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Mathematical Methods of Engineering Analysis - Mathematical...

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Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis - Mathematical...

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Unformatted text preview: Mathematical Methods of Engineering Analysis Erhan C inlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . 2 Disjoint Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Products of Sets . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Functions and Sequences . . . . . . . . . . . . . . . . . . . . . . . . 4 Injections, Surjections, Bijections . . . . . . . . . . . . . . . 4 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 On the Real Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Positive and Negative . . . . . . . . . . . . . . . . . . . . . . 9 Increasing, Decreasing . . . . . . . . . . . . . . . . . . . . . 9 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Supremum and Infimum . . . . . . . . . . . . . . . . . . . . 9 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Convergence of Sequences . . . . . . . . . . . . . . . . . . . 11 5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Ratio Test, Root Test . . . . . . . . . . . . . . . . . . . . . . 16 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Absolute Convergence . . . . . . . . . . . . . . . . . . . . . 18 Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . 19 Metric Spaces 23 6 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Inner Product and Norm . . . . . . . . . . . . . . . . . . . . 23 Euclidean Distance . . . . . . . . . . . . . . . . . . . . . . . 24 7 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Distances from Points to Sets and from Sets to Sets . . . . . . 26 Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Interior, Closure, and Boundary . . . . . . . . . . . . . . . . 30 i Open Subsets of the Real Line . . . . . . . . . . . . . . . . . 31 9 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Convergence and Closed Sets . . . . . . . . . . . . . . . . . 36 10 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . 37 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . 38 11 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Compact Subspaces . . . . . . . . . . . . . . . . . . . . . . 40 Cluster Points, Convergence, Completeness . . . . . . . . . .Cluster Points, Convergence, Completeness ....
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