APPLICATIONS OF TENSOR ANALYSIS
(formerly titled: Applications of the Absolute Differential Calculus)
by A. J. McCONNELL
Dover Publications, Inc., Neiv York
CONTENTS.
PART I. — ALGEBRAIC PRELIMINARIES/
CHAPTER I.
NOTATION AND DEFINITIONS.
1. The indioial notation * • 1 2. The summation convention - - - 3 3. Addition, multiplication, and contraction of systems . . . 5 4. Symmetric and skew-symmetric systems 6 5. The skew-symmetric three-systems and the Kroneoker deltas - 7
DETERMINANTS.
6. The determinant formed by a double system oj - • • • 10 7. The cofactors of the elements in a determinant - - 12 8. Linear equations • - . . - - 14 9. Corresponding formula for the system amn - - - - 15 10. Positive definite quadratic forma. The determinantal equation 16
CHAPTER H.
TENSOR ANALYSIS.
1. L i n e a r t r a n s f o r m a t i o n s . . . . . . . . . . IQ 2 . I n v a r i a n t s , c o n t r a v a r i a n t a n d c o v a r i a n t v e c t o r s . . . . 2 0 3 . T e n s o r s of a n y o r d e r - - - '.- - - - 2 2 4 . A d d i t i o n , m u l t i p l i c a t i o n a n d c o n t r a c t i o n of t e n s o r s - 2 4 6. T h e q u o t i e n t l a w of t e n s o r s - - » - 2 6 6. R e l a t i v e o r w e i g h t e d t e n s o r s - - • - 2 8 7 . G e n e r a l f u n c t i o n a l t r a n s f o r m a t i o n s - - - 3 0 8. T e n s o r s w i t h r e s p e c t t o t h e g e n e r a l f u n c t i o n a l t r a n s f o r m a t i o n r - 3 2
PART II. — ALGEBRAIC GEOMETRY.
CHAPTER m .
RECTILINEAR COORDINATES.
1. Coordinates and tensors - • - 35 2. Contravariant veotors and displacements - - - • - - 37 3. The unit points and the geometrical interpretation of rectilinear
coordinates • • - • • - - 38 vii
viii CONTENTS. 7- 4. The distance between two points and the fundamental double'
sor. The e-systems
5. The angle between two directions; orthogonality - - •"-;
6. Associated tensors - - i,
7. Scalar and vector products of vectors - - - - "*
8. Areas and volumes . , f
CHAPTER IV. *
THE PLANE. \
1/ The equation of a plane - - - " j 2. The perpendicular distance from a point to a plane • \i 3. The intersection of two planes '- \ 4. The intersection of three planes
5. Plane coordinates - - ^ 6. Systems of planes ',•:•
7. The equation of a point - • *'.' CHAPTER V. T
THE STRAIGHT LINE. j
1. The point equations of the straight line - - - - * ~- 2. The relations of two straight lines - - • - . "- 3. The six coordinates of ft straight line . . . . , ' . 4. The plane equations of a straight line • - • - • •
CHAPTER VI.
THE QUADRIC CONE AND THE CONIC.!
1. T h e e q u a t i o n of a quadrio cone . . . . . . 'I 2 . T h e e q u a t i o n of a conic - - . . . »v -.
3 . T h e t a n g e n t p l a n e ' t o a cone \ 4. Poles a n d p o l a r planes w i t h respect t o a cono • 6. T h e canonical e q u a t i o n of a cone - - - - . •••
6 . T h e p r i n c i p a l a x e s o f a c o n e
7 . T h e c l a s s i f i c a t i o n o f c o n e s . . . ' • •
C H A P T E R V I I .
SYSTEMS OF CONES AND CONICS. ?
1. T h e e q u a t i o n of a s y s t e m of c o n e s w i t h a c o m m o n v e r t e x - 2 . T h e c o m m o n p o l a r d i r e c t i o n s of a f a m i l y of c o n e s -
3 . T h e c a n o n i c a l f o r m s of t h e e q u a t i o n of a f a m i l y of c o n e s -' 4 . T h e t h e o r y of e l e m e n t a r y d i v i s o r s . . . . 5 . A n a l y t i c a l d i s c r i m i n a t i o n of t h e cases - - . . - •
CONTENTS. a CHAPTER VHI.
CENTRAL QUADRICS.
1. The point equation of a central quadric - - • • . . . . 104 2. T h e tangential equation of a central quadric - ' - - - - 105 3. Canonical form of t h e equation of a quadric. Principal axes - 107 4. Classification of t h e central quadrics 108 5. Confocal quadrics . . . . ' 1 1 0
C H A P T E R I X .
THE GENERAL QUADRIC.
1. T h e general equation of a quadric , - . . . 113 2. The centre - - - - . , - - - 114 3. The reduction of t h e equation of a quadrio • • ' - - - 115
C H A P T E R X .
AFFINE TRANSFORMATIONS.
1. A f f i n e t r a n s f o r m a t i o n s . . . 1 2 0 2 . T h e q u a d r i c of a t r a n s f o r m a t i o n - • • - • - - - 1 2 1 3 . P u r e s t r a i n - - 1 2 3 4 . R i g i d b o d y d i s p l a c e m e n t s - - 1 2 4 5 . I n f i n i t e s i m a l d e f o r m a t i o n s . . . 1 2 6
PART. III. — DIFFERENTIAL GEOMETRY.
CHAPTER XI.
CURVILINEAR COORDINATES.
1. General coordinate systems - 130 2. Tensor-fields - - ^ 133 3. The line-element and the metric tensor. The e-systems - - 134 4. The angle between two directions 136
CHAPTER XH.
COVARIANT DIFFERENTIATION.
1. A parallel field of vectors. The Christoffel symbols - - - 140 2. The intrinsic and covariant derivation of vectors - . - - - 143 3. The intrinsic a n d covariant derivatives of tensors - - - - 146 4. Conservation of t h e rules of t h e ordinary differential calculus.
Ricci's lemma 148 5. The divergence and curl of a vector. T h e Laplacian . . . 151 6. The Riemann-Christoffel tensor. The Lame relations - 152
* CONTENTS.
- CHAPTER XIII.
CURVES IN SPACE.
1. The tangent vector to a curve 156 2. Normal vectors. The principal normal and binormal - • • 157
3. The Frenet fSrmulae - - " 1 5 9
4. Parallel vectors .along* a curve. The straight line - - - 160
"CHAPTER XIV.
INTRINSIC GEOMETRY OF A SURFACE.
1. Curvilinear coordinates on a surface - - 163 2. The conventions regarding Greek indices. Surface tensors - - 164 3. The element of length" and t h e metrio tensor . . . . 167 4. Directions on a surface. Angle between two directions - - - 168 5. The equations of a geodesic 171 6. The transformation of t h e Christoffel symbols. Geodesic coordin-
ates / . - " ' 175 7. Parallelism with respect t o a surface 178 8. Intrinsic a n d covariant differentiation of surface tensors - - - 180 9. The Riemann-Christoffel tensor. The Gaussian ourvature of a sur-
face - - - -. . . . -_ - . - . . - - 182 10. The geodesic curvature of a curve on a surface - . . . . 184 11. Beltrami's differential parameters ~ 186 12. Green's theorem on a surface - - . . . 188
• •' [' C H A P T E R X V .
THE. FUNDAMENTAL FORMULAE OF A SURFACE.
1. Notation - - _ - - : - ; 193
2. The tangent vectors^ t o a surface 294 3. The first groundform of a surface 295 4. The normal vector to the; surface . ~- - ' . . . jgg 5. The tensor derivation'of-tensors - - „ 297 6. Gauss's formulae. 'The second groundform of a surface - . - 200 7. Weingarten's formulae.. The'third groundform of a surface - - 201 8. The equations of Gauss a n d Codazzi - - ' 203
; C H A P T E R X V I . CURVES ON A SURFACE.
1. The equations of a curve on a surface - 2 0 7
2. Meusnier's theorem • " " " , ' • " " 208 3. The principal ourvatures. Gauss s theorem 2 1 0
4 The lines of curvature , - - - . . 2l l 5 The asymptotic lines. Enneper's formula - . . _ ' * 6. The geodesic torsion of a ' c u r v e on a s u r f a c e . . . . . 2 U
CONTENTS. xi
PART IV.— APPLIED MATHEMATICS.
' CHAPTER XVH.
DYNAMICS OF A PARTICLE.
1 . T h e e q u a t i o n s o f m o t i o n - - - . . . . 2 1 8 2 . W o r k a n d e n e r g y . L a g r a n g e ' s e q u a t i o n s o f m o t i o n . . . . 2 2 0 3 . P a r t i c l e o n a c u r v e 2 2 3 , 4 . P a r t i c l e o n a s u r f a c e 2 2 6 6 . T h e p r i n c i p l e o f l e a s t a c t i o n . T r a j e c t o r i e s a s g e o d e s i e s . . . 2 2 8
CHAPTER
DYNAMICS OF RIGID BODIES.
SECTION A — RECTILINEAR COORDINATES.
1. Moments of Inertia 233 2. The equations of motion - - - 235 3. Moving axes. Euler's equations • - - • - • • - 238
SECTION B — T H E GEOMETRY OF DYNAMICS.
4. Generalised coordinates of a dynamical system . . . . 240 5. T h e equations of motion in generalised coordinates • - - 242 6. The manifold of configurations . - . - . . . . 245 7. The kinematics! line-element - 246 8. The dynamical trajectories of t h e manifold of configurations - - 247 9. The principle of stationary action. T h e action line-element - - 249
C H A P T E R XTX.
ELECTRICITY AND MAGNETISM.
1. G r e e n ' s t h e o r e m - - . . . . - . . . 2 5 5 2. S t o k e s ' s t h e o r e m 2 5 8 3. T h e e l e c t r o s t a t i c f i e l d 2 5 9 4. D i e l e c t r i c s 2 6 1 5. T h e m a g n e t o s t a t i c field - 2 6 3
• 6 . T h e e l e c t r o m a g n e t i c e q u a t i o n s - - - . . . - 2 6 5 C H A P T E R X X .
MECHANICS OF CONTINUOUS MEDIA.
1. Infinitesimal strain - • - - - 271 2. Analysis of stress 274 3. Equations of motion for a perfect fluid 276 4. The equations of elasticity . . . 278 6. The motion of a viscous fluid • - 280
xii CONTENTS.
CHAPTER XXI.
THE SPECIAL THEORY OF RELATIVITY.
1. The. four-dimensional manifold 285 2. Generalised coordinates in space-time r , - 286 3. The. principle of special relativity. T h e interval and t h e funda-
mental quadratic form - •"- - - 288 4. Local coordinate systems and their transformations - 292 5. Relativistic dynamics of a particle . - . . - . . 294 6. Dynamics of a continuous medium 296 7. T h e electromagnetic equations . - 298
A P P E N D I X .
ORTHOGONAL OTEVffilNEAR COORDINATES IN MATHEMATICAL PHYSICS.
1 . T h e c l a s s i c a l n o t a t i o n - - - . . . 3 0 3 2 . T h e p h y s i c a l c o m p o n e n t s o f v e c t o r s a n d t e n s o r s . . . 3 0 4 3 . D y n a m i c s 3 0 5 4 . E l e c t r i c i t y 3 0 6 5 . E l a s t i c i t y - 3 0 7 6 . H y d r o d y n a m i c s . . . 3 0 9 B I B L I O G R A P H Y - - • • - 3 1 4 I N D E X - . . . 3 1 f i
