20e-lecture17

20e-lecture17 - Lecture 17 Friday May 8th [email protected]

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Unformatted text preview: Lecture 17 - Friday May 8th [email protected] Key words: Multiple integrals Key concepts: Know how to evaluate multiple integrals 17.1 Multiple Integrals A rectangular parallelepiped in R n is a set of the form [ a 1 ,b 1 ] × [ a 2 ,b 2 ] × ··· × [ a n ,b n ] where a i < b i are real numbers. We refer to this also as a Box . If B is such a box, then B consists of all vectors ( x 1 ,x 2 ,...,x n ) such that a i ≤ x i ≤ b i for i = 1 , 2 ,...,n . We begin with the definition of the multiple integral of a function over a box. Let P i be a partition of [ a i ,b i ] for i = 1 , 2 ,...,n . A grid partition of a box B consists of all boxes of the form I 1 × I 2 ×···× I n where I 1 ,I 2 ,...,I n are intervals in partitions P 1 ,P 2 ,...,P n of [ a 1 ,b 1 ] , [ a 2 ,b 2 ] ,..., [ a n ,b n ]. An example of such a partition is shown below: A grid partition of a box B = [ a 1 ,b 1 ] × [ a 2 ,b 2 ] × [ a 3 ,b 3 ] P 1 = { [ a 1 ,b 1 ] } , P 2 = { [ a 2 ,c 2 ] , [ c 2 ,b 2 ] } and P 3 = { [ a 3 ,b 3 ] } For a grid partition P of a box B , let k P k be the volume of the largest box in P and let | I | denote the volume of a box I ∈ P . Now we can define multiple integrals....
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20e-lecture17 - Lecture 17 Friday May 8th [email protected]

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