Do all calculus books contain this?

In summary, Lieb's book on Analysis expects readers to have a strong understanding of vector spaces, limits, lim inf, lim sup, open, closed, and compact sets in ]Rn, continuity and differentiability of functions, convergence and uniform convergence, the Riemann integral and integration by parts. These are typically covered in an undergraduate analysis course and not in a standard freshman calculus course. Books such as Spivak, Apostol, and Courant would meet these prerequisites, but more introductory calculus books like Stewart, Larson, and Anton would not. It ultimately depends on the major and intended focus of the course as to which book would be more suitable.
  • #1
theoristo
153
3
This is from Analysis by Lieb :
''Things the reader is expected to know : While we more or less start from 'scratch' , we do expect the reader to know some elementary facts, all of which will have been learned in a good calculus course . These include : vector spaces , limits , lim inf, lim sup , open , closed and compact sets in ]Rn, continuity and differentiability of functions ( especially in the multivariable case ) , convergence and uniform convergence ( indeed , the notion of 'uniform ', generally ) , the definition and basic prop erties of the Riemann integral , integration by parts ( of which Gauss' s theorem is a special case ) . ''
Is this covered in all rigourous calculus books?
Is this book any harder than the by Carothers or Berberian?
 
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  • #2
Those are analysis topics (those that look strange). A good calculus course would focus on applications, not on things like uniform convergence.
 
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  • #3
Lieb's book is a graduate-level text. The prereqs he lists sound more like what you would learn in an undergrad analysis class than the normal freshman calculus sequence, at least in the US.

There are books with "Calculus" in their title that include analysis, notably those by Spivak, Apostol, and Courant, but they are very seldom used for a first course in calculus. They may be used for freshmen at MIT or Cal Tech, but the students in those classes typically have already had a less rigorous calculus course in high school.
 
  • #4
Yes ,those are the books I'm reading,do they contain this?
 
  • #5
theoristo said:
Yes ,those are the books I'm reading,do they contain this?
Yes, reading Spivak, Apostol, or Courant should provide you with the prerequisites that Lieb discusses. On the other hand, more introductory Calculus books like Stewart, Larson, and Anton wouldn't meet those prerequisites.
 
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  • #6
verty said:
Those are analysis topics (those that look strange). A good calculus course would focus on applications, not on things like uniform convergence.

Why in the world would that make it a "good" calculus course? Uniform convergence is an extremely important concept and is easy to introduce in an honors calculus class; Spivak has an entire chapter on it. Not all calculus classes are made equal, some are made for the sole purpose of doing calculus correctly from freshman year itself instead of just hand-waving things and jumping into applications (to wit not all students care about applications-some care about rigorous theory).
 
  • #7
lugita15 said:
Yes, reading Spivak, Apostol, or Courant should provide you with the prerequisites that Lieb discusses. On the other hand, more introductory Calculus books like Stewart, Larson, and Anton wouldn't meet those prerequisites.
So would you think that a school with a calculus series using only the Larson books for calculus 1,2,3 would be a little lacking in the grand scheme of things?
 
  • #8
462chevelle said:
So would you think that a school with a calculus series using only the Larson books for calculus 1,2,3 would be a little lacking in the grand scheme of things?

It depends a lot on the major. For an engineering major, a book like Larson would be very suitable. A very theoretical approach is not the best idea, since engineering should focus on applications anyway. For math or physics majors however, I guess only Larson is something that is severely lacking.
 
  • #9
R136a1 said:
It depends a lot on the major. For an engineering major, a book like Larson would be very suitable. A very theoretical approach is not the best idea, since engineering should focus on applications anyway. For math or physics majors however, I guess only Larson is something that is severely lacking.

Yea so far engineering is my intended major. I have the Larson and McKeague trig book and I have to say I prefer the Larson over the McKeague. Were using the lial book (pearson) for trig next semester though. I don't know how that will compare. I like my Lial college algebra book. How does Larson calculus compare to Stewart?
 
  • #10
WannabeNewton said:
Why in the world would that make it a "good" calculus course? Uniform convergence is an extremely important concept and is easy to introduce in an honors calculus class; Spivak has an entire chapter on it. Not all calculus classes are made equal, some are made for the sole purpose of doing calculus correctly from freshman year itself instead of just hand-waving things and jumping into applications (to wit not all students care about applications-some care about rigorous theory).

I was wrong. A theory course can be good if it gives the student the ability to apply the theory rigorously. An applied course can be good if it gives the student the ability to solve problems.
 
  • #11
This is actually a very good example of how English is a tonal language (tone carries meaning) and we don't have punctuation signs to indictate the tone. I'll use an accent sign (á, í) to indicate the tone of what I wrote, how it would have sounded if spoken:

verty said:
I wás wrong. A theory course cán be good if it gives the student the ability to apply the theory rigorously. An applíed course can be good if it gives the student the ability to solve problems.
 

Related to Do all calculus books contain this?

1. What is the purpose of including this topic in calculus books?

The purpose of including this topic in calculus books is to provide a comprehensive understanding of calculus principles and techniques. This topic is commonly found in calculus books because it is a fundamental concept that is used in various applications of calculus.

2. Is this topic necessary for understanding calculus?

Yes, this topic is necessary for understanding calculus. It is a fundamental concept that is used in various applications of calculus, and without understanding this topic, it would be difficult to fully grasp the principles and techniques of calculus.

3. Are there any alternative resources for learning this topic besides calculus books?

Yes, there are alternative resources for learning this topic besides calculus books. Some examples include online tutorials, videos, and interactive websites that provide explanations and practice problems for this topic.

4. Can this topic be skipped while studying calculus?

No, this topic should not be skipped while studying calculus. As mentioned before, it is a fundamental concept that is used in various applications of calculus. Skipping this topic may result in a gap in understanding and hinder the ability to solve more complex calculus problems.

5. Are there any common misconceptions about this topic in calculus books?

Yes, there can be common misconceptions about this topic in calculus books. It is important to carefully read and understand the explanations provided in the book to avoid any misunderstandings. Additionally, seeking clarification from a teacher or tutor can help clear up any misconceptions.

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