Mathematical analysis for engineers

This book follows an advanced course in analysis (vector analysis, complex analysis and Fourier analysis) for engineering students, but can also be useful, as a complement to a more theoretical course, to mathematics and physics students.The first three parts of the book represent the theoretical aspect and are independent of each other. The fourth part gives detailed solutions to all exercises that are proposed in the first three parts.ForewordForeword (71 KB)

Foreword	p. ix
I Vector analysis	p. 1
1 Differential operators of mathematical physics	p. 3
1.1 Definitions and theoretical results	p. 3
1.2 Examples	p. 5
1.3 Exercises	p. 7
2 Line integrals	p. 9
2.1 Definitions and theoretical results	p. 9
2.2 Examples	p. 10
2.3 Exercises	p. 11
3 Gradient vector fields	p. 13
3.1 Definitions and theoretical results	p. 13
3.2 Examples	p. 14
3.3 Exercises	p. 18
4 Green theorem	p. 21
4.1 Definitions and theoretical results	p. 21
4.2 Examples	p. 23
4.3 Exercises	p. 24
5 Surface integrals	p. 27
5.1 Definitions and theoretical results	p. 27
5.2 Examples	p. 29
5.3 Exercises	p. 31
6 Divergence theorem	p. 33
6.1 Definitions and theoretical results	p. 33
6.2 Examples	p. 34
6.3 Exercises	p. 37
7 Stokes theorem	p. 39
7.1 Definitions and theoretical results	p. 39
7.2 Examples	p. 41
7.3 Exercises	p. 43
8 Appendix	p. 45
8.1 Some notations and notions of topology	p. 45
8.2 Some notations for functional spaces	p. 49
8.3 Curves	p. 50
8.4 Surfaces	p. 52
8.5 Change of variables	p. 64
II Complex analysis	p. 67
9 Holomorphic functions and Cauchy-Riemann equations	p. 69
9.1 Definitions and theoretical results	p. 69
9.2 Examples	p. 71
9.3 Exercises	p. 74
10 Complex integration	p. 77
10.1 Definitions and theoretical results	p. 77
10.2 Examples	p. 78
10.3 Exercises	p. 79
11 Laurent series	p. 83
11.1 Definitions and theoretical results	p. 83
11.2 Examples	p. 86
11.3 Exercises	p. 88
12 Residue theorem and applications	p. 91
12.1 Part I	p. 91
12.1.1 Definitions and theoretical results	p. 91
12.1.2 Examples	p. 92
12.2 Part II: Evaluation of real integrals	p. 93
12.3 Exercises	p. 97
13 Conformal mapping	p. 101
13.1 Definitions and theoretical results	p. 101
13.2 Examples	p. 102
13.3 Exercises	p. 104
III Fourier analysis	p. 107
14 Fourier series	p. 109
14.1 Definitions and theoretical results	p. 109
14.2 Examples	p. 113
14.3 Exercises	p. 116
15 Fourier transform	p. 121
15.1 Definitions and theoretical results	p. 121
15.2 Examples	p. 123
15.3 Exercises	p. 125
16 Laplace transform	p. 127
16.1 Definitions and theoretical results	p. 127
16.2 Examples	p. 129
16.3 Exercises	p. 132
17 Applications to ordinary differential equations	p. 135
17.1 Cauchy problem	p. 135
17.2 Sturm-Liouville problem	p. 137
17.3 Some other examples solved by Fourier analysis	p. 140
17.4 Exercises	p. 143
18 Applications to partial differential equations	p. 145
18.1 Heat equation	p. 145
18.2 Wave equation	p. 150
18.3 Laplace equation in a rectangle	p. 152
18.4 Laplace equation in a disk	p. 155
18.5 Laplace equation in a simply connected domain	p. 159
18.6 Exercises	p. 162
IV Solutions to the exercises	p. 167
1 Differential operators of mathematical physics	p. 169
2 Line integrals	p. 177
3 Gradient vector fields	p. 181
4 Green theorem	p. 189
5 Surface integrals	p. 199
6 Divergence theorem	p. 203
7 Stokes theorem	p. 219
9 Holomorphic functions and Cauchy-Riemann equations	p. 233
10 Complex integration	p. 239
11 Laurent series	p. 247
12 Residue theorem and applications	p. 263
13 Conformal mapping	p. 277
14 Fourier series	p. 291
15 Fourier transform	p. 303
16 Laplace transform	p. 309
17 Applications to ordinary differential equations	p. 317
18 Applications to partial differential equations	p. 331
Bibliography	p. 353
Table of Fourier Transform	p. 355
Table of Laplace Transform	p. 356
Index	p. 357

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