By Sheldon Axler
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This e-book is dedicated to an in depth improvement of the divergence theorem. The framework is that of Lebesgue integration — no generalized Riemann integrals of Henstock–Kurzweil style are concerned. partly I the divergence theorem is demonstrated by way of a combinatorial argument concerning dyadic cubes. purely ordinary homes of the Lebesgue indispensable and Hausdorff measures are used.
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2 · (−3) = −6 1 2 1 −2 • The product of a positive number and a negative number is negative. is positive • The multiplicative inverse of a positive number is positive. is negative • The multiplicative inverse of a negative number is negative. 3 Inequalities, Intervals, and Absolute Value 25 Lesser and Greater We say that a number a is less than a number b, written a < b, if a is left of b on the real line. Equivalently, a < b if and only if b − a is positive. In particular, b is positive if and only if 0 < b.
This example shows the importance of paying careful attention to the order of operations, even when using a calculator. 9 and then on most calculators you need to press the enter button or the = button. 09 as the result. 72. 9. 9 should mean. 09. Make sure you know how your calculator interprets the kind of input shown in the example above, or you may get answers from your calculator that do not reflect the problem you have in mind. 2 Algebra of the Real Numbers 17 Many calculators have parentheses buttons, which are often needed to control the order in which operations are performed.
Suppose b = 0 and d = 0. Explain why a c ad − bc − = . b d bd 60. Suppose b = 0 and d = 0. Explain why a c = if and only if ad = bc. b d worked-out solutions to Odd-numbered Exercises Do not read these worked-out solutions before first attempting to do the exercises yourself. Otherwise you may merely mimic the techniques shown here without understanding the ideas. For Exercises 1–4, determine how many different values can arise by inserting one pair of parentheses into the given expression. 1. 19 − 12 − 8 − 2 solution Here are the possibilities: Best way to learn: Carefully read the section of the textbook, then do all the odd-numbered exercises (even if they have not been assigned) and check your answers here.