Mathematical Physics Vol 1

First page Table of contents Next page Last page

Cover	1
TOC	5
Preface	11
Part I — Vector algebra and analysis	13
1 Vector algebra	15
1.1 Introduction - On scalars, vectors and tensors	15
1.2 Coordinate system	15
1.3 Vector algebra	17
1.4 Operations on vectors	18
1.5 Algebraic model of linear vector space	34
1.6 Gram-Schmidt orthogonalization procedure	37
2 Vector analysis	41
2.1 Vector analysis	41
2.2 Integration	49
3 Examples	57
3.1 Vector algebra	57
3.2 Vector analysis	71
Part II — Field theory	79
4 Field theory	81
4.1 Scalar field	81
4.2 Vector field	91
4.3 Examples of some fields	105
4.4 Generalized coordinates	110
4.5 Special coordinate systems	116
4.6 Examples	121
Part III — Solving differential equations	223
5 Series Solutions of Differential Equations. Special functions	225
5.1 Functional series. Power series	225
5.2 Series Solutions of Differential Equations	229
5.3 Legendre: equation, function, polynomial	230
5.4 Bessel equation. Bessel functions	233
5.5 Some other special functions	245
5.6 Special functions that are not a result of the Frobenius method	246
5.7 Mittag-Leffler functions	257
5.8 Elliptic integrals	259
5.9 Orthogonal and normalized functions	265
5.10 Examples	272
Part IV — Trigonometric Fourier series. Fourier integral	307
6 Trigonometric Fourier series. Fourier integral	309
6.1 Periodic functions	309
6.2 The fundamental convergence theorem for Fourier series	313
6.3 Examples	326
Part V — PDE	333
7 Partial differential equations	335
7.1 Definitions and notation	336
7.2 Formation of partial differential equations	337
7.3 Linear and quasilinear first order PDE	343
7.4 Linear second order PDE	357
7.5 A formal procedure for solving LDE	365
7.6 The variable separation method	366
7.7 Green formulas	372
7.8 Examples	384
Part VI — Fractional Calculus	435
8 Introduction to the Fractional Calculus	437
8.1 Brief History of Fractional Calculus	437
8.2 Basic Definitions of Fractional Order Differintegrals	443
8.3 Basic Properties of Fractional Order Differintegrals	446
8.4 Some other types of fractional derivatives	448
Appendices	455
Appendix A Fractional Calculus: A Survey of Useful Formulas	457
A.1 Introduction	457
A.2 Notation and Special Functions	457
A.3 Fractional Derivatives and Integrals	461
A.4 Analytical Expressions of Some Fractional Derivatives	465
A.5 Laplace and Fourier Transforms	466
A.6 Systems of Fractional Equations	469
A.7 Transfer Functions	470
A.8 An Introduction to Fractional Vector Operators	472
Bibliography	475
Index	481

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