Mathematical Physics Vol 1

Contents

Vector algebra and analysis

I

11

Chapter 1 Vector algebra

13 1.1 Introduction - On scalars, vectors and tensors . . . . . . . . . . . . . . . . . . . 1. 3 1.2 Coordinatesystem .................................1.3 1.3 Vectoralgebra....................................1.5 1.4 Operationsonvectors................................1.6 1.4.1 Additionofvectors.............................1.6 1.4.2 Multiplication of a vector by a real number (scalar) . . . . . . . . . . . . 2. 0 1.4.3 Projectiononanaxisandonaplane . . . . . . . . . . . . . . . . . . . . 2.0 1.4.4 Scalar (dot or internal) product of two vectors . . . . . . . . . . . . . . . 2. 2 1.4.5 Vector (cross) product of two vectors . . . . . . . . . . . . . . . . . . . 2. 4 1.4.6 Reciprocal (conjugate) system of vectors . . . . . . . . . . . . . . . . . 2. 6 1.4.7 Linear dependence of vectors. Dimension of a space . . . . . . . . . . . 2. 7 1.5 Algebraicmodeloflinearvectorspace . . . . . . . . . . . . . . . . . . . . . . . 3.2 1.6 Gram-Schmidt orthogonalization procedure . . . . . . . . . . . . . . . . . . . . 3. 5 39 2.1 Vectoranalysis ...................................3.9 2.1.1 Vectorfunction ...............................3.9 2.1.2 Hodographofavectorfunction . . . . . . . . . . . . . . . . . . . . . . 4.0 2.1.3 Limitprocesses.Continuity . . . . . . . . . . . . . . . . . . . . . . . . 4.0 2.1.4 Derivative of a vector function of one scalar variable . . . . . . . . . . . .41 2.1.5 Propertiesofthederivative . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 2.1.6 Differential of the vector function . . . . . . . . . . . . . . . . . . . . . 4. 3 2.1.7 Higher order derivatives and differentials . . . . . . . . . . . . . . . . . 4. 4 2.1.8 Partial derivative of a vector function of several independent variables . . 4. 4 2.1.9 Differential of a vector function of n scalarvariables . . . . . . . . . . . 4.4 2.2 Integration......................................4.7 2.2.1 Indefinite integral of a vector function . . . . . . . . . . . . . . . . . . . 4. 7 2.2.2 Definiteintegral ..............................4.7 2.2.3 The line integral of a vector function . . . . . . . . . . . . . . . . . . . . 4. 8 2.2.4 Surfaceintegral................................51 55 3.1 Vectoralgebra....................................5.5 3.2 Vectoranalysis ...................................6.9

Chapter 2 Vector analysis

Chapter 3 Examples