I do not have specific books in mind as I took/learned math long ago. But I would likely use two books with one being a basic textbook and one being a book of problems with solutions for Algebra, and similarly up the line.
My guess is that your struggle with Calculus I3 could have been due to weaknesses in the basics, and if so, doing as I suggest could get you started in Calculus with a more solid foundation.
Another thought  in addition, not instead, of the above is to invest in a set of DVD's from The Great Courses on relevant mathematical subjects  for example the following (I haven't seen that one myself but I have a lot of The Great Courses DVD's and love them.)
"Mathematics Describing the Real World: Precalculus and Trigonometry"
https://www.thegreatcourses.com/courses ... metry.html
I would do something like that before Calculus !.
I don't know anything.
 triceratop
 Moderator
 Posts: 5257
 Joined: Tue Aug 04, 2015 8:20 pm
 Location: la la land
Re: Relearning Calculus
You are correct that most people have no need for this kind of math.patrick013 wrote: ↑Fri Feb 02, 2018 8:53 pmYeah, see I'm not a math major. Whatever the practical calculus in the GMATtriceratop wrote: ↑Fri Feb 02, 2018 8:38 pmOne of these has a formulaic answer in the sense that there are a series of steps one can carry out to arrive at the result; the other is more abstract and foundational. For one thing, maybe the fourier transform converges to nothing at all? You have to prove each piece of the argument, in general terms without relying on specifics of the function other than its properties.
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.
By the way the answer to my last question about the fourier transform is "zero", and the result is known as the RiemannLebesgue lemma.
"To play the stock market is to play musical chairs under the chord progression of a bidask spread."
Re: Relearning Calculus
For what you're wanting, Spivak or Apostol. My recollection from several years ago was that Caltech at the time required all freshman to take the year of Calculus at Caltech, regardless of AP test scores. They used Apostol. I had the chance to look through both books once, and bought Apostol. It seems to be pretty cheap on Amazon.
 patrick013
 Posts: 2253
 Joined: Mon Jul 13, 2015 7:49 pm
Re: Relearning Calculus
Sounds like an MSFinance would be easy for triceratops.
age in bonds, buyandhold, 10 year business cycle
Re: Relearning Calculus
I went through Khan Academy multivariable calc section to refresh my memory around optimization methods. It's very good at getting an understanding, but doesn't have exercises and isn't rigorous in the proving lemmas sense.
 triceratop
 Moderator
 Posts: 5257
 Joined: Tue Aug 04, 2015 8:20 pm
 Location: la la land
Re: Relearning Calculus
It is still used at Caltech. Apostol is a good text, I would recommend it as well in addition to Spivak. It is also easily found on the internet.Jimmei1 wrote: ↑Fri Feb 02, 2018 9:24 pmFor what you're wanting, Spivak or Apostol. My recollection from several years ago was that Caltech at the time required all freshman to take the year of Calculus at Caltech, regardless of AP test scores. They used Apostol. I had the chance to look through both books once, and bought Apostol. It seems to be pretty cheap on Amazon.
@patrick013: my college roommate did an MS Computational Finance. Given what I heard I doubt I want to do that, but I have selfstudied some of the books. Options pricing is fun! I doubt it would be 'easy' for me, though.
"To play the stock market is to play musical chairs under the chord progression of a bidask spread."