Vector Analysis ：A Physicist's Guide to the Mathematics of Fields in Three Dimensions

Publication subTitle ：A Physicist's Guide to the Mathematics of Fields in Three Dimensions

Author： N. Kemmer

Publisher： Cambridge University Press‎

Publication year： 1977

E-ISBN: 9780511865091

P-ISBN(Paperback): 9780521290647

Subject： O183.1 vector analysis

Keyword：数学

Language： ENG

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Vector Analysis

Description

Vector analysis provides the language that is needed for a precise quantitative statement of the general laws and relationships governing such branches of physics as electromagnetism and fluid dynamics. The account of the subject is aimed principally at physicists but the presentation is equally appropriate for engineers. The justification for adding to the available textbooks on vector analysis stems from Professor Kemmer's novel presentation of the subject developed through many years of teaching, and in relating the mathematics to physical models. While maintaining mathematical precision, the methodology of presentation relies greatly on the visual, geometric aspects of the subject and is supported throughout the text by many beautiful illustrations that are more than just schematic. A unification of the whole body of results developed in the book - from the simple ideas of differentiation and integration of vector fields to the theory of orthogonal curvilinear coordinates and to the treatment of time-dependent integrals over fields - is achieved by the introduction from the outset of a method of general parametrisation of curves and surfaces.

Chapter

1 Summary of vector algebra

1 Addition and multiplication of vectors

2 The cartesian components of a vector

3 The quotient theorem

Exercises A

2 The geometrical background to vector analysis

1 The point

2 The curve

3 The surface

4 The three-dimensional region of space

3 Metric properties of Euclidean space

1 The length of a curve and the total mass on it

2 The area of a surface and the total mass on it

3 The volume of, and mass in a three-dimensionalr egion

Exercises B

4 Scalar and vector fields

1 Scalar fields

2 Vector fields

Exercises C

5 Spatial integrals of fields

1 The work integral

2 The flux integral

3 The quantity integral

6 Further spatial integrals

1 Vector integrals

2 More general integrals over fields

Exercises D

7 Differentiation of fields. Part 1: the gradient

1 The operator V

2 The gradient of a scalar field

3 The field line picture of a gradient field

4 An alternative definition of grad 0

5 The fundamental property of a gradient field

8 Differentiation of fields. Part 2: the curl

1 Definition of the curl of a vector field

2 Evaluation of curl/. Stokes' theorem

3 The fundamental property of a curl field

4 The quantitative picture of a solenoidal field

9 Differentiation of fields. Part 3: the divergence

1 Definition of the divergence of a vector field

2 Evaluation of div/. The divergence theorem

3 A survey of results

10 Generalisation of the three principal theoremsand some remarks on notation

1 General integral theorems

2 A general notation

3 Generalisation of ranges of integration

Exercises E

11 Boundary behaviour of fields

1 Surface discontinuities

2 Singularities at lines and points

3 A special discontinuity: the double layer

4 Discontinuities of scalar fields

Exercises F

12 Differentiation and integration of productsof fields

1 Differentiation

2 Integration by parts

13 Second derivatives of vector fields; elements ofpotential theory

1 The Laplace operator. Poisson's and Laplace's 145equations

2 The 'Newtonian' solution of Poisson's equation — 149a digression

Exercises G

14 Orthogonal curvilinear coordinates

1 The basic relations

2 Definition of A for scalars and for vectors. An alternative approach

3 Fields with rotational symmetry. The Stokes' stream function

4 Solenoidal fields in two dimensions — a digression

Exercises H

15 Time-dependent fields

1 The equation of continuity

2 Time-dependent relations involving the velocity field v(r)

Exercises I

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