2015 - Real Functions in Several Variables
SPE, scientific pdf ebooks is presenting this great serie of publications about multiple variables. It is composed of 11 books and is clearly a better study of all the topics about that. We hope you will find it very useful and handful.
Good lecture!
Leif Mejlbro
Real Functions in multiple Variables - Leif Mejlbro- Tome I - Point sets in Rn
Real Functions in multiple Variables - Leif Mejlbro - Tome II - Continuous Functions in Several Variables
Real Functions in multiple Variables - Leif Mejlbro - Tome III - Differentiable Functions in Several Variables
Real Functions in multiple Variables - Leif Mejlbro - Tome IV - Differentiable Curves and Differentiable Surfaces in Several Variables
Real Functions in multiple Variables - Leif Mejlbro - Tome V - The range of a function Extrema of a Function in Several Variables
Real Functions in multiple Variables - Leif Mejlbro - Tome VI - Antiderivatives and Plane lntegrals
Real Functions in multiple Variables - Leif Mejlbro - Tome VII - Space lntegrals
Real Functions in multiple Variables - Leif Mejlbro - Tome VIII - Line lntegrals and Surface lntegrals
Real Functions in multiple Variables - Leif Mejlbro - Tome X - Vector Fields I - Tangential Line lntegral and Gradient Fields GauB's Theorem
Real Functions in multiple Variables - Leif Mejlbro - Tome XI - Vector Fields Il - Stokes's Theorem Nabla Calculus
Introduction to volume I,
In this first volume of the series of books on Real Functions in Several Variables we start in Chapter 1 by giving a small theoretical introductory to what is more needed so as to started on this subject. We shall work in Euclidean space En , which in rectangular coordinates is similar to the vector space !Rn , also called the coordinate space. The difference may at the first glance seem very small, and yet this difference is qui te important. If we ever prove something in En , then this is done geometrically without any coordinate axes. This may be very strange to most younger readers, who have never learned Geometry in school using only ruler and compasses. For that reason I have in lack of better words called objects in En for "abstract" or "theoretical", though they are neither "abstract" nor "purely theoretical”
Introduction to volume II,
Continuous Functions in Several Variables
This is the second tome in this serie on Real Functions in Multiple Variables. We start in Chapter 5 with the necessary theoretical background. Here we assume that the theory of volume I is known by the reader.
We introduce maps and functions, including vector functions, and we give some guidelines on how to visualize such functions. This is not always an easy task, because we easily are forced to consider graphs lying in spaces of dimension 2c: 4, where very few human beings have a geometrical understanding of what is going on.
Then we introduce the continuous functions, starting with defining the basic concept of what we understand by taking a limit. We must apparently have some sense of "distance" in order to say that two points are close to each other. After, we use some topological concepts of norm and distance already studied in tome I.
Continuous functions are then defined as functions, for which "the image points are lying close together, whenever the points themselves are close to each other". We then make this more clear in the main text.
The 1st application of continuous functions is the introduction of continuous curves. The safest description of such curves, though it is not always necessary, is to use a parametric description of them. This is also done in MAPLE, and at the same time we get a sense of direction of a motion along the curve from an initial point to a final point.
Then we use the continuous curves to define ( curve) connected sets, which are the only connected sets we shall consider here. (There exist sets which are connected, but not curve connected; but they will not be of interest to us). One set “A” is (curve) connected, if any two points x and y in “A” can always be connected with a continuous curve, which lies entirely in “A”. If “A” C IRn is open, then any two points can always be connected by a continuous curve of a very special and convenient structure. The curve consists of concatenated line segments, where each of them is parallel to one of the axes in IRn . This characteristic will be very handful in the integration’s theory later on.
If furthermore, two curves connecting any two given points x and y E A can be transformed continuously into each other without leaving A during this transformation process, then A in some sense "does not contain holes", and A is called simply connected. As one would expect, simply connected sets have better properties than sets, which are only connected.
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