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SCHAUM'S OUTLINE OF

THEORY AND PROBLEMS

0'

VECTOR ANALYSIS

114324

and an introduction to

TENSOR ANALYSIS

SI (METRIC) EDITION

BY

MURRA Y R. SPIEGEL, Ph.D. ProfeUiOr 0/ MalMmatic8

RensselMr Polyted,n;c i1utilute

1 • •••••

IIIIIIIII

Y I A • S 1 • • • • • •

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Contents

CHAPTER PAGE

1. VECTOR 5 AND SCALA R 5.............................................................. 1 Vectors. Sc.l.l.l.u. Vector algebr&. Laws of vector algC'brll. Unit vectors. Rectangub.r unit v«tors. Components ol a veclOr. Scalar fields. Vector fields.

2. THE DOT AND CROSS PRODUCT.. ............................................... 16 Dot or scalar products. erou or vector products. Triple products. Reciprocal sets of vectors.

3. VECTOR DIFFERENTIA TlON........................................................ 35 Ordinary derivatives of vectors. Space curves. Continuity and differentiability. Differen­tiation for mulae. Partial derivatives of vectors. Differcntiab of vectors. Differential g«lmetry. Mechanics.

4. GRADIENT, DI VERGENCE AND CUR L..................... .............. .. 57 The vector differential operator dd. Gradient. Divergence. Curl. Formulae involving del. Invariance.

5. VECTOR INTEGRA TlON................................................................ 82 Ordinary integrals of v«IOlS. Line integrals. Surface integrals. Volume integrals.

6. THE DIVERGENCE THEOREM, STOKES ' THEOREM , AND RELATED INTEGRAL THEOREMS ............................. 106

The diver~nce th~rem of Gauss. Stokes' theorem. Green's theorem in the plane. Re· lated integral theorems. Integral operator form for del.

7. CURVILINEAR COORDINATES ..................................................... 135 Transformation of coordinates. Orthogonal curvilinear coordinates. Unit vectors in curvilinear systems. Arc length and volume elements. Gradient, divergence and curl. Special orthogonal coordinate systems. Cylindrical coordinates. Spherical coordinates. Parabolic cylindrical coordinates. Paraboloidal coordinates. Elliptic cylindrica l coordinates. Prolate spheroidal coordinates. Oblate spheroidal coordinates. Ellipsoidal coordi nates. Bipolar coordinates.

8. TENSOR ANA L Y SI 5 ........................................................................ 166 Physical laws. Spaces of N dimensions. Coordinate tnnsformalions. The summation convention. Contra variant and covarian! vectors. Contravariant, covariant and mixed tensors. The Kronecket delta. Tensors of rank greater tMn two. Scalars or invariants. Tensor fields. Symmetric and skew-symmet ric tensors. Fundamental operations with tensors. Matrices. Matrix algebra. The line element :and metric tensor. Conjugate or reciprocal tensors. Associated tensors. Lt-ngth of a vector. Angle betwe-en vectors. Physical components. Christoffel's symbols. Transformation laws of Christoffel's symbols. Geo­d~sic$. Cov,uiant derivatives. Permutation symbols and tensors. Tensor form of gradient, dIvergence and curl. The intrinsic or absolute derivative. Relative and absolute tensors.

I N DE X................................................. ................................................... 218

Index

Absolute derlnUve. 1'74 Absolute motion. 53 Absolute tenaar. 1'75 Acceleration. alon, • space cune. 35.39.40.50.56

centripetal. 43.50.53 CorloUs. 53 in cyUndrlcal coordinates. 143. 204 In ,eneral coordinates. 204. 205 iD polar coordl.Dates. 56 in .pherIcal coordiDates. 160. 212 of. panicle. 38.42.43.50,52. M. 203. 203 reWi .... to fixed and mo,.-iD, ob.erYer •. 52.53

Addition. of matrices. 1'70 of tensor.. 169

Addition. of Tector.. 2.4.5 ••• 0claUnla." for. 2.5 commutathe la." for. 2.5 par.nelo,ram la." for. 2." trlan,le 1 • ." for, 4

Aerodynamics, 82 Affine tranaforl!llltion. 59, 210, 213 Al,ebra, of matrices. 170

of ,.-ector., 1,2 An,le, bet."een two surfaces. 63

bet."een t."o .... ctors, 19. 1'72, 190 solid, 124, 125

Ancular momentum. 50,51.56 Ancular speed and velocity, 26. 43. 52 Arbltr.ry conatant Tector, 82 Are lencth. 3'7. 56. 136. 148

In cun1l1near coordinates, 56, 148 In ortho,ona1 cunlUnear coordinates, 136 on. surface, 56

Areal .... locib', 85. 86 Area, bounded by a simple cloeed cur,.... 111

of elllpse. 112 of paraUelocram, 1'7. 24 of surface, 104, 105, 162 of trlanlle, 24, 25 vector. 25, 83

Aasoelated tensor.. 1'71, 190. 191. 210 Associative 1 • .". 2, 5, 1'7

Bue .... ctor •• 7, 8, 136 unitary, 136

Binormal. 38. 45. 47. 48 BIpolar coordinate., 140, 160

Box product, 1 '7 Brabe. TYcbo. 86

C.lculu. ot ,.-ariaUons. 1'73 Can_Ian tenso,., 210 Central torce, 56, S5 Centripetal acceleration, 43, 50, 53 Centroid, 15 CbaiD rule, '7'7, 1'7'7, 1'79 CharacteruUc equation, 210 Characteristic ,.-aluea, 210 Cbar,. density, 126 Chrlatottel' •• )'1IIbol •• 1'72, 192-195, 211

ullISformation la.". ot, 1'72, 193, 194 Circulation, 82, 131 Circumcentre, 33 Clock."l.e direction, 89 Cofactor. 1'71, 187, 188 Collln .. r Yectors, 8, 9

non-, 7, 8 Columa _ttU or .... ctor, 169 Commutatl,... la.". 2, 5, 16. 1'7 Component ,...ctor., 3, 7, 8

rec:tanlU!ar, 3

218

Component., contranr1ant, 136. 156, 15'7, 16'7, I" coyarlant. 136 of a d,ad, 73 otateuor, 157,187, 168 of a vector, 3, 136. 156.157, 158, 16'7 pbJ"alcal, (eee Ph]'slcal compon.nta)

CondueUdty. tbermal, 126 Conformable matrices, 1'70 Conic section, 87 ConJU(ate metric tenaor, 1'71. 188, 189 Con,ju,a.te tensors, I'll Conaeryation of enerlP', 94 ConaernU,... !leld. '73. 83. 90. 91, 93

motion of particle in. 93, 94 necessary.nd sufticlent condition tOt, 90. 91

Continuity, 38, 3'7 equaUon of. 6'7, 126,212

Contraction. 169, 181, 182 Contr.,.-arllllDt components, 138, 156. 15'7, 18'7, 188

et. tensor, 15'7, 18'7, 168 of. ,.-ector, 131S, 158. 15'7, 16'7

Contranrlant tensor, of f1rat rank. 15'7, 187 ot secoDd arid hl,ber luk, lIS8

INDEX 219

Contravarl.nt ,ector, (see Contr ..... d.nt compo­nents of s vector)

Coordinate curves or lines, 135 Coordinates, curvlUneu, (see Cun1l1neu coordi-

nates) Coordinate surf.ces, 135 Coordinate tr.llIIformatiollS, 58, 59, 76, 135, 166 Copl.anar vectors. 3

neceslluy and suftlc1ent condition for. 21 non-. 1, 8

CodoUs .cceleraUon, 53 Cosines, direction. 11, 58

law- of, for plane trlanglell, 20 ta'lf of, for spherical trlandes, 33

Counterclock'lflae direction, 89 Covartant components, 136, 151, 158, 161

of s tensor, 161, 168 of a ...ector, 136, 151, 158, 161

Covulant curvature tensor, 201 Co,ulant derivative, 113, 191-199, 211 Cova,!ant tenaor, of first rank, 158 Co,ulant ...ector, (see Covulant components of a

vector) CrOIlS-C.,lt, 113 Croas product, 16, n, 22-26

commutative ta'lf failure for, 16 determinant form for, 11, 23 distributive law for, 16, 22, 23

Cubic, twlated, 55 Curl, 51, 58. 61-12

In c,.Undrlcal coordinates, 153, 154 In ortOOlonal curvU1neu coordinates. 131, 150 In puaboUc cyUndrtcal coordlnatea, 161 In spherical coordinates, 154 intell'al definition of, 123, 152, 153 Invartance of, 81 of tbe padient, 58, 69, 211 ph.Tslcalsilnilicance of. 72. 131 tensor torm of. 174, 200

Current density, 126 Clln'ature, 36, 45, 47, 113

radius ot, 38, 45, 46, SO Rlemann-Chrlstoftel, 206 ten.sor, 207

Curve, space, (lIee Space curves) CurvllInear coordinates, 135-165

acceleration In, 143, 204, 205. 212 arc lenlth In. 56, 136. 148 definition of, 135 leneral, 148, 156-159 orthOlonal, 49, 135 surface. 48, 49, 56, 155 volume slements In, 138, 137, 159

Cycloid, 132 Cylindrical coordinates, 137, 138, 141, 142, 160.161

arc len(th In, 14.3 Chrlstorrel's symbols In, 195,211 cOrUUlate metric tenaor In. 189

CyUndrlcal coordinates, continuity equation In, 212 curl In, 153, 154 diverlence in, 153. 200. 201 elUptlc, (lIee Elliptic cyllnddcal coordlnate~ \

,eodeslcs In, 211 ll'ad.1ent In, 153, 154 Jacoblan in, 161 Laplacian In, 153, 154, 201 metric tensor In, 187 parabolic, (see Pu.bolic c,.Undrical coordinates) velocit,. and acceleration In, 143, 204, 205 volume element In, 144, 145

V, (see Del) ~, (lIee Lapl.clan operator) Del IV>, 57.58, (see also Gradient, Diver,ence and

Curl)

rormulae involvinll. 58 Intearal operator form for, 101, 123 In'I'Brlance of, 81

Delt., Kronecller. 168, 179. 180. (see also Kron­ecller's symbol)

DensU,., 126 char,e, 126 current , 126 tensor, 115, 203

Dependence, linear. 10.15 DerlvaUu, absolute, 114

coT&rlant, 113, 197-199,211 directional. 51. 61-63 Intrinsic, 114. 202, 211

Deriv.Uves, ot yectors. 35-56 ordinary. 3S. 36, 39-43 partial, 36, 37, 44. 45

Deacutes, tol1um of, 132 Determinant. cofactor of. 171, 181, 188

cross product expressed as, 11, 23 curl expressed as, 57, 58 differentiation of, 41 Jacobian, (see J.coblan) of a matrix, 110, 209 scalar triple product expressed as, 17, 26, 27

Determinants, multiplication of. 159 Dextral syatem. 3 Dlaaonal of a square matrix, 169 Difference, of matrices, 110

of tensors, 169 of Yectors, 2

DlrterenU.ble, scalar field, 51 Victor field, 51

DirterenUabtUty, 36, 31 Differential equations, 54. 104 Differential ,eometr)" 37, 38, 45·50, 54·56, 166,212-13 Differentials, 37

exact. (aee Exact differentials ) Differentiation of vectors, 35·56

formulae ror. 36. 37 . 40. 41

220 INDEX

DuterenUaHoD of ,..ectora, order of, 37, 69 ordinar,., 3:1, 38 partial, 38, 37

DUfuaIy!ty. 127 DirectIonal derlvath-e, .57, 61-63 DllecUon cosines, 11, .58 Dl.etance between two points, 11 Dlatrlbutin 1.'111', 2

for ero •• products, 16, 22, 23 tOt dot products, 18. 18 tor dndica, '4 for mlltrlcea, 170

DIy, (Bee Dlnt,enca) DIYerc.nce, 57. 64-67

In cun1l1neu coordinates, 137. 150 In cl11ndrlca.l coordln&tea, 153, 200. 201 1n parabolic cylindrical coordinate., 161 10 apherlcIJ coordinates, 181, 200, 201 In .... rt .. nce of. 81 of the curl. 58, 69, 70, 211 of the ,n,dlent, 58, 64 ph1alcl.l al,nitteance of, 66, 67. 119. 120 tensor form of, 174,200,20] theorem, (.ea Dlyereance theorem)

Dhet,ence theorem, 108, llD, Ill, 115-127 expressed In 'lll'orda, 115 Oreen '8 theorem a. IS special case of, 108. 110, III physical al,nltteance of, 118, 111 proof of, 117, 118 rectl.n,uiar form of. 116 tenaor rorm of. 206

Dot product, 16, 1&-21 commutaUn la. for. 16. 18 dlstrlbuth' e la. for, 18, 18

Dummy Index. 167 Dyad, 73

,

Dyadlc, 73·75, 81 D,namlca, 38. (aee also Mechanica)

La,ranle'a equations in, 196, 205 Newton'a la. in. (eee Newton's law)

Eccentricity. 87 El,envalues. 210 Einstein, theory of relatI1'ity of, 148.207.213 Electromacnetic theor" 54. 72, 208 Element, line, 170, 187·189

l'olume, 138, 137. 159 Elements. of a matrix, 189 ElUpee, 63, 139

area. of, 112 motion of planet In. 86. 87

Elllpsoldal coordinates. 140, 160 Ell1ptlc c,Uncirtcal coordinates. 139, US, 160.161,

21l EDltrc, 94

cOlUlerntIon of, 94 kinetic. 94, 204

EDerc, potential, 94

EQualit,. or matrices, 170 of vectors, 1

Equtubn,nt, 8 Euclldean spacn, 170

N dimensional, 171 Euler'a eQuatIona, 196 Exact dlUerentW.a, 83, 93, III

necessar, and aumcieDt condition for. 93 Extremum, 198

Fictitloua forcea, 53 FIeld, (a" Seal., and Vector neld)

conservative, (see Conaernth'e field) lrrotatlonal, 72, 73, 90 aink, 13, (aee also Sink) solenoldal, 67, 73, 120, 126 source, 13, (Bee also Source) tenaors, 168 l'ortex, 72

Fbed and mo.in, Bystema. obaerTers In, 51·53 Fluid mechanic., 62 Fluid motion, 86, 67, 72, 118, 117, 125. 128

Incompressible, 67. 126 Flux. 83, 120 Force, central. 56, 85

CorloUs, 53 moment of, 25, 26, 50 on a particle, 203, 205 repulain, 85 univeraal cravltatlonal, 88

Forces, tlctltloua, 53 real, 53 resultant of, 11

Framea of reference. 58. 166 Free Index, 167 Frenel-8erret formulae, 38, 45, 213 Fundamental Quadratic form, 148 Fundamental tenaor, 17l

Gausa' dlverlence theorem. (aee Dlnr,ence theorem) Gauss' law, 134 Gauss' theorem, 124, 125 Geodesics, 172, 173. 196, 197.211 Geometry. differential, (see Differential seomet".) Orad, (see Oradlent) Oradlent, 57, 58, 59--83, 177

In cylindrical coordinates. 153, 154 In orthOlonal curvlllneat coordinateB,l37. 148, 149 In parabolic c,Undrical coordinates, 181, 211 In spherical coordinates, 161 Imecral detlDIUon of, 122, 123 Invariance of, 77 of a nctor, 13 tensor form of, 174, 200

Graphical, addition of nctors, " representation of a nctor, 1

INDEX 221

Gr..,UaUon, Ne"ton'l universal law of, 88

Green's, first Identity or theorem, 107, 121 second Identity or symmetrical theorem, 107,121 tl'leorem In space, (see Dlversence theorem)

Green's theorem In the plane, 108, 10&-11~ as special case of stokes' theorem, 108, 110 as special case of the dlver,ence theorem, 108,

110, III tor mulUply·connected relions, 112·114 tor slmply-connected rellona, 108·110

HamUton-Cayley theorem, 210 Hsmllton's principle, 205 Heat, 126, 127

specific, 126 Heat equation, 128, 127, 161

In elliptic cylindrical coordinates, 15~ In spherical coordinates, 161

Heat now, stead;y-state, 127 Helix, circular, 45 Hyperbola, 87 Hyperplane, 176 Hypersphere, 176 Hypersudl.ce, 176 Hypocyclold, 132

Independence, of orl,ln, 9 of path ot Intelration, 83,89,90,111,114,129,130

Independent, Unearl.J', ID, 15 Index, dumlU.Y or umbrl.l, 167

free, 167 Inertll.l systems, ~3 Initll.! point ot a vector, 1 Inner multiplication, 189, 182 Inner product, 169, 182 rntearal operator form tor V. 107, 123 Intecrala, of 'fectors, 82-105

definite, 82 Indefinite, 82 line, (see Une IntecrtJs) ordinary, 82 surface, (lee Surface Intecrals) theorems on, (see IntelUal theorems) yolume, (see Volume Intepals)

Intepal theorems, 107, 120, 121, 124, 12~, 130, (see also Stokes' theorem and Dlver,ence theorem)

Integration, (see lnte,rals, of vectors) Intrinsic derlvati'fe, 174, 202, 211 Inndance, 58,59,76, n, 81, (see also lnndant) Invariant, 59, 168, 190, (see also Invariance) Inverse of a matrix, 170 Irrotatlool.l field, 72, 73, 90

Jl.coblan, 79, 133, 146, 147, 148, 159,181,182,175,202-3

Kepler's laws, 88, 87, 102 Kinematic., 38, (see Ilso Dynamics soo Mechanica) Kinetic enerlY, 94, 204

Kronecker delta, 168, 119, 180 Kronecker's symbol, 77, 208

Lacrl.nlean, 205 La,ranle's equations, 196, 205 Laplace's equation, 65, 127, 134

In para~lIc cyUndrlcal coordinates, 154, 155 IAplace transforms, 182

Laplaclan operator (~), 58, 64,81. 200 In curvilinear coordinates, 137, 150, 151 In cylindrical coordlnatea, 1~3, 154, 201 In parabolic cylindrical coordinates, 154, 155,211 In spherical coordinates, 154, 201 Invariance of, 81 tensor form of, 174, 200

Laws of vector al,ebrl., 2, 18 Lemnlscate, 132 Lencth, of a vector, 171, 172, 190 U,ht r.,.s. 63 Li,ht, l'8loclty of, 81 Linearly dependent vectors, 10, 15 Line element, 170, 187-189 Line, equation of, 9, 12

parametric equations of, 12

sink, 13 aource, 13 symmetric form for equation of, 9

Line Intepals, 82, 87-94, III circulation In terms of, 82, 131 eniuation of, 87·89, 111 Green's theorem and evaluation of, 112 Independence of path, 83,89,90, Ill, 114, 129, 130 work expressed In terms of, 82, 88

Lorentz·F1tzaerald contraction, 213 Lorentz transformation, 213

Ma,nitude, of a nctor, 1 Maln dia,onal, 169 Mappln" 162 Matrices, 169,110.185,186, (see also Matdx)

addition of, 110 conformable, 170 equal1tyof, 170 operations ,,!th, 170

Matrix, 73, 169, Csee also MAtrices) al,ebra, 170 column, 189 determinant of, 170, 209 elements ot, 169 Inverse ot, 170, 209, 210 main or principal dla,onal of, 189 null , 189 order of, 169 principal dla,onal of, 169 row , 169 slnrular, t 70 square, 169 transpose of, 170,210

222 INDEX

Ml.xwell's equations, 72, 81 In tensOf form, 206

Mechanics, 38, 56, (see alao Dynamica) nuid, 82

Metrie coefficients, lU Metric form, 148 WetTle teneor, 170, 111, 187·189 Mixed tensor. 167, 168 Moeblu8 strip. 99 Moment of force, 25, 26. 50 Momentum, 38

ancuiar. 50, 51. 56 Motion, absolute, 53 Notion, of nuld, (lIee Fluid motloo)

of planets, as.8'l Moyln, and f1I.ed aystems. observers In, 51·53 Movlo, trlhedral, 38 Multiplication, (see Product) MulUplJ'-connected re&1on, 110. 112-114

Nabla, (aee Del) Ne,ul't'8 direction, 89 Newton'. law, 38, 50, 53

In tenaor form, 203 of universal ,ruU_tlon, 86

Normal plane, 38, 48 Normal, principal, 38, 45. 47. 48, 50

bI-. 38, 45, 47, 48 Normal, to .. surface. 49, 50. 56, 61

positive or outwarrl drawn, 49, 83 Null matrix, 169 Null vector. 2

Oblate spheroidal coordinates. 140, 145, 160, 161 Operations, with tenaors, 169. 179-184 Operator. del. 57. (aee also Del)

Laplaclan, (see Laplaelan operator) time derivative, In nxed and movinc systema,

51. 52 Order, of a matrix, 189

of a tensor, 167 Orlentable awface, 99

OrICin, of a Teetor, 1 Independence ot vector equation on, 9

OrthOd!ntre,33 Ortoogonal coordInates. special, 137·141

bipolar, 140, 160 cylindrIcal, 137, 138, (see Cylindrical coordinates) elllpaoldal, 140, 160 elliptic cyllndrlcal, 139, 155, 160, 161, 211 oblate spheroidal, 140, 145, 160, un parabolic cyllDdr1cal, 138, (see also Parabolic

cyUndrlcal coordinates) paraboloidal, 139, 160, 161, 211 prolate spheroidal, 139, 160, 161 epherical, 137,138, (ue Spherical cOOfdlnates) toroldal, 141

OrthoConal CIlf'fWnear coordinate systems, 49,135, 191

apeclal, 13'7-141 Ortooconal transformation, 59 oaculatlnc plane, 38, 48 OUter muItlpUcalloD, 169 Outer product, 169 OUtward drawn normal, 49, 83

ParabOla, 87, 138 Pvabol1c cyUndrical coordinates, 138, 144, 145, 154,

155, 160, 161. 211 vc lenph In, 144 Christortel's s)'lnbOls In, 211 curl'ln, 161 dIvergence In, 161 gradIent In, 161, 211 Jacoblan In, 161 Laplaclan In, 154, 155, 211 Schroedinaer's equation in, 161 'folume element In, 145

Pvabololdal coordinates, 139, 160, 181, 211 Parallelogram, area of, 17, 24 Parallslogram law of vector addition, 2, 4 Pvametrlc eQuatloDS, of a curve, 39, 40

of a line, 12 of a surface, 48, 49

Periods, of planets, 102 Permutation s)'lnbols and tensors, 173, 174, 211

Physical componenta, 172. 200, 201, 205, 211 Plane, distance from orlcln to, 21

equation of, 15, 21, 28 normal, 38, 48 oaculatinc, 38, 48 lecttrytOl, 38, 48 tancent, 49, 50, 61 Teetor perpeo;licular to, 28 vectors 10 a, (see Coplanv vectors)

Planets, motion of, M-87 Point function, sealv and vedor, 3 Polslon's equation, 134 Polv coordinates, 98 P081Uon 't'ector, 3 Pos1tln direction, 89, 106, 113 Posltin normal, 83 Potential ellerD', 94 Potential, scalar, 73, 81. 83, 91, 92

Tector,81 PrincIpal diqonal, 169 Principal normal, 38, 45, 47, 46, 50 Product, box, 17

ero .. , (see CroIS product) dot, (see Dot product) Inner, 169, 182 of a Tecter by a scalar, 2 of determ1nants, 159 of matrices, 170

Product, of tensors, 169 outer, 169, 181 scalar, 182, (see also Dot product) vector, (see Cross product)

Projectile, 102 Projection, of a vector, 18, 20

of surfaces , 95 , 96 Prolate spheroidal coordinates, 139, 160, 161 Proper vector, 2 Pytha,orean theorem, 10

Quadratic form, fundamental, 148 Quantum mechanics , 161 Quotient law, 169, 184

Radius, of curvature, 38, 45, 46, 50 of torsion, 38, 45

Radius vector, 3 Rank, of a tensor, 167 Rank zero tensor, 168 Real forces, 53 Reciprocal sets or systems of vectors, 1'7, 30, 31

34, 136, 147 Reciprocal tensors, 171 Rectanlular component vectors, 3 Rectanlular coordinate systems, 2 Rectifyin, plane , 38, 48 Region, multiply-connected, 110, 112-114

simply-connected, 110, 113, 114 Relath'e acceleration, 53 Relative tensor, 175 , 202, 203, 212 Relative velocity, 52 Relativity, theory of, 148, 201, 213 Resultant of vectors , 2, 4, 5, 6, 10 Rtemann-Chr1stotrel tensor, 207, 212 R1emannlan space, 171 , 112

geodesics In, 172, 196, 19'7 Right-handed coordinate systems, 2, 3

localized, 38 Rh:td bod,., motion of, 59

velocity of, 26, 33 Rot, (see Curl) RataUn, coordinate systems, 51, 52 Rotation, Invariance under, (see Invariance)

of axes, 58, 16, 71 pure, 59

Row matrix or vector, 169

Scalu, 1,4, 168 field , 3, 12, 168 fUnction of pOSition, 3 point fUnction, 3 potential, 13, 81 , 83 , 91, 92 product, 182, (see also Dot product) triple products, (see Triple products) vulable , 35

Scale factors, 135

INDEX

Schroedln&et's equation, 161 SImple closed curve, 82, 106

area bounded b,., III SlrtlDlY-connected resion, 110, 113, 11 4 Slnes, law of, fo r plane tr1an,les, 25

for spherical trianlles , 29, 30 Slnlular matrix, 110 Slnlulu points, 141 SInk, 13, 61, 120 Sink field, 13, (see also Sink) Solenoldal field, 67, 73, 120, 126 Solid an,le, 124, 125 Sound rSJ'S, 63 Source, 13,61, 120 Source field, 13, (see also Source) Space curves, 35

&cceieration alonl, 35, 39, 40, 50, 56 arc lencth of, 37, 56, 136, 148 blnormal of, 38, 45, 47, 48 curvature of, 38, 45, 47, 113 principal normal of, 38, 45, 47, 48, 50 radius of curvature of, 38, 45, 46, '0 radius of torsion of, 38, 45 tanlent to, 37, 38, 40, 45, 41, 48, 50

Space intelrals, (see Volume Intepals) Spaces, Euclldean, 110

Riemannlan, 171 Space, N dimensional, 166 Special theory of relativity, 213 Speed, 4

ancular , 26, 43, 52

223

Spherical coordinates, 131, 138, 141, 147 , 160, 161 arc length In, 144 Ctlrlstoffel's symbols 1n, 195, 211 cOnjulate metric tensor In, 189 continuity equation in , 212 covarlant components In, 17'7, 178 curl!n, 154 divergence in, 161, 200, 201 ,eodeslcs In, 211 r;rsdlent in, 161 heat equation In, 161 Jacoblan In, 161 Laplacian In, 154,201 metric tensor In, 187 velocity and acceleration In, 160, 212 volume element In, 144, 145

Spheroidal coordinates , oblate, 140, 145 , 160, 161 prolate, 139, 160, 161

Stationary scalar fleld, 3 Stationary-state , (see Steady-state) Steady-state, heat flow, 121

scalar field, 3 vector field , 3

Stokes' theorem, 106, 1l0, 127-131 Green 's theorem as special case of, 110 proof of, 127-129 tensor form of, 212

224 INDEX

Subtraction, of tensors. 169 of vectors, 2

Summation convention, 167. 175, 178.207 Superscripta, 166 Surface, area or. 104. 11)5, 162 Surface cUlviUneu coordinates, 48. 49. 56, 155

arc len&lh In, 58, 118 Surface integrals, 83, 94-99

defined u Umlt of .. sum, 94, 95 enluaUon of. 83

Surface., 37 an,le behreen. 63 arc length on, 58 coordinate, 135 one-sided, 99 orlentable, 99 outward drawn normal to, 83 two-sIded. 83

Symmetric form, of equation of .. line, 9

Tan,ent, to aPACe cur"" 37, 38, 40, 45, 47, 48, 50 Tan,ent plane. 49. 50, 61 Tensor analYsIa, 73. 137. 158, 166-217 Tensor, absolute, I'Hi

associated, 171, 190, 191,210 Cartesian. 210 conju,ate, I'll

contr.,..rtant, Caee Contr.n,Iant components) conl'l.nt, (aee Conriant components) cun.ture. 207 density. 175, 203 field, 168 fundamental, 111 metric, l'J0 mixed, 167, 168 order of, 167 rank of, 167 reclllrocal, 171 relative, 175, 202, 203, 212 akew-symmetrlc, 168, 169 symmetric, 168

Tensors, fundamental operations with, 169, l'J9-IU Terminal point or Terminus, 1,2, 5,11 Thermal conductivtt,)', 126 Toroldal coordinates, 141 Torque, 50, 51 Torsion, 38, 45, 47, 213

radius ot, 38, 45 Transformation, artine, 59,210,213

of coordinates, 58, 59, 76, 135, 166 ortho,onal, 59

Translation, 59 Ttansllcse, of a matrix, 170, 210 Triad, 38 Triadic, 73 Trlan,le, area of, 24, 25 Trlan,le law of vector addition, 4 Trlhedral, movtna, 38

Trlllle llroduch, 17, 26-31 Twisted cubic, 55

Umbraltndu, 167 Unit dJads, 73 Unit tratril, 169 UnU nclors, 2, 11

reclanrulu, 2, 3

Variable, 35, 36 Vector, area, 25, 83

column, 169 equations, 2, 9 field, 3, 12, 13, 168 function of position, 3 malnltude of a, I, 10 nUll, 2 operator V , (see Del) point function, 3 position, 3 potential, 81 product, (see Cross product) radius, 3 row, 169 time derlnUve of a, 51, 52 triple product, (see Triple products)

Vectors, I, 4

addition of, 2, 4 al,ebra of, 1,2 analytical representation aI, an,le between, 19, 172, 190 base, 7, 8, 136 coll1near, (see Collinear vectors) component, 3, 7, 8 contravarlant components of, 136,156,157, 167 coplanar, <see Copianar nctora) covarlant components of, 13/1, 157, 158, 167 differentiation of, 35-56 equality of, 1 graphJcal representaUon of, 1,4 Initial paint of, 1 ori,ln of, 1 reciprocal, l'J

resultant of, 2, 4, 5, 6, 10 terminal point of, terminus of, 1 unit, 2

unltar,)'. 136 VelocltJ, alon, a space curn, 35. 39, 40

ancula.r, 26, 43, 52 areal, 85, 86 llneat, 26 of a nuld, 179 of a particle, 42, 52, 203, 204 of U,ht, 81 reiatln to nxed and movln. obae"ers , 52, 53

Volume, elements of, 136, 137, 159 In cW'vllinear coordinates, 136, 137

Volume, In ,ener.} coordInates, U9 of plll'allelepiped. 1', 28

Volume Inteplls, 83, 9!HOl defined as limit of a Bum, 99, 100

Vortex tield, 72

INDEX

Wau equation, 72 Wel,ht, of .. tensor, 17~ Work. 21, 82. 88, 89, 90. 91

u a line 1ntelt.I, 88, 89, 90. 91

Zero vector. 2

225