Study Maths & Physics: How to Proceed?

[kripkrip420]

Hello there! I will be studying Mathematics and Physics in University in approximately 2 months. I really enjoy Mathematics and have done some introductory Calculus. I have looked into Spivak's book "Calculus" and, although written very well, I just tent to find Calculus boring. However, when looking at books like Hardy's "A Course of Pure Mathematics" or Rudin's "Principles of Mathematical Analysis", I find these to be far more entertaining. Now, it is my understanding that Real Analysis is generally a more formal approach to Calculus (which I very much prefer). Is it possible to simply skip the "formalized" (from high school at least) Calculus in books like Spivak's and start an Analysis course in something like the books mentioned above. Is that a wise move or will I be missing important topics not found in Analysis books? Thank you.

 

[micromass]

Despite his name, I consider Spivak's calculus to be an analysis book. At the very least, it's some kind of intro to analysis. It is far more rigorous than things like Stewart.

However, everybody likes different styles of books. It might be that you consider Spivak to be boring and Rudin to be very entertaining (although many people are exactly the other way around). In that case, I would suggest you to read the book you enjoy most.

If you start reading Rudin now, you won't miss anything important. I guess it only misses some computational exercises.

Be warned though, Rudin is quite a difficult book. It doesn't explain intuition at all. He does his proofs in the most elegant ways, and these ways are often not the most understandable. For example, while reading a proof of Rudin, you may say things like: wow, we're lucky we had this little trick otherwise the proof will not have worked.
Furthermore, the exercises of Rudin are very hard (but Spivak's exercises are also hard).

But if you feel ready for Rudin, then go for it!

 

[Jorriss]

For most people, even though it is boring, it is good to do some computational calculus exercises. Those give intuition and lots and lots of examples/counter examples for analysis.

 

[kripkrip420]

Despite his name, I consider Spivak's calculus to be an analysis book. At the very least, it's some kind of intro to analysis. It is far more rigorous than things like Stewart.

However, everybody likes different styles of books. It might be that you consider Spivak to be boring and Rudin to be very entertaining (although many people are exactly the other way around). In that case, I would suggest you to read the book you enjoy most.

If you start reading Rudin now, you won't miss anything important. I guess it only misses some computational exercises.

Be warned though, Rudin is quite a difficult book. It doesn't explain intuition at all. He does his proofs in the most elegant ways, and these ways are often not the most understandable. For example, while reading a proof of Rudin, you may say things like: wow, we're lucky we had this little trick otherwise the proof will not have worked.
Furthermore, the exercises of Rudin are very hard (but Spivak's exercises are also hard).

But if you feel ready for Rudin, then go for it!

Thank you very much for your response Micromass. I actually started doing a lot of work in Spivak's book and the exercises are fun, there is no question. However, as soon as I opened Hardy's or Rudin's books, I just find the building of the Real numbers and the introductions to Set Theory so exciting. Personally, I feel that some of the problems in Spivak's book are too computational (although the first chapter had a lot of what I consider to be Number Theory problems which I also find extremely fun). The lack of intuitive approach in Rudin's books does not really bother me. If I ever find myself not totally grasping a concept, I rarely stick to the book I'm reading to find a solution. Rather, I go online and look at articles or videos regarding such a concept. I then dwell on it in my bed for some time until something "clicks". I will likely keep moving through Spivak's book but I will also be working through Rudin's or Hardy's simultaneously simply because I find it adrenaline pumping (like when I found out that there are infinite sets that vary in size through "Introductory Real Analysis" by the two Russian authors). Regardless, thank you both for your responses.

 

[Matrix0]

Hello there! It's great to hear that you will be studying Mathematics and Physics soon. Both of these subjects are highly interconnected and can provide a solid foundation for a career in science.

Regarding your question about skipping the "formalized" Calculus in books like Spivak's and starting with Real Analysis, it is definitely possible. Real Analysis is a more rigorous and formal approach to Calculus, as you mentioned, and it can be a more enjoyable and interesting way to learn the subject. However, it is important to note that Calculus is a crucial foundation for Real Analysis, so it is important not to completely skip over it.

I would recommend reviewing the fundamental concepts and techniques of Calculus before diving into Real Analysis. This will ensure that you have a strong understanding of the basics before moving on to more advanced topics. Additionally, it is always helpful to have a solid understanding of the fundamentals when studying more complex subjects.

As for the books you mentioned, Hardy's "A Course of Pure Mathematics" and Rudin's "Principles of Mathematical Analysis" are both excellent resources for learning Real Analysis. They are highly regarded in the mathematical community and cover a wide range of topics. However, it is important to supplement your learning with other resources and practice problems to fully grasp the material.

In summary, it is possible to skip the "formalized" Calculus and start with Real Analysis, but it is important to review the fundamentals first. Additionally, using multiple resources and practicing problems will help solidify your understanding of the material. Good luck with your studies!