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chap09 - integration on Rn

chap09 - integration on Rn - Integration on Rn 1 Chapter 9...

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Integration on R n 1 Chapter 9 Integration on R n This chapter represents quite a shift in our thinking. It will appear for a while that we have completely strayed from calculus. In fact, we will not be able to “calculate” anything at all until we get the fundamental theorem of calculus in Section E and the Fubini theorem in Section F. At that time we’ll have at our disposal the tremendous tools of differentiation and integration and we shall be able to perform prodigious feats. In fact, the rest of the book has this interplay of “differential calculus” and “integral calculus” as the underlying theme. The climax will come in Chapter 12 when it all comes together with our tremendous knowledge of manifolds. Exciting things are ahead, and it won’t take long! In the preceding chapter we have gained some familiarity with the concept of n -dimensional volume. All our examples, however, were restricted to the simple shapes of n -dimensional parallelograms. We did not discuss volume in anything like a systematic way; instead we chose the elementary “definition” of the volume of a parallelogram as base times altitude. What we need to do now is provide a systematic way to think about volume, so that our “definition” of Chapter 8 actually becomes a theorem. We shall in fact accomplish much more than a good discussion of volume. We shall define the concept of integration on R n . Our goal is to analyze a function D f -→ R , where D is a bounded subset of R n and f is bounded. We shall try to define the integral of f over D , a number which we shall denote Z D f. In the language of one-variable calculus, we are going to be defining a so-called definite integral. We shall never even think about extending the idea of indefinite integral to R n . We pause to explain our choice of notation. Eventually we shall perform lots of calculations of integrals, and we shall then use all sorts of notations, such as Z D f ( x ) dx, Z D f ( x 1 , . . . , x n ) dx 1 . . . dx n , Z · · · Z D f ( x 1 , . . . , x n ) dx 1 . . . dx n , and more. But during the time we are developing the theory of integration, it would be inconvenient to use an abundance of notation. Therefore we have chosen the briefest notation
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2 Chapter 9 that includes the function f and the set D and an integration symbol, Z D f. Analogy: in a beginning study of Latin the verb amare (to love) is used as a model for learning the endings of that declension, rather than some long word such as postulare. And in Greek a standard model verb is λυω (to loose). In the special case f = 1, we obtain the expression for n -dimensional volume, vol( D ) = vol n ( D ) = Z D 1 . A. The idea of Riemann sums Bernhard Riemann was not the first to define the concept of a definite integral. However, he was the first to apply a definition of integration to any function, without first specifying what properties the function has. He then singled out those functions to which the integration process assigned a well defined number. These functions are called “integrable.” He then derived a necessary and sufficient condition for a function to be integrable. This is contained
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