Tom Apostol Calculus Vol 2 Complete Solutions
Gaynelle Alnutt
Which one of those is the best for a person interested in pure mathematics and who wants to have a deep understanding of calculus? Apostol or Spivak? Could you guys tell me some differences between the approaches of them? What about the exercises? I would like to be challenged, but in a constructive way.
While both books have complete proofs and a good emphasis on theory, Spivak's book is better as an introduction to rigorous math because many of its problems are more difficult and theoretically oriented than Apostol's. (I assume you mean his Calculus). Spivak's book also has a solution manual, which is very useful when you're studying on your own.
On the other hand, Apostol actually covers more material, even just within Volume 1. Volume 2 of Apostol is actually one of the best introductions to multivariable calculus. Apostol's book also has a greater variety of exercises involving applications of calculus to physics.
Both books give you an introduction to calculus, being: set and number theory, some induction, series and notation. Big difference can be the order in which topics are introduced, for Apostol integration comes first due to historical reasons, while for Spivak derivatives comes first.
For me, Apostol provides an interesting framework to learn about how calculus' ideas evolved historically, as well volume 1 introduces lineal algebra, providing a more gentle progressions towards multivariable calculus and differential equations in volume 2. On the hand, Spivak goes deeper into proofs with rigour, which is essential for every mathematician or even physicist. Therefore, I decided to use Apostol as my "guide" book and Spivak to delve deeper into concepts in which I'm interested.
I'm a math student. I work with both Spivak's and Apostols calculus books. There is a solutions manual for Spivak and there is a blog for Apostol Vol I. However, I haven't been able to find any solutions manual for Vol II. Does anybody know where you can get it or if it doesn't exist?
So if you have questions about specific problems, by all means ask them here. But posting a complete list of solutions will not be doing anyone a favor. Many instructors assign those problems as homework, and if complete solution sets become readily available, it makes the problems (and therefore the book) far less useful.
It's interesting to note that when I've written chapters with everything proved and few or no problems at the end, readers invariably ask me to provide some problems for them to work on. If you want problems with solutions already written down, they're already there -- the theorems and examples in the book! Just look at the statement of a theorem or the claims made in an example, close the book and try to prove the theorem on your own, and then go back and compare your work to the proof in the book. (And if you find a better proof that the one I wrote, please let me know about it!)
In response to the counter-arugment "What about people who are self-studying, or for whom mathematics is just a hobby?", I think that the advice is even more relevant. If you are studying a subject for a class, you are rewarded and penalized for your work, hence there is a very strong incentive to get it done correctly under the pressure of a deadline. The hobbiest or self-studier is under no such pressure, and has the time to be "stuck" on difficult problems. There is no penalty for late work.
Moreover, if one is taking a class, then there is a ready-made structure for expanding upon and providing context for solutions to problems. This structure is not provided by a solutions manual, but can be found through conversation (e.g. on MSE). Such conversation is going to help one to understand the errors in their thinking or underlying assumptions much more readily than a solutions manual.
With regard to "checking one's work," I think it is worth pointing out that a solutions manual may not actually be all that useful. If you are really uncertain as to whether or not your proof is sound, a solutions manual may not help all that much, because the approach in the manual may be different from the approach of a given student. Again, the student is going to benefit more from conversation and interaction than from a solution written from a particular point of view at a particular point in time.
The solutions to the exercises in "Apostol calculus volume 1" can typically be found in the back of the book or in a separate solutions manual. Some universities or libraries may also have digital access to the solutions.
While having access to the solutions can be helpful for practice and understanding, it is not necessary to have them for studying for exams. It is important to focus on understanding the concepts and practicing problem-solving techniques.
Yes, there are various websites and forums where students and educators may share solutions or offer assistance with exercises from "Apostol calculus volume 1". However, it is important to use these resources as a supplement and not rely on them entirely for understanding the material.
No, using the solutions in "Apostol calculus volume 1" for cheating is not ethical or beneficial in the long run. It is important to work through the exercises and understand the concepts yourself in order to fully grasp the material and succeed in the course.
For over six years now, I've been studying mathematics on my own in my spare time - working my way through books, exercises, and online courses. In this post I'll share what books and resources I've worked through and recommend and also tips for anyone who wants to go on a similar adventure.
Self-studying mathematics is hard - it's an emotional journey as much as an intellectual one and it's the kind of journey I imagine many people start but then drop off after a few months. So I also share (at the end) the practices and mindset that have for me allowed this hobby to continue through the inevitable ups and downs of life (raising two young boys, working at a startup, and moving states!)
I used to love mathematics. Though I ended up getting an engineering degree and my career is in software development, I had initially wanted to study maths at university. But the reality is, that's a very tough road to take in life - the academic world is, generally speaking, a quite tortuous path with low pay, long hours, and rife with burnout. So I took the more pragmatic path and as the years went by never really found the time to reconnect with math. That was until about six years ago when I came across Robert Ghrist's online course Calculus: Single Variable (at the time I took it, it was just a Coursera course but now it's freely available on YouTube). Roughly 12 weeks and many filled notebooks later, I had reignited my interest in math and felt energized and excited.
Growing up I always loved puzzles and problem solving. I would spend hours working my way through puzzle books, solving riddles, and generally latching on to anything that gives you that little dopamine hit.
If you're similar, mathematics might just be for you. Mathematics is hard. Seriously hard. And then suddenly, what was hard is easy, trivial, and you continue your ascent on to the next hard problem. It deeply rewards patience, persistence, and creativity and is a highly engaging activity - it's just you quietly working away, breaking down seemingly impossible problems and making them possible. I can't say enough how deeply satisfying and personally enriching it is to make the impossible, possible through your own hard work and ingenuity.
One thing many people don't know as well is that the mathematics you learn at most high schools is actually quite different from what you're exposed to at the university level. The focus turns from being about rote computation to logic, deduction, and reasoning. A great quote I read once is that for most of us, when we learn mathematics at school, we learn how to play a couple of notes on a piano. But at university, we learn how to write and play music.
As a self-learner, it's critical to pick books with exercises and solutions. At some point later on you can swap to books without exercises and/or solutions, but in the beginning you need that feedback to be able to learn from your mistakes and move forward when you're stuck.
The books you pick as a self-learner are also sometimes different from what you would work use if you were engaged in full-time study at a university. Personally, I lean more towards books with better exposition, motivation, and examples. In a university setting, lecturers can provide that exposition and complement missing parts of books they assign for courses, but when you're on your own those missing bits can be critical to understanding.
I recommend avoiding the Kindle copies of most books and always opting for print. Very few math books have converted to digital formats well and so typically contain many formatting and display errors. Incidentally, this is often the main source for bad reviews of some excellent books on Amazon.
I'd be remiss as well if I didn't mention the publisher Dover. Dover is a well known publisher in the math community, often publishing older books at fantastically low prices. Some of the Dover books are absolutely brilliant classics - I own many and have made sure to make note of them in my recommendations below. If you don't have a big budget for learning, go for the Dover books first.
In several places I also recommend courses from MIT OpenCourseware. These courses are completely free and often have full recorded video lectures, exam papers with solutions, etc. If you like learning by video instruction and find at various points that you're getting a bit lost in a book, try looking up an appropriate course on MIT OpenCourseware and seeing if that helps get you unstuck.
Pretty much all my books I recommend below focus on undergraduate level math, with an emphasis on pure vs applied. That's just because that's the level that I'm at and also the kind of maths I like the most!
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