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07Ma1cAnalyticalNotesChap1

# 07Ma1cAnalyticalNotesChap1 - Notes on Vector Calculus...

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Notes on Vector Calculus Dinakar Ramakrishnan March 26, 2007

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Chapter 1 Subsets of Euclidean space, vector fields, and continuity Introduction The aims of this course are the following: (i) Extend the main results of one-variable Calculus to higher dimensions (ii) Explore new phenomena which are non-existent in the one-dimensional case Regarding the first aim, a basic step will be to define notions of continuity and differ- entiability in higher dimensions. These are not as intuitive as in the one-dimensional case. For example, given a function f : R R , we can say that f is continuous if one can draw its graph Γ f without lifting the pen (resp. chalk) off the paper (resp. blackboard). For any n 1, we can still define the graph of a function (here called a scalar field ) f : R n R to be Γ( f ) := { ( x, y ) R n × R | y = f ( x ) } , where x denotes the vector ( x 1 , . . . , x n ) in R n . Since R n × R is just R n +1 , we can think of Γ( f ) as a subset of the ( n + 1)-dimensional space. But the graph will be n -dimensional, which is hard (for non-constant f )to form a picture of, except possibly for n = 2; even then it cannot be drawn on a plane like a blackboard or a sheet of paper. So one needs to define basic notions such as continuity by a more formal method. It will be beneficial to think of a lot of examples in dimension 2, where one has some chance of forming a mental picture. 1
Integration is also subtle. One is able to integrate nice functions f on closed rectangular boxes R , which naturally generalize the closed interval [ a, b ] in R , and when there is spherical symmetry, also over closed balls in R n . Here f being nice means that f is bounded on R and continuous outside a negligible set . But it is problematic to define integrals of even continuous functions over arbitrary subsets Y of R n , even when they are bounded , i.e., can be enclosed in a rectangular box. However, when Y is compact, i.e., closed and bounded, one can integrate continuous functions f over it, at least when f vanishes on the boundary of Y . The second aim is more subtle than the first. Already in the plane, one is interested in line integrals , i.e., integrals over (nice) curves C , of vector fields , i.e., vectors of scalar fields, and we will be interested in knowing when the integrals depend only on the beginning and end points of the curve. This leads to the notion of conservative fields , which is very important also for various other subjects like Physics and Electrical Engineering. Somehow the point here is to not blindly compute such integrals, but to exploit the (beautiful) geometry of the situation. This chapter is concerned with defining the basic structures which will be brought to bear in the succeeding chapters. We start by reviewing the real numbers . 1.1 Construction and properties of real numbers This section intends to give some brief background for some of the basic facts about real numbers which we will use in this chapter and in some later ones.

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