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.Stephen Tashi said:(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.
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?Santiago Perini said: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.
I'd appreciate if you can give me some advice for improve my proof writing. Thanks for answering.
Thanks, i'll try with it.jim mcnamara said: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 said: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?
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.Stephen Tashi said: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.
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.Stephen Tashi said: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.
The main focus of the book is to provide a rigorous and comprehensive introduction to the fundamental concepts of calculus and analysis, including topics such as limits, derivatives, integrals, sequences, and series. It also covers more advanced topics such as multivariable calculus and vector analysis.
While the book is quite comprehensive, it is not recommended for complete beginners as it assumes a basic understanding of algebra and trigonometry. It is better suited for students who have already taken a pre-calculus course and are looking to deepen their understanding of calculus and analysis.
Yes, the book does include real-world applications of calculus and analysis, particularly in the later chapters. These applications range from physics and engineering to economics and biology. However, the main focus of the book is on developing a strong theoretical foundation in calculus and analysis.
This book is known for its rigorous and comprehensive approach to teaching calculus and analysis. It also includes challenging exercises and problems, as well as historical notes and biographies of mathematicians. Additionally, the book has been widely praised for its clear and concise writing style.
The book can be used for self-study, but it is recommended to have a strong background in mathematics and to have access to a teacher or tutor for assistance with difficult concepts. It is also commonly used as a textbook in university-level calculus and analysis courses.