32Ahandout4

32Ahandout4 - MATH 32A/1 WINTER QUARTER 2009 HANDOUT 4...

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MATH 32A/1 WINTER QUARTER 2009 HANDOUT 4 CONTENTS HOMEWORKS #7, #8 INTEGRATION WITH TABLES OF INTEGRALS AND SOFTWARE INTEGRATION WITH MATHEMATICA PARTIAL DIFFERENTIAL EQUATIONS APPENDIX VECTOR ANALYSIS APPENDIX VECTOR FIELDS AND LINE INTEGRALS

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32A/1 - WINTER QUARTER 2009 HOMEWORK ASSIGNMENTS HOMEWORK #7 (Due Friday March 6) PARTIAL DIFFERENTIAL EQUATIONS , Problems 3.1, 3.3, 3.4, 3.5, 3.6, 4.2, 4.4, 4.8, 5.4, 6.1 (all p&p), 4.1, 4.3, 5.1, 5.2, 5.5, 5.6, 5.7, 6.2 (all ( M )) . HOMEWORK #8 (due Friday March 13) VECTOR ANALYSIS Problems 3.1 (p&p), 3.2, 3.4 (parametric plots with MATHEMATICA, calculations p&p) 3.5 ( M ) , 5.1, 5.2, 5.3 (all p&p) 5.5, 5.6, 5.7 (in all three, part ( a ) p&p, part ( b ) ( M ) , 6.1, 6.8, 6.9, 6.11 (all p&p). Problems 5.6 and 5.7 are done in VECTOR FIELDS AND LINE INTEGRALS. 1

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INTEGRATION WITH TABLES OF INTEGRALS AND SOFTWARE 1. Tables of integrals. One way to compute a defnite integral Z b a f ( x ) dx is to fnd an indefnite integral (or primitive, or antiderivative ), F ( x )= Z f ( x ) dx (that means a Function such that F 0 ( x f ( x )) and then use the Fundamental Theorem o± Calculus: Z b a f ( x ) dx = F ( b ) F ( a ) . There are many tricks For computing the indefnite integral F ( x )o Fa Function f ( x ) . One oF these tricks is change o± variables (also called substitution ). Other tricks are integration by parts and partial ±raction expansion (the latter suitable For rational Functions). ±inally, a number oF integrals can be computed by introducing new Functions (For instance, the inverses oF trigonometric and hyperbolic Functions). In this way we can do simple integrals like 1 Z xdx x 2 +1 = 1 2 log | x 2 | , Z xdx x 2 = 1 2 p x 2 , Z dx x 2 1 = 1 2 log | x 1 | | x | , Z (cos x ) 2 dx = x 2 + sin2 x 4 , (1 . 1) and many others. Most oF us eventually Forget integration tricks, thus iF we have to compute an indefnite integral we use a table o± integrals. Every calculus textbook contains a small table oF integrals (perhaps three or Four pages long), but they only contain a Few integrals. The standard table since the 1960s has been I. S. GRADSTEIN and I. M. RYZHIK, Tables o± integrals, sums, series and derivatives The original (1963) is in Russian. It has been translated into English (Academic Press, N. Y., 1979). It is more than 1100 pages long and contains more than 15,000 integrals (both defnite and indefnite). You can fnd this table in the EMS Library, together with many other large tables. 1 It is customary to write integrals adding an arbitrary constant C on the right side, to emphasize that the result oF doing an indefnite integration is unique only up to a constant. ±or instance, the frst integral (1.1) is usually written Z xdx x 2 = 1 2 log | x 2 | + C. We don’t bother with the constant. Neither do tables oF integrals or soFtware. 1

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A table of integrals is used more or less like a dictionary, but searches are not as simple. In a dictionary, you are absolutely sure to Fnd a word (if it’s there) using the alphabetical or lexicographical order (words ordered by Frst letter, by second letter if the Frst coincides, ...
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This note was uploaded on 06/03/2011 for the course MATH 32A 32A taught by Professor Moshchovakis during the Spring '10 term at UCLA.

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32Ahandout4 - MATH 32A/1 WINTER QUARTER 2009 HANDOUT 4...

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