# "Special" Calculus and Analysis book

• paalfis
In summary, the speaker is looking for a recommendation for a book on Analysis II, which covers topics such as topology, functions, differential calculus, and multiple variable functions. They mention that the course focuses on both theoretical and practical aspects, similar to a honors course in math and physics. They also mention needing a book that covers double and triple integrals. The speaker recommends the book "Edwards and Fleming" for this subject.
paalfis
I used 'special' for the title, because the book I need is for a particular course that has a very good reputation, but it is quite unique in the way its go through mathematics, it does a very deep analysis of theorems and demonstrations, but also very practical; it would be the equivalent in my country to a honor course for majoring in Math and Physics in America. I will denote all the topics that are covered in the course, and I hope you can recommend the book you think is the best, for each or all of the topics. (I am hoping I can find something like the book from Klepnner and Kolenkow is to Mechanics, but for this topics):

Before, I must said that the final goal of this course (called Analysis II) is to be able to show proves and demonstrations for particular cases and examples of all of the topics named below.

The topics:
1-Topology in R and Rn : Completeness of R, Distance, open and closed disks disks. Interior points. Interior of a set. Open sets, closed sets, bounded sets. Limits in Rn
2-Functions of Rn in Rk ,( n, k = 1, 2, ... ): Graphing. Domain of definition. Curves and level surfaces. Limit functions in Rn Rk. Limit along lines and curves. Continuous functions. Composition of continuous functions. Properties of continuous functions.
3-Differential calculus with multiple variables:Partial derivatives. Linear approximation. Differential of a function. Jacobian matrix. Tangent plane to the graph of a function. Chain Rule. General theorems of the inverse function and implicit function. Scalar product in Rn. Equation of plane orthogonal to a vector. Directional derivatives. Gradient. Relationship between the level surfaces and the direction of maximum growth. Plane tangent to a level surface. Mean value theorem in several variables. Higher derivatives. Polynomial approximation order 2. Hessian matrix (or Hessian) of a function.
4-Extremes of multiple variable functions : Critic points and extremes of a function. Quadratic forms, associated matrix. Analysis of critical points in several variables from the Hessian: maximum, minimum, saddle points. Ligated ends: ends of a function over a set given by an equation G = 0 . Condition for a point to be critical . Lagrange multipliers.
5-Double and Triple Integers: Review: definite integral, Riemann sums, Fundamental Theorem of Calculus, Barrow rule. Improper integrals: definition, properties, convergence criteria, absolute convergence. Application: convergence of series. The double integral over rectangles. The double integral over more general regions. Changing the order of integration: Fubini Theorem. The triple integral. The Change of variables theorem. Applications of double and triple integrals.

Double and Triple Integers, huh? That must be a 'special calculus'.

SteamKing said:
Double and Triple Integers, huh? That must be a 'special calculus'.

Steamking, I hope that is not sarcasm, it would be kind of rude from your part. By special I meant that the focus of the course require one to have the formality of a math major and the "practical view" of a physicist or even an engineer.

It was just a typing mistake from google translate.. Of course I meant integrals

## 1. What is "Special" Calculus and Analysis?

"Special" Calculus and Analysis is a branch of mathematics that focuses on advanced techniques for solving problems in calculus and analysis. It goes beyond the basics of calculus and delves into more complex topics such as multivariable calculus, differential equations, and vector calculus.

## 2. How is "Special" Calculus and Analysis different from regular Calculus?

"Special" Calculus and Analysis covers more advanced topics and techniques that are not typically covered in a basic calculus course. It also places a greater emphasis on the theoretical foundations of calculus and how it relates to other branches of mathematics.

## 3. Who should use a "Special" Calculus and Analysis book?

This type of book is typically used by students who have a strong foundation in calculus and are looking to further their understanding of the subject. It can also be useful for professionals in fields such as physics, engineering, and economics who use advanced calculus techniques in their work.

## 4. What are some common applications of "Special" Calculus and Analysis?

"Special" Calculus and Analysis techniques are used in various fields, including physics, engineering, economics, and computer science. They can be used to model and solve complex problems involving rates of change, optimization, and differential equations.

## 5. Are there any prerequisites for using a "Special" Calculus and Analysis book?

A strong understanding of basic calculus is a prerequisite for using a "Special" Calculus and Analysis book. It is also helpful to have knowledge of linear algebra and differential equations, as these topics are often covered in this type of book.

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