"Introduction to Calculus and Analysis" by Courant & Fritz

[Santiago24]

Hi, I'm reading the volume 1 of "Introduction to calculus and analysis" by Courant and Fritz but the problems are hard for me, i understand what he say but i can't solve many problems of the chapter one. It's normal or should i try with other book?

 

[Stephen Tashi]

(I don't own the book. I looked at the table of contents as shown on its Amazon page.)

If your goal is to learn calculus to study engineering, there are simpler calculus books. If your goal is to study calculus as part of a pure mathematics degree, I don't know if there are better books. I haven't looked at math textbooks in many years. Other forum members can probably give you advice about good texts for "analysis".

In USA educational terminology, "analysis" implies a more formal type of mathematics than "calculus".

In the USA, most students begin the study of calculus with texts that don't emphasize proofs and are suitable for engineering students. After that, mathematics majors take a course in "analysis", which emphasizes proofs and rigorous mathematical thinking.

If you have not studied calculus before or if you have not taken courses that require you to do proofs then it is normal to have trouble with a book like Courant and Fritz.

 

[Santiago24]

(I don't own the book. I looked at the table of contents as shown on its Amazon page.)

If your goal is to learn calculus to study engineering, there are simpler calculus books. If your goal is to study calculus as part of a pure mathematics degree, I don't know if there are better books. I haven't looked at math textbooks in many years. Other forum members can probably give you advice about good texts for "analysis".

In USA educational terminology, "analysis" implies a more formal type of mathematics than "calculus".

In the USA, most students begin the study of calculus with texts that don't emphasize proofs and are suitable for engineering students. After that, mathematics majors take a course in "analysis", which emphasizes proofs and rigorous mathematical thinking.

If you have not studied calculus before or if you have not taken courses that require you to do proofs then it is normal to have trouble with a book like Courant and Fritz.

Hi, I'm reading this book because i want to study physics and I've seen in internet that this book is in the middle of analysis and calculus. This isn't my first exposure to calculus, i know how to do derivates, integrals, Taylor series, etc. But this is my first exposure to analysis and doing proofs, i'd like to do proofs, but when i have to do it simply i can't see how to prove it. I'd appreciate if you can give me some advice for improve my proof writing. Thanks for answering.

 

[jim mcnamara]

There is an old book, which has a Dover reprint, 'A book of Abstract Algebra' by Charles Pinter. The original helped me get going on proofs a long time ago for a project I got involved with. I'm a biologist...

 

[Stephen Tashi]

Hi, I'm reading this book because i want to study physics and I've seen in internet that this book is in the middle of analysis and calculus.

What physics have you studied? Proofs are relevant to theoretical physics. Do you want to do theoretical physics - as opposed to the more concrete and applied sort?

I'd appreciate if you can give me some advice for improve my proof writing. Thanks for answering.

My thoughts on developing skill with proofs.

1) Proofs and formal mathematics are legalistic. You must learn to sympathize with the legalistic approach and appreciate the need for it. When you hear a mathematical claim, you must want it to be a precise statement. Instead of relying on intuition to decide if it's true, you must desire to see a proof of it.

This may involve a big change in your mental life style! The average person is not sympathetic to a legalistic style of thinking. You have to appreciate the need for it in mathematics and learn the unreliability of intuitive thinking.

Many different mathematical subjects can be used to learn this legalistic outlook. In the USA, it is common for a course in abstract algebra or "linear algebra" to be an introduction to writing proofs. Another good choice is a course in "point set topology". I don't know how well Courant and Fritz teach doing proofs or whether that is one goal of their text. Perhaps another forum member can suggest books on analysis that take an introductory approach to proof writing.2) Writing proofs requires understanding basic mathematical logic - how to interpret logical connectives "and", "or", "implies" - how to interpret logical quantifiers "for each", "there exists". Some people develop this understanding without studying mathematical logic as separate topic. In my opinion, most people would learn mathematical logic quicker by making a short study of it from a textbook on logic. Consider doing that. A study of mathematical logic is also a good way to develop a sympathy and appreciation for the legalistic nature of math.

3) Logic is necessary for doing math but each different field of mathematics has its own style and tricks.

 

[Santiago24]

There is an old book, which has a Dover reprint, 'A book of Abstract Algebra' by Charles Pinter. The original helped me get going on proofs a long time ago for a project I got involved with. I'm a biologist...

Thanks, i'll try with it.

 

[Santiago24]

What physics have you studied? Proofs are relevant to theoretical physics. Do you want to do theoretical physics - as opposed to the more concrete and applied sort?

I'll start in April with my first year in physics, I'm very interested in theoretical physics so i think that i need to understand maths very well.

My thoughts on developing skill with proofs.

1) Proofs and formal mathematics are legalistic. You must learn to sympathize with the legalistic approach and appreciate the need for it. When you hear a mathematical claim, you must want it to be a precise statement. Instead of relying on intuition to decide if it's true, you must desire to see a proof of it.

This may involve a big change in your mental life style! The average person is not sympathetic to a legalistic style of thinking. You have to appreciate the need for it in mathematics and learn the unreliability of intuitive thinking.

Many different mathematical subjects can be used to learn this legalistic outlook. In the USA, it is common for a course in abstract algebra or "linear algebra" to be an introduction to writing proofs. Another good choice is a course in "point set topology". I don't know how well Courant and Fritz teach doing proofs or whether that is one goal of their text. Perhaps another forum member can suggest books on analysis that take an introductory approach to proof writing.

Thanks, i have to change my style of learn and think maths because i use my intuition a lot. I'll try with abstract algebra and see if i can improve my writing proofs skills. You know if "Understanding Analysis" by Stephen Abbot is a good book for that? I know that is about real analysis but i see that a lot of people recommend it.

2) Writing proofs requires understanding basic mathematical logic - how to interpret logical connectives "and", "or", "implies" - how to interpret logical quantifiers "for each", "there exists". Some people develop this understanding without studying mathematical logic as separate topic. In my opinion, most people would learn mathematical logic quicker by making a short study of it from a textbook on logic. Consider doing that. A study of mathematical logic is also a good way to develop a sympathy and appreciation for the legalistic nature of math.3) Logic is necessary for doing math but each different field of mathematics has its own style and tricks.

I read about logic but in books of other topics that make a small introduction. I think that my big problem is that i don't know how to start with the proof.