Understanding Calculus

[Psycho1990]

I want to understand the logic behind Calculus, can you refer me to some material available online? I have done calculus before (about 2 years ago), and I'm looking for a book on the foundations of calculus, not a book on practical calculus.

 

Also, what terms should I search for in search engines?

 

If you would like to talk a bit about the foundations of calculus, that would be really appreciated.

[DrRocket]

I want to understand the logic behind Calculus, can you refer me to some material available online? I have done calculus before (about 2 years ago), and I'm looking for a book on the foundations of calculus, not a book on practical calculus.

 

Also, what terms should I search for in search engines?

 

If you would like to talk a bit about the foundations of calculus, that would be really appreciated.

 

You are better off with a good book than something questionable on line

 

Any book with something like "Real Analysis" in the title should do.

 

Two very good ones are:

 

Elements of Real Analysis by Robert Bartle

 

Principles of Mathematical Analysis by Walter Rudin

 

Earlier editions are affordable at Amazon or Alibris and are fully adequate.

Edited February 15, 2012 by DrRocket

[Shadow]

I'm reading "A Working Excursion to Accompany Baby Rudin" (Evelyn M. Silvia, 1999), which is as far as I can tell is a "gentler" version of Rudin, and while I haven't gotten as far as calculus yet, it has been demanding enough for me to wonder if Rudin is the correct choice if one wants to understand the "logic" behind calculus. To understand calculus formally, sure, but if one seeks to develop his intuition, other books might be advisable. But then again, I have no sense whatsoever of what is out there, so maybe Rudin is a good choice after all.

[doG]

I'm looking for a book on the foundations of calculus, not a book on practical calculus.

I quite enjoyed Newton's Principia. There's probably an english version on Google books.

[ydoaPs]

I want to understand the logic behind Calculus, can you refer me to some material available online? I have done calculus before (about 2 years ago), and I'm looking for a book on the foundations of calculus, not a book on practical calculus.

 

Also, what terms should I search for in search engines?

 

If you would like to talk a bit about the foundations of calculus, that would be really appreciated.

 

Khan Academy has some good videos.

[D H]

I quite enjoyed Newton's Principia. There's probably an english version on Google books.

That is the last place you want to look to understand calculus. For one thing, there's a dirty little secret of science and math education. Unlike the humanities, we hardly ever look back to the original formulation of some concept. It took a couple of hundred years after Newton to get classical physics nailed down tight -- right about at the time physics was turned topsy-turvy by quantum mechanics and relativity. It took almost as long a time to get calculus nailed down tight.

 

There's an even bigger problem with looking to Newton's Principia to get a handle on calculus. You will find that is almost completely devoid of calculus! There isn't even very much algebra. What you will see instead is wall-o'-text page after wall-o'-text page of geometric reasoning.

[doG]

That is the last place you want to look to understand calculus.

I'll second that but it is still a good read and gives insight on the beginnings of calculus and what Newton was thinking when he began working on it. It is not written in the language of calculus as we know it now and the notation used is quite different so it can be difficult to understand.

[Xittenn]

If DrRocket's suggestion is not what you thought you meant a more comprehensive version of what you had is probably in order. I read Bittinger's Calculus and it was very relaxed, but I grabbed Briggs & Cochran's Calculus and it goes over things in much more detail. I'm not sure where Spivak's Calculus fits in between Briggs & Cochran's Calculus and Walter Rudin's Principles of Mathematical Analysis because I have only started reading Spivak's book and haven't picked up Rudin's book yet--I will be. . . .

 

I think the point I'm trying to get at is and I really hate to ask but, could you be a little more specific?

 

I prefer Briggs & Cochran over James Stewart

[doG]

Hahn's Basic Calculus: From Archimedes to Newton to its Role in Science also gives a good history and foundation of calculus.

[DrRocket]

I quite enjoyed Newton's Principia. There's probably an english version on Google books.

 

The Principia is a tour de force that is a monument to the genius of Newton.

 

It is also nearly unreadable and is probably the very worst place one could go for an understanding of calculus.

 

A great deal of understanding of the foundations of mathematics has emerged since calculus originated in the minds of Newton and Leibniz. Modern formulations and explanations are more elegant, more easily understood and far deeper than what was available in 1687.

[JohnStu]

Calculus is all about derivatives. Derivatives are used to calculating area of a curve, which you will learn much later on.

 

The core of calculus is really finding the anti-derivatives (to calculate area of that shape made by the equation). All the rules such as multiplication rules, division rules of calculus are short cuts.

 

 

[DrRocket]

Calculus is all about derivatives. Derivatives are used to calculating area of a curve, which you will learn much later on.

 

The core of calculus is really finding the anti-derivatives (to calculate area of that shape made by the equation). All the rules such as multiplication rules, division rules of calculus are short cuts.

 

 

 

This represents a rather superficial understanding of the derivative, the integral and calculus in general.

 

The gist of differential calculus is that one can learn a lot about a fairly general class of functions by studying their local behavior in terms of linear approximations. For instance when the best linear approximation at a point is a flat line neither increasing nor decreasing in either direction, the function itself has a local maxima or minima.

 

The gist of elementary integral calculus is the development of a concept called the Riemann integral which is obtained as a limit of a sum each term of which is the product of the length of an interval and a value of a function at a point on that interval. This is generalized and made much more clear in more advanced classes on measure and integration and the theory of the Lebesgue integral. The net result is concept of "area under the curve" and a means whereby that area can be computed exactly for functions that have "nice" expressions that allow exact values of the integral to be computed. (While calculus books are full of examples chosen to be "nice" there are in fact relatively few functions for which the anti-derivative is easily found in closed form.)

 

Integral and differential calculus are united by the Fundamental Theorem of Calculus which, for functions which are derivatives (not all functions are the derivative of anything) relates the integral over an interval to the values of the anti-derivative at the end points of the interval.

 

While (poor) elementary calculus classes emphasize symbol-pushing and "evaluation" of integrals and derivatives, that approach quite commonly results in little or no understanding of the subject of calculus itself, or the conceptual framework involved in its development and use.

 

To understand the basis for calculus one needs read a good book on real analysis -- any of the previously recommended books are adequate and there are certainly others.