Is there a better calculus textbook for self-study than Courant's?

[Dembadon]

I am at a frustrating point in my education. I am pursuing a B.Sc. in Engineering Physics and am only in my first year. I am taking Precalc (Algebra III) and will take Trig next semester. I am loving maths, and my frustration is not due to a lack of understanding, but rather a lack of knowledge. I am anxious to start learning calculus and I have very strong algebra and geometry skills. I want to participate in analysis discussions, become proficient at writing proofs, excel in mathematical modeling, et cetera, but I do not have the prerequisite knowledge to enable me to do so.

So I guess my frustration lies in the fact that I want to pursue maths as a hobby on my free time, but can't due to where I'm at in my education. Is there something I can do in the meantime to satiate my desire? I do all of the exercises in my Precalc textbook, even if they are not part of the assignment, just because I like to solve problems, but most of them are quite trivial to me and do not provide much of a challenge so I'm still left with a desire for more. Is there something which I can read/do on my "free time" that will stretch my understanding of maths?

I realize that I may just need to settle down and be patient about my education; know that the more interesting material will arrive in time. I just don't know what to do in the interim with my anxiousness.

Thanks for making it through my ramblings!

 

[Mensanator]

Have you thought about answering questions on this forum as a hobby?

If you can't answer them, what does that tell you?

 

[Dembadon]

Have you thought about answering questions on this forum as a hobby?

That's a good idea, however, the topics about which I have sufficient knowledge are slim and are usually answered rather quickly. I'll check in more often.

If you can't answer them, what does that tell you?

I'm assuming this is a rhetorical question, as the literal answer was precisely the motivation for my post. Would you mind indulging me and be a little more direct with what you are trying to communicate?

 

[Gear300]

You could independently read and advance through material...you don't necessarily have to be bound by classes.

 

[dx]

Hi Dembadon,

I know exactly how you feel. Personally, I discovered that when I try to learn too much too fast, I get frustrated. So my advice is take it slowly. Do little projects on your own, get to know some of the history of the subject etc. (btw this is not just for fun, knowing the context in which the ideas developed is a great help to understanding the subject itself.)

 

[Mensanator]

That's a good idea, however, the topics about which I sufficient knowledge are slim and are usually answered rather quickly. I'll check in more often.

I'm assuming this is a rhetorical question, as the literal answer was precisely the motivation for my post. Would you mind indulging me and be a little more direct with what you are trying to communicate?

Well, as a beginner, you need not actually post an answer, the important thing is whether or not you can come up with the answer. Sure, actually posting an answer before anybody else does carries a certain sense of accomplishment, but that's irrelevant. I read forums such as this regularly. Sometimes I know the answer, sometimes maybe I know but would have to work it out, sometimes I haven't a clue. I don't even bother to read the calculus sub-forum.

Take for example, the thread about the Search Algorithm. Would you know how to write a 3D version using Ulam's Spiral? If not, that's a great example problem, possibly not covered in any book. If it takes too long to solve, so what? Don't reply, no one will hold it against you. But now your knowledge has increased. Perhaps you'll never have an opprotunity to use it, but you never know.

I once spent an inordinate amount of time trying to figure out how to box lottery numbers and print out all the combinations. Useless knowledge? Not when my job required me to figure out how to create sample labels for 7 sites, each with 7 wells, each sample requiring 7 analyses, each analysis requiring 7 bottles. (Actually, the reality is vastly more complicated.) When I actually needed to do it, I was glad I had that experience with lotto combinations.

 

[Mandark]

You could try some competition maths problems. They tend to require only basic maths but can be challenging nonetheless.

 

[Dembadon]

Hi Dembadon,

I know exactly how you feel. Personally, I discovered that when I try to learn too much too fast, I get frustrated. So my advice is take it slowly. Do little projects on your own, get to know some of the history of the subject etc. (btw this is not just for fun, knowing the context in which the ideas developed is a great help to understanding the subject itself.)

I've actually been meaning to finish reading a book from a friend that came highly recommended called A Tour of the Calculus by David Berlinski. Your post reminded me that I have this book and am only about 10 pages into it. I think its goal is to provide precisely what you are recommending: a history and conceptual view of Calculus. Thanks dx!

Well, as a beginner, you need not actually post an answer, the important thing is whether or not you can come up with the answer. Sure, actually posting an answer before anybody else does carries a certain sense of accomplishment, but that's irrelevant. I read forums such as this regularly. Sometimes I know the answer, sometimes maybe I know but would have to work it out, sometimes I haven't a clue. I don't even bother to read the calculus sub-forum.

Take for example, the thread about the Search Algorithm. Would you know how to write a 3D version using Ulam's Spiral? If not, that's a great example problem, possibly not covered in any book. If it takes too long to solve, so what? Don't reply, no one will hold it against you. But now your knowledge has increased. Perhaps you'll never have an opprotunity to use it, but you never know.

I once spent an inordinate amount of time trying to figure out how to box lottery numbers and print out all the combinations. Useless knowledge? Not when my job required me to figure out how to create sample labels for 7 sites, each with 7 wells, each sample requiring 7 analyses, each analysis requiring 7 bottles. (Actually, the reality is vastly more complicated.) When I actually needed to do it, I was glad I had that experience with lotto combinations.

That's a really good perspective. I'll spend some more time in this forum. I feel a little silly that the answer was right in front of my face.

You could try some competition maths problems. They tend to require only basic maths but can be challenging nonetheless.

Where does one go to find competition problems? I haven't looked into before, but I guess I would start at my university.

 

[Mandark]

Great forum for competition problems: http://www.artofproblemsolving.com/Forum/index.php

You can browse the forum or click on "Contests" which has an archive of thousands of problems from different competitions.

 

[epenguin]

So you're in the first year of an engineering physics (splendid subject) course and you have the time and energy to do something additional? Good, good.

Just reading ahead of where you are, as some have suggested, could do you some good. Either in the books you have or find out the recommended ones for next semester and year. Somehow a lot of the stuff read when you don't have to goes in better. You get the ideas without having to worry just yet about being proficient.

Or if, like you sound, you actually want to amuse yourself with some math stretching without all the weight of your main math syllabus of which you have several years ahead of you, I suggest you might try and look at graph theory. It has advantages of there being a lot that is fairly independent of all other math so not demanding much other knowledge, e.g. not calculus, yet being applicable to your subject (e.g. elec. circuits) though it may only be covered cursorily in your course. You might even find a little aspect to develop a little project. Perhaps others can suggest books for it. Better start with a slim one like Wilson.

But then does it have to be math? Or only math?

You're doing engineering physics, what do you know of these subjects in the world? Of the histories of engineering and physics? Of the philosophy of physics? More generally of philosophy of Science? More generally of philosophy? Of history? Or recent history or ongoing world affairs? Politics? Everyone should take an interest. Or more lasting, political ideas of politics... When I was at school I read Machiavelli and am sure some of it has stayed with me, now decades later am reading for first time Burke, must go on to Paine. For something up to date and highly praised there is Sen's The Idea of Justice. These guys are all surprisingly readable.

What about literature? When I was a bit younger than you and at a loose end in the summer I remember a man said to me "Read the classics now - you'll never have time later." (I think I got through The Three Musketeers that summer - but that's a lot of pages.)

Some of these ideas will be probably inappropriate for you, just brainstorming, though none of them excludes the others. For a choice of books or studies get advice from someone who knows you or you can talk to.

You can't do any math without a certain level of strenuousness. For all general reading there is so much there you can afford to follow:
"A man ought to read just as inclination leads him; for what he reads as a task will do him little good." Samuel Johnson. Good luck.

 

[Dembadon]

Great forum for competition problems: http://www.artofproblemsolving.com/Forum/index.php

You can browse the forum or click on "Contests" which has an archive of thousands of problems from different competitions.

Thank you, Mandark.

So you're in the first year of an engineering physics (splendid subject) course and you have the time and energy to do something additional? Good, good.

Just reading ahead of where you are, as some have suggested, could do you some good. Either in the books you have or find out the recommended ones for next semester and year. Somehow a lot of the stuff read when you don't have to goes in better. You get the ideas without having to worry just yet about being proficient.

I've just picked up What Is Mathematics? An Elementary Approach to Ideas and Methods, by Courant, Robbins, and Stewart. As you've said, I won't be able to understand everything in the book yet, but It'll stretch me and at the same time offer some exposure to topics I'll be encountering later.

Or if, like you sound, you actually want to amuse yourself with some math stretching without all the weight of your main math syllabus of which you have several years ahead of you, I suggest you might try and look at graph theory. It has advantages of there being a lot that is fairly independent of all other math so not demanding much other knowledge, e.g. not calculus, yet being applicable to your subject (e.g. elec. circuits) though it may only be covered cursorily in your course. You might even find a little aspect to develop a little project. Perhaps others can suggest books for it. Better start with a slim one like Wilson.

Good idea; I'll check it out.

What about literature? When I was a bit younger than you and at a loose end in the summer I remember a man said to me "Read the classics now - you'll never have time later." (I think I got through The Three Musketeers that summer - but that's a lot of pages.)

I've finished a couple of Melville's short stories and I think that Moby-Dick is going to be my next adventure in literature. If I can get in 20-30 minutes of reading a day, I'm happy. Before I started school I read a lot more.

Some of these ideas will be probably inappropriate for you, just brainstorming, though none of them excludes the others. For a choice of books or studies get advice from someone who knows you or you can talk to.

Thanks for the time you invested in your post! I really appreciate your input.

 

[n!kofeyn]

Just get a good calculus textbook, for example the one by https://www.amazon.com/dp/354065058X/?tag=pfamazon01-20 and start to go through it. It might be difficult, but when you get stuck, you know you've hit a gap in your education. Then find out how to fill in that gap either through you pre-calculus or trigonometry courses until you can proceed again. This is a rather focused way to learn, and struggling through something on your own spare time can be very rewarding. There isn't any reason to beat around the bush with popularizations of math. Just get to it.

 

[JYouker]

Hello,

I am currently a dual degree student in mathematics and engineering. My appreciation and interest in mathematics has really matured while I've been in college and I'm looking to explore mathematics further, in my free time. What paths could I take to pursue further knowledge in mathematics? I could solve problems on mathematics forums, but, what else is there to do? Are there any good books for self-taught learners? What areas of mathematics are useful to learn for the fields of engineering or, my current interest, bioinformatics? Could I start doing a research project and, if so, how do I go about getting started? So far, I have completed Calculus II and Differential Equations. Now, I am in Calculus III and Linear Algebra courses.

 

[sponsoredwalk]

Hello,
Are there any good books for self-taught learners? What areas of mathematics are useful to learn for the fields of engineering or, my current interest, bioinformatics? Could I start doing a research project and, if so, how do I go about getting started? So far, I have completed Calculus II and Differential Equations. Now, I am in Calculus III and Linear Algebra courses.

There are tons of good books. I have a list of @ least 50 books ahead of me, all of which I have made sure are possible to learn from by self study. Here are a few that I'm working on/planning to work on. Hopefully they will @ least motivate you to look at what's there. The reviews on amazon are extremely valuable reads to scope the book out. These books all seem more or less relevant to the topic & it took me a long time to find some of these gems.

https://www.amazon.com/dp/0471154962/?tag=pfamazon01-20
https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20
https://www.amazon.com/dp/0486649407/?tag=pfamazon01-20
https://www.amazon.com/dp/0070854238/?tag=pfamazon01-20
https://www.amazon.com/dp/354065058X/?tag=pfamazon01-20
https://www.amazon.com/dp/0387982582/?tag=pfamazon01-20
https://www.amazon.com/dp/038797606X/?tag=pfamazon01-20
https://www.amazon.com/dp/0521679710/?tag=pfamazon01-20
https://www.amazon.com/dp/0471198269/?tag=pfamazon01-20

All I can remember off the top of my head :p. There are way way more though...

 

[JYouker]

Thank you! Thanks for taking the time to respond with all those links. I will now be exploring them and finding ones I like best. Maybe, they're in my library!

 

[VKint]

If I were you, I would look into high school math competitions. It sounds trivial, but honestly, some of the hardest math problems I've ever done were from the USAMO (US Mathematical Olympiad) and the IMO (International Math Olympiad). These competitions are intended for the best young mathematicians in the world, and make use of nothing more advanced than precalculus mathematics. Check out <http://www.artofproblemsolving.com/> for archives, or just google USAMO or IMO, for problems. (There are also printed compendiums available from Springer.) Once you've learned some basic calculus, you should definitely check out the Putnam competition for undergraduates. It's pretty intense; the median score is frequently 0. You can find a problem-solution archive at <http://www.unl.edu/amc/a-activities/a7-problems/putnamindex.shtml>.

Before doing all this, though (or while doing it), you should read some books on general problem-solving. Two of my favorites are Arthur Engel's "Problem Solving Strategies" and Paul Zeitz's "Art and Craft of Problem Solving." Each of these has excellent explanations of elementary methods and strategies that don't often get taught in introductory courses, as well as lots of interesting (and HARD) problems (especially Engel).

 

[avarmaavarma]

You might like these links : well written articles on people who treat math as a hobby

http://www.3quarksdaily.com/3quarksdaily/2011/08/mathematical-learning-and-math-as-a-hobby.html

http://www.anujvarma.com/mathematics-as-a-hobby-yes-really/

 

[Robert1986]

I've actually been meaning to finish reading a book from a friend that came highly recommended called A Tour of the Calculus by David Berlinski. Your post reminded me that I have this book and am only about 10 pages into it. I think its goal is to provide precisely what you are recommending: a history and conceptual view of Calculus. Thanks dx!.

I tried to read this book and it was one of the most frustrating books I have ever laid (lain?) eyes on. It isn't a bad thing to read Tour Of Calculus, but I certainly understand why you are only 10 pages in. Tour is like a book that a not-so-brilliant father of a brilliant math genius would read to have a remote idea of what is son is doing.

I don't know how far along you are in pre-calc but I would suggest going to the library and getting an actual calc textbook. Stay away from anything written by anyone named Courant for now. If there are parts you don't understand due to lack of pre-req knowledge, try to look them up in your pre-calc book. Lots of calc books have a Chapter 0 and a ton of appendices that cover pre-req stuff if you need it.


Another option is to just take pre-calc, then start do trig next semester, and just wait to do calc. In the meantime, start learning Abstract Algebra (you said you have good algebra skills, but I'm guessing you don't mean abstract, right?) and Linear Algebra. Neither of these require you to know Calculus, though some examples might require a knowledge of calculus. Really, all you need to know is that differentiation and integration are linear operators, and it would help to know how to differentiate and integrate a polynomial (this can be, and usually is, defined formally, which means you don't have to understand what differentiation and integration is to get the examples in Abstract/Linear Algebra).


Yet another option is to get a good introductory book on Combinatorics. Here is a good online book written by Prof. Tom Trotter and one of his former graduate students Prof. Mitch Keller: http://people.math.gatech.edu/~trotter/ (the link is on this page.) This does not require a knowledge of calculus, either, and is a very fun thing to do.

 

[Robert1986]

Just get a good calculus textbook, for example the one by https://www.amazon.com/dp/354065058X/?tag=pfamazon01-20 and start to go through it. It might be difficult, but when you get stuck, you know you've hit a gap in your education. Then find out how to fill in that gap either through you pre-calculus or trigonometry courses until you can proceed again. This is a rather focused way to learn, and struggling through something on your own spare time can be very rewarding. There isn't any reason to beat around the bush with popularizations of math. Just get to it.

I respectfully disagree. While I agree he needs to abandon the awful Tour book, Courant is a little too much to read as a beginner in a self-study program, don'tcha think? I know Courant was not the first calc. book I read, and I'm guessing it isn't the first one you read, was it? Don't get me wrong, Courant is good, but not as a first-time book for someone doing self-study.