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Vector analysis

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fullscreen: Vector analysis

Multivolume work

Persistent identifier:: 1048861678

Title:: VIIIth International Congress of Photogrammetry

Sub title:: Stockholm, Sweden, July 17 - 26, 1956

Type of content:: Konferenzschrift

Info on language/writing:: Beiträge teilweise deutsch

Year of publication:: 1957

Place of publication:: Stockholm

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 1048861678

Reihe:: International archives of photogrammetry (12)

Language:: English

Other Title:: Paralleltitel: VIIIe Congrès International de Photogrammétrie

Editor:: Fagerholm, P. O.

Editor:: Board of the VIIIth International Congress of Photogrammetry

Author:: International Congress of Photogrammetry, 8.; 1956; Stockholm

Document type:: Multivolume work

Volume

Persistent identifier:: 1048862747

Title:: Proceedings of the Congress

Sub title:: VIIIe Congrès International de Photogrammétrie

Scope:: 399 Seiten

Type of content:: Konferenzschrift

DOI:: 10.14463/GBV:1048862747

Year of publication:: 1957

Place of publication:: Stockholm

Publisher of the original:: [Verlag nicht ermittelbar]

Identifier (digital):: 1048862747

Illustration:: Illustrationen

Reihe:: International archives of photogrammetry (12,1)

Signature of the source:: ZS 312(12,1)

Language:: French

Other Title:: Paralleltitel: Compte rendu du Congrès

Usage licence:: Attribution 4.0 International (CC BY 4.0)

Contributor:: Ternryd, C. O.

Editor:: Fagerholm, P. O.

Editor:: Board of the VIIIth International Congress of Photogrammetry

Author:: International Congress of Photogrammetry, 8.; 1956; Stockholm

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2019

Document type:: Volume

Collection:: Earth sciences

Chapter

Title:: GENERAL ASSEMBLY - OPENING SESSION

Document type:: Multivolume work

Structure type:: Chapter

Contents

Table of contents

Vector analysis
Cover
ColorChart
Title page
Pale Bicentennial publications
Title page
PREFACE BY PROFESSOR GIBBS
GENERAL PREFACE
TABLE OF CONTENTS
Title page
CHAPTER I ADDITION AND SCALAR MULTIPLICATION
1.] - [3.] SCALARS AND VECTORS]
4.] [EQUAL AND NULL VECTORS]
5.] [THE POINT OF VIEW OF THE CHAPTER]
6.] - [7.]] Scalar Multiplication
8.] - [10.]] Addition and Subtraction.
11.] [SUBTRACTION]
12.] [LAWS GOVERNING THE FOREGOING OPERATIONS]
13.] - [16.]] Components of Vectors
17.] The Three Unit Vectors i, j, k.
18.] - [19.]] Applications
20.] - [22.]] Vector Relations independent of the Origin
23.] - [24.]] Centers of Gravity
25.] The Use of Vectors to denote Areas
SUMMARY OF CHAPTER I
EXERCISES ON CHAPTER I
CHAPTER II DIRECT AND SKEW PRODUCTS OF VECTORS
27.] - [28.]] Products of Two Vectors
29.] - [30.] THE DISTRIBUTIVE LAW AND APPLICATIONS]
31.] - [33.] THE SKEW, VECTOR, OR CROSS PRODUCT OF TWO VECTOERS]
34.] - [35.] THE DISTRIBUTIVE LAW AND APPLICATIONS]
36.] Products of More than Two Vectors
37.] - [38.] THE SCALAR TRIPLE PRODUCT A B X C or [A B C]]
39.] - [40.] THE VECTOR TRIPLE PRODUCT A X (B X C)]
41.] - [42.] PRODUCTS OF MORE THAN THREE VECTORS WITH APPLICATIONS TO TRIGONOMETRY]
43.] - [45.]] Reciprocal Systems of Three Vectors. Solution of Equations
46.] - [47.] SOLUTION OF SCALAR AND VECTOR EQUATIONS LINEAR IN AN UNKNOWN VECTOR]
48.] - [50.]] SUNDRY APPLICATIONS OF PRODUCTS Applications to Mechanics
51.] [KINEMATICS OF A RIGID BODY]
52.] [CONDITIONS FOR EQUILIBRIUM OF A RIGID BODY]
53.] Applications to Geometry
54.] [PROBLEMS IN GEOMETRY. PLANAR COÖRDINATES]
SUMMARY OF CHAPTER II
EXERCISES ON CHAPTER II
CHAPTER III THE DIFFERENTIAL CALCULUS OF VECTORS
55.] - [56.]] Differentiation of Functions of One Scalar Variable
57.] [CURVATURE AND TORSION OF GAUCHE CURVES]
58.] - [59.]] Kinematics
60.] [THE INSTANTANEOUS AXIS OF ROTATION]
61.] [INTEGRATI0N WITH APPLICATIONS TO KINEMATICS]
62.] Scalar Functions of Position in Space. The Operator [...]
63.] - [67.] THE VECTOR DIFFERENTIATING OPERATOR [...]]
68.] [THE SCALAR OPERATOR A [...]]
69.] Vector Functions of Position in Space
70.] [THE DIVERGENCE [...] AND THE CURL [...] X]
71.] [INTERPRETATION OF THE DIVERGENCE [...]]
72.] [INTERPRETATION OF THE CURL [...] X]
73.] [LAWS OF OPERATION OF [...], [...], [...] X]
74.] - [76.] THE PARTIAL APPLICATION OF [...]. EXPANSION OF a VECTOR FUNCTION ANALOGOUS TO TAYLOR'S THEOREM. APPLICATION TO HYDROMECHANICS]
77.] [THE DIFFERENTIATING OPERATORS OF THE SECOND ORDER]
78.] [GEOMETRIC INTERPRETATION OF LAPLACE'S OPERATOR [...] [...] AS THE DISPERSION]
SUMMARY OF CHAPTER III
EXERCISES ON CHAPTER III
CHAPTER IV THE INTEGRAL CALCULUS OF VECTORS
79.] - [80.] LINE INTEGRALS OF VECTOR FUNCTIONS WITH APPLICATIONS]
81.] [GAUSS'S THEOREM]
82.] [STOKES'S THEOREM]
83.] [CONVERSE OF STOKES'S THEOREM WITH APPLICATIONS]
84.] [TRANSFORMATIONS OF LINE, SURFACE, AND VOLUME INTEGRALS. GREEN'S THEOREM]
85.] [REMARKS ON MULTIPLE-VALUED FUNCTIONS]
86.] - [87.]] The Integrating Operators. The Potential
88.] [COMMUTATIVE PROPERTY OF POT AND [...]]
89.] [REMARKS UPON THE FOREGOING]
90.] [THE INTEGRATING OPERATORS "NEW," "LAP," "MAX"]
91.] [RELATIONS BETWEEN THE INTEGRATING AND DIFFERENTIATING OPERATORS]
92.] Poisson's Equation
93.] - [94.] SOLENOIDAL AND IRROTATIONAL PARTS OF A VECTOR FUNCTION. CERTAIN OPERATORS AND THEIR INVERSE]
95.] [MUTUAL POTENTIALS, NEWTONIANS, LAPLACIANS, AND MAXWELLIANS]
96.] [CERTAIN BOUNDARY VALUE THEOREMS]
SUMMARY OF CHAPTER IV
EXERCISES ON CHAPTER IV
CHAPTER V LINEAR VECTOR FUNCTIONS
97.] - [98.] LINEAR VECTOR FUNCTIONS DEFINED]
99.] [DYADICS DEFINED]
100.] [ANY LINEAR VECTOR FUNCTION MAY BE REPRESENTED BY A DYADIC. PROPERTIES OF DYADICS]
101.] The Nonion Form of a Dyadic
102.] [THE DYAD OR INDETERMINATE PRODUCT OF TWO VECTORS IS THE MOST GENERAL. FUNCTIONAL PROPERTY OF THE SCALAR AND VECTOR PRODUCTS]
103.] - [104.]] Products of Dyadics
105.] - [107.]] Degrees of Nullity of Dyadics
108.] The Idemfactor; Reciprocals and Conjugates of Dyadics
109.] - [110.] RECIPROCAL DYADICS. POWERS AND ROOTS OF DYADICS]
111.] [CONJUGATE DYADICS. SELF-CONJUGATE AND ANTI-SELF-CONJUGATE PARTS OF A DYADIC]
112.] - [114.]] Anti-self-conjugate Dyadics. The Vector Product
115.] - [116.]] Reduction of Dyadics to Normal Form
117.] Double Multiplication
118.] - [119.] THE SECOND AND THIRD OF A DYADICS]
120.] [CONDITIONS FOR DIFFERENT DEGREES OF NULLITY]
121.] Nonion Form. Determinants. Invariants of a Dyadic
122.] [INVARIANTS OF A DYADIC. THE HAMILTON-CAYLEY EQUATION]
SUMMARY OF CHAPTER V
EXERCISES ON CHAPTER V
CHAPTER VI ROTATIONS AND STRAINS
CHAPTER VII MISCELLANEOUS APPLICATIONS
136.] - [142.]] Quadric Surfaces
143.] - [146.]] The Propagation of Light in Crystals
147.] - [148.]] Variable Dyadics. The Differential and Integral Calculus
149.] - [157.]] The Curvature of Surfaces
158.] - [162.]] Harmonic Vibrations and Bivectors
Cover

Full text

THE INTEGRAL CALCULUS OF VECTORS 255 
Exercises ox Chapter IV 
l. 1 If V is a scalar function of position in space the line 
integral 
L vd * 
is a vector quantity. Show that 
V d r = — J J s W x d a = J ' f da. xW. 
That is ; the line integral of a scalar function around a 
closed curve is equal to the skew surface integral of the deriv 
ative of the function taken over any surface spanned into 
the contour of the curve. Show further that if V is constant 
the integral around any closed curve is zero and conversely 
if the integral around any closed curve is zero the function V 
is constant. 
Hint; Instead of treating the integral as it stands multiply 
it (with a dot) by an arbitrary constant unit vector and thus 
reduce it to the line integral of a vector function. 
2. If W is a vector function the line integral 
H = JjN x dr 
is a vector quantity. It may be called the skew line integral 
of the function W. If c is any constant vector, show that if 
the integral be taken around a closed curve 
H. c = /J s (c V*W-c*VW)^a = c* J Wxdr, 
1 The first four exercises are taken from Föppl’s Einführung in die Max- 
well’sche Theorie der Electricität where they are worked out.

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Wilson, Edwin Bidwell, and J. .Willard Gibbs. Vector Analysis. Scribner u.a., 1902.

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