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## Advanced calculus of several variables Devendra Kumar

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## MATH 535 - Multivariable Advanced Calculus

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Updating Results. If you wish to place a tax exempt order please contact us. In this chapter we discuss R n in some detail, as preparation for the development in subsequent chapters of the calculus of functions of an arbitrary number of variables. This generality will provide more clear-cut formulations of theoretical results, and is also of practical importance for applications.

For example, an economist may wish to study a problem in which the variables are the prices, production costs, and demands for a large number of different commodities; a physicist may study a problem in which the variables are the coordinates of a large number of different particles. Thus a "real-life" problem may lead to a high-dimensional mathematical model.

Fortunately, modern techniques of automatic computation render feasible the numerical solution of many high-dimensional problems, whose manual solution would require an inordinate amount of tedious computation.

## Advanced Calculus of Several Variables

As a set, R n is simply the collection of all ordered n -tuples of real numbers. That is,. The geometric representation of R3, obtained by identifying the triple x1, x2, x3 of numbers with that point in space whose coordinates with respect to three fixed, mutually perpendicular "coordinate axes" are x1, x2, x3 respectively, is familiar to the reader although we frequently write x, y, z instead of x1, x2, x3 in three dimensions. By analogy one can imagine a similar geometric representation of R n in terms of n mutually perpendicular coordinate axes in higher dimensions however there is a valid question as to what "perpendicular" means in this general context; we will deal with this in Section 3.

The elements of R n are frequently referred to as vectors. Thus a vector is simply an n -tuple of real numbers, and not a directed line segment, or equivalence class of them as sometimes defined in introductory texts. The set R n is endowed with two algebraic operations, called vector addition and scalar multiplication numbers are sometimes called scalars for emphasis.

Given a [member of] R , the scalar multiple a x is defined by.

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The familiar associative, commutative, and distributive laws for the real numbers imply the following basic properties of vector addition and scalar multiplication:. Here x, y, z are arbitrary vectors in R n , and a and b are real numbers.

V1—V8 are all immediate consequences of our definitions and the properties of R. Thus V1—V8 may be summarized by saying that R n is a vector space. For the most part, all vector spaces that we consider will be either Euclidean spaces, or subspaces of Euclidean spaces.