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Which self-contained calculus book explains the math in order...

  • #1
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Main Question or Discussion Point

Which calculus book self contains experiences in order, and is stable to the max, for all the problems known in the subject?
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I have got confused now. Math history seems to allow knowing all the data on how math formed, for any of a particular utility (?), and seems to allow having experiences as they were had in the history. This seems to allow having experiences on the principles of the creation of math too.

And textbooks seem to give no historical data, often. They seem to give experiences in a particular order, as needed, and might eliminate problems which may have been faced before, in the history, from not having them, in the needed time. I am now knowing calculus, for knowing particle physics, quantum mechanics, etc. properly; so I seem to know the utility of math I am studying.

Reading history, seems to confirm knowing the subject and even allow knowing the recurring conformations or principles, which allow creation of math, but seems to take time.
I don't have time now; in college, they are running. I am far behind. I have stopped, since knowing zero dimensional points, irrational numbers location on the number line, zeno's paradoxes, and non-clarity in me on knowing integration/differentiation process.

I don't know the consequences of leaving historical development; if they are any, we may be able to know conformations for eliminating any of them, which would not allow attaining any of the particular intended utilities.

And I searched text books which would be self contained in all the data, and which would give all the experiences, to solve any of the problems as zeno, irrationality, etc. I have till now seen:
Courant, Richard, Differential and integral calculus.
Apostol, Tom M., Calculus. Vol. I: One variable calculus, with an introduction to linear algebra.
Spivak, Calculus.
Lang, Serge, A first course in calculus.
being mentioned often. Which textbook would you recommend for me? Or would you recommend reading any other set of books in a particular order?
 

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
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Which textbook would you recommend for me?
What is your level of education? What math courses have you taken?

Or would you recommend reading any other set of books in a particular order?
What is your native language? (You only mentioned books in English.)

I don't have time now; in college, they are running. I am far behind. I have stopped, since knowing zero dimensional points, irrational numbers location on the number line, zeno's paradoxes, and non-clarity in me on knowing integration/differentiation process.
That passage appears to say that you are now in college and that you are taking calculus, and that you are falling behind. Is that correct?

Keeping up with a typical calculus course requires accepting certain facts about without proof. Detailed proofs and definitions of the properties of the real numbers and advanced theories of integration are treated in later courses. Many people hope to learn each topic in math in great depth before proceeding to the next topic. However, I don't know any calculus textbooks written in that style.
 
  • #3
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Sorry, for being late in replying here, and thank you very much for helping.

What is your level of education? What math courses have you taken?
I am studying MSc Mathematics, in India. I am in the first semester of the first year of my two year 4 semester MSc.

Here in India, syllabus will be decided by the college; we will have to study only those topics in the subject. In my syllabus, for the current semester, I have
1. Algebra (Group theory, Ring Theory, Field Theory and Galois Theory),
2. Real Analysis, Metric Spaces, Function of several variables,
3. Ordinary Differential Equations,
4. Discrete Mathematics,
5. Linear Algebra and
6. Statistics (Random variable and Expectations)

What is your native language? (You only mentioned books in English.)
My native language is Kannada; all here, read English books.

That passage appears to say that you are now in college and that you are taking calculus, and that you are falling behind. Is that correct?
Yeah, they are teaching all the higher math, and am still not having proper understanding of number line (location of irrational numbers), zeno's paradoxes, zero dimensional point. This had somehow not allowed me understand calculus processes.

I am not able to read what they are teaching, as....(I seem to have not explored precisely on this before, though I seem to have had notions)…..I am always getting new data which is to be known for having clear view of zeno, irrationality, or calculus, and am tempted to not leave them. I had before thought on studying data given in college, and to use those data to get clear view of zeno, etc. But, seem to have not been. I will try being precise on this, as I go on trying to study college data now. I seem to still not have a stable algorithm. I will think on what you have said in the below quote.

Keeping up with a typical calculus course requires accepting certain facts about without proof. Detailed proofs and definitions of the properties of the real numbers and advanced theories of integration are treated in later courses. Many people hope to learn each topic in math in great depth before proceeding to the next topic. However, I don't know any calculus textbooks written in that style.
My Sir too said what you are saying, i.e. accepting certain facts without proof. He asked to assume things for now; from my Sir's saying, it seems as if, the problems I am having as to be not really problems in math, but seems to be only problems for now, from me not having complete data.
 
  • #4
Stephen Tashi
Science Advisor
7,021
1,244
.I am always getting new data which is to be known for having clear view of zeno, irrationality, or calculus, and am tempted to not leave them.
We must distinguish between questions that have a precise mathematical statement versus questions that have a less precise meaning. Zeno's paradoxes (in their original form) are not precise mathematical questions. In ancient times the language Zeno used may have been considered perfectly clear and precise. However, the historical trend in mathematics is that statements and language considered precise an earllier epic in history are considered vague and ambiguous in a later epic. (For example, the derivative of a function as described by the language of Newton or Leibnitz was found by later mathematicians to be too vague.)

In studying modern mathematics, we often encounter definitions that seem strange. Yes, studying the history of mathematics can explain why language and definitions in modern textbooks are complicated. Yes, studying history can be a great distraction from studying one's coursework.

Studying the history of mathematics in a casual way need not take up a large amount of time. However, becoming fascinated by the history of mathematics leads to spending hours and hours in studying and thinking about it. There are interesting books and articles written about the history of mathematics, especially about the history of calculus. I myself don't recall specific titles, but other forum members can probably recommend specific books.

There is a saying: "ontogeny recapitulates phylogeny", which is obsolete as a theory of biology. However, as a theory of intellectual development, there is some truth to it. Individuals who study mathematics (such a calculus) are puzzled by the same conceptual dilemmas that faced the general population of mathematicians in earlier times. Many individuals' personal understanding of mathematics follows a path similar to how mathematical ideas evolved in history. (Of course, we also see individuals who develop eccentric personal conceptions of mathematical ideas. For example, we encounter people who think the modern definition of a derivative or limit is "wrong" and propose their own language for defining it.)

If history is a guide, the key to understanding mathematics to a greater degree is to use increasingly precise language. Paradoxes are often due to fundamental vagueness and ambiguity in the language used to state them. If you don't have time to study the history of a mathematical idea, you can at least criticize the language that you yourself use to ask questions about the idea.
 
  • #5
StoneTemplePython
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Gold Member
2019 Award
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While it is not purely a "calculus book" per se, though calculus is discussed in a large chunk of the book, it occurs to me that Courant's What is Mathematics? could be just the book that OP is looking for.
 
  • #6
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Thank you very much StoneTemplePython and Stephen Tashi; I am thinking on what you all have said. I will report back later again.
 

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