517

518

519

520

521

522

523

524

525

526

527

528

529

517

576 MATHEMATICS FOR ENGINEERS AND PHYSICISTS

INDEX

578 MATHEMATICS FOR ENGINEERS AND PHYSICISTS

Elliptio intégrais, 47-55 complété, 48

580 MATHEMATICS FOR ENGINEERS AND PHYS1CISTS

WDEX

EERS AND PHYSICISTS

Minima (see Maxima and minima) Minimax, 162

584 MATHEMATICS FOR ENGINE ERS AND PIIYSICISTS

Région, multiply connected, 205, 212, 455

586 MATHEMATICS FOR ENGINEERS AND PHYSICISTS

INDEX

577

Differential équations, general solution of, 230, 290, 292, 350, 358 homogeneous, 259, 261 homogcneous linear, 290 intégral curve of, 226, 228 integratmg factor of, 265 Laplace’s, 369, 382, 385, 386, 439, 451, 470, 481 Legendre’s, 342, 384 linear, 226, 283-349, 357 numerical solution of, 346 of electnc circuits, 301, 305, 386 of beat conduction, 367 of membrane, 377 of vibrating spring, 308 of vibrating string, 361 order of, 225 ordmary, 225-349 partial, 225, 350-391 partieular intégral of, 290, 292, 297, 318, 359 partieular solution of, 230 second order, 269, 295 séparation of variables in, 257 siinultaneous, 312-315 singular solution of, 279 solution ni senes, 228, 325, 349, 364

Flow, of electricity in a cable, 386 of heat, 256, 368-377, 423, 425 seepage, 483

581

Parallelogram law of addition, 394 Parametcrs, 277, 280 intégrais containing, 167 variation of, 318

Solution, of differential équations, singular, 279 of équations, 85-122 algebraic, 86, 95 graphical method of, 83 transcendental, 85, 97 of Systems of linear algebraic équations, 102-122 steady-state, 309, 310 Source and sink, 221, 411, 416, 424, 427, 429

587

Coordinates, ellipsoidal, 433 parabolie, 439

first kind, F(k, <p), 48-55, 238 second kmd, E{k, <p), 48-54 third kmd, II(w, k, <p), 50 Empirical formulas, 525-560 Entropy, 224 Envelope, 279 Equation, auxiliary, 292 Bernoulli’s, 286 Bessel’s, 332, 380 characteristic, 292 oubic, 86 Euler, 322 Fourier, 425 mdicial, 334 intégral, 347

Homogeneous équations, differential, 259, 261

Modulus, of complex number, 441, 442

of intégration, 173 eimply connected, 205 Régula falsi, 101 Regular functions, 451 Remainder in Taylor’s sériés, 36-37 Remainder theorem, 92 Repeated trials, 501 Représentation, applications of conformai, 479-491 Rcsiduals, 534, 537 Résonance, 310 Riemann surface, 473 Right-handed System of axes, 397 Rod, flow of heat in, 373 vibiâtions of, 366, 367 Roots, of équations, 83-102 isolation of, 92 theorems on, 92-94 of umty, oi, a»2, 87 Rot (see Curl)

Variation, of parameters, 318, 323 Vector analysis, 392-439 Vector équation of line, 395 Vector field, 406, 408, 409, 412, 418, 423

Cable, flexible, 244 flow of eleetricity in, 386 supportmg horizontal roadway, 242

solution of, 226

Fluid motion, 220, 424, 428, 475-478, 480—484 Flux, 416

Ko(s), 337

Parametric équations, 143, 149, 150, 199, 215, 247

Space curves, 149-152 Spherical coordinates, 152, 185, 382, 386, 434, 439 Spring, 299, 308, 313 oscillation of, 299 Standard déviation, 523 Steady heat flow, 256, 368, 369, 427 Steady-state solution, 309, 310 Stereographic projection, 479 Stirling’s formula, 508 Stokes’s theorem, 421 Stream function, 221, 432, 453, 481 Stream Unes, 277, 432, 467 Stresses, shearing, 485 String, vibration of, 361 Sum, of a sériés, An. of vectors, 393

Absolute convergence of sériés, 16,

polar, 183, 184, 276, 279, 386, 438 spherical, 152, 185, 382, 386, 434, 439

Laplace’s, 195, 369, 382, 385, 386, 439, 451, 470, 481 Legcndre’s, 342, 384 of eontmuity, 221, 429, 481 of plane, 147 wave, 432

linear algebraïc, 119-122 linear differential, 290 Homogeneous function, 136, 259 définition of, 136 Euler’s theorem on, 136 Hooke’s law, 241, 299 Horner’s method, 95 Hydrodynamics, 221, 422, 428-433, 480-484 Hyperbola, 247 Hyperbolic functions, 247-256 Hyperbolic paraboloid, 162

of elliptic function, k, 51 Moment, bending, 241 Moment of inertia, 177, 180, 182, 183, 187, 190, 191, 196, 241 Moments, method of, 544 Most probable value, 505 approximation for probability of, 511

Rotational field, 418 Rule, Cramer’s, 113 Simpson’s, 556 trapézoïdal, .556

Vector point function, 406 Vector product, 400 applications of, 404 Vector relationships, 402 Vectors, 144, 152, 392 addition of, 393 base, 396 curl of, 418 décomposition of, 396 divergence of, 411 magnitude of, 393 multiplication of, 399 ongin of, 393 orthogonal, 398 radius, 195 unit, 394, 397 zéro, 393

Cartography, 479 Catenary, 247, 252. Cauchy-Riemann équations, 221, 450, 455

witli constant coefficients, 287-315, 357

Flux density, magnetic, 52 Force, 217, 239, 392, 406, 409 components of, 217 conservative field of, 219 electrostatic, 487 Force function, 411 Forced vibrations, 308, 310 Formula, asymptotic, 509 Cauchy’s intégral, 461 empirical, 525-560 interpolation, 550 Lagrange’s, 552 Poisson, 512 Stirling’s, 508 Wallis’s, 45 Founer coefficients, 65 Fourier équation, 425 Fourier sériés, 63-82 complex form of, 78 différentiation of, 80 expansion in, 65-82 intégration of, 80 solution of équations with, 364 Functional dependence, 2 Functional déterminant, 183 Functions, 1 analytic, 451 Bessel, 336 Beta, 276

Partial dérivatives, 125-143, 153 Partial differential équation, 350-391

Superposition of effects, 129, 223 Surface, équation of, 144 level, 406

17, 20, 21

cos x, 46, 250 cosh, 247

Equations, Cauchy-Biemann, 221, 450, 455

Motion, dead-beat, 304 fluid, 220 l*ws of, 231, 234 of a membrane, 377 oscijlatory, 304 pendulum, 48, 234 simple harmonie, 233, 301, 314, 380

Velocity, 404, 424

Cauchy’s équation, 322«.

with variable coefficients, 284, 315-349

complementary, 290 conjugate, 468, 470 continuous, 23, 448 elementary, 315 expansion of, 35, 65, 155 Gamma, 272-277 holomorphic, 451 homogeneous, 136, 259 hyperbolic, 247-256 of a complex variable, 444-491 of several variables, 123, 160 orthogonal, 81, 339, 345 periodic, 64

Lagrange’s interpolation formula, 552

dérivation of, 351

normal to, 147, 407 Surface intégral, 188-196, 415, 421 Surfaces, coordinate, 434 Riemann, 473

Absolute value of coniplex number,

Cosine, hyperbolic, 247 powcr sériés for, 38, 40 Cosine séries, 73

consistent, 117-122 dépendent, 105 difïerential, 225-391 Euler’s, 430

Functions, potential, 219, 411 power, 30 real, 2, 123 regular, 451 scalar point, 406 singularities of, 222 stream, 221, 432, 453, 'toi vector point, 406

•

under gravity, 232 Multiple intégrais, 172-196 définition and évaluation of, 173, 179

Scalar field, 406, 408 Scalar point funetion, 406, 418 Scalar product, 399 application of, 404 Scalars, 392

angular, 61, 191, 236, 404, 424 cntical, 61

Cauchy’s intégral formula, 461 Cauchy’s intégral test, 12 Cauchy’s intégral theorem, 455 Center of gravity, 177, 182, 183, 187, 190, 191, 196, 522

Differential expression, 225 Differential operators, 287-299, 357, 406

Lagrange’s method of multipliera, 163-167

Fourier, 425

Systems of équations, consistent or inconsistent, 117-122 linear algebraic, 107-122

441

Cosines, direction, 146, 147, 151, 188, 194, 398 coth, 249 Cramer’s rule, 113 Cross product, 400 Cubic équation, algebraic solution of,*86

mconsistent, 105, 117-122 normal, 537, 540

Fundamcntal principlc, of combina-tory analysis, 493 of sequences, 6

Imaginary roots, 94 Implicit functions, 132, 137-142 Inclined plane, 280, 282, 306 Incompressible fluid, 424, 430 Inconsistent équations, 105, 117-122 Independence, linear, 116, 317 of path, 208, 216, 452, 455 Independent events, 495 Independent trials, 501 Indicial équation, 334 Infinité sériés, 1-62

géométrie interprétation of, 177 Multiplication, of complex numbers, 442

Schwartz transformation, 478, 485, 491

of earth’s rotation, 61 of escape, 61 terminal, 59, 254

Change of variables, in dérivatives, 154

Différentiation, of implicit fonctions, 132-142

Lamellar field, 423 Laplace’s approximation, 515 Laplace’s équation, 195, 369, 382, 385, 386, 439, 451, 470, 481 Law, Bernoulli-Euler, 241 binomial, 502, 512 of attraction, 218 of conservation of matter, 429 of cooling, 254 of dynamics, 231 of error, 520, 536 of gravitation, 232 of small numbers, 512 Least squares, method of, 536 theory of, 521

intégration of, 353

Velocity potential, 221, 222, 277, 430, 432, 453, 467, 480 Vibration, forced, 308, 310 of elastic rod, 366, 367 of membrane, 377 of spring, 308 of string, 361

Addition, of séries, 21 of vectors, 393 parallelogram law of, 394 Adiabîftic process, 224 Aerodynamics, 133, 431 Algebra, fondamental theorcm of, 92 Algebraic theorems, 92-94 Alternatmg séries, 15 am «, 51

graphical solution of, 83 Curl, 418, 422, 423, 438 Current, 386, 427 Curve, distribution, 504, 516 elastic, 240, 307 map of, 466 Curve fitting, 525-560 Curves, intégral, 226, 228, 279 orthogonal, 277, 468 Curviiinear coordinates, 433-439 Cylinder functions (see Bessel func-tions)

parametric, 143, 149, 150, 199, 215 representmg spécial types of data, 528

Fundamental theorem, of algebra, 92 of intégral calculus, 172, 457

absolute convergence of, 16, 17, 20 conditional convergence of, 16, 17 définition of, 4 expansion m, 35-46, 155-158 of constants, 6-22 of functions, 23-62 of power functions, 30 of trigonométrie functions, 63-82 operations on, 21, 29, 33-35 sum of, 4n.

of déterminants, 110 of sériés, 21 of vectors, 399 Multiplicity of root, 93, 294 Multiplier, Lagrangian, 165 Multiply connected région, 205, 212, 455

Seepage flow, 483 Séparation of variables, 257 Sequences, 2

in intégrais, 183-188 Charactcristic équation, 292 Charge, distribution of, 487 Charts, distribution, 506 Chemical reaction, 258 Circular functions, 247 Circulation, of a liquid, 475, 477 of a vector, 418, 419 Closed curve, area of, 199-201 direction around, 200 intégral around, 201, 203, 206, 216, 421 simple, 200 en u, 51

of senes, 29, 33, 34, 80 partial, 123-171 term by term, 33, 34, 80 under intégral sign, 167 Diffusion, 369, 427 Diffusivity, 368n.

Legendre polynomials, 344, 384 expansion in, 346 Legendre’s équation, 342, 384 Leibnitz’s rule (see Différentiation, under intégral sign)

Laplace’s, 369, 382, 385, 386, 439

Tangent line, 143, 147, 151 Tangent plane, 146-149 tanh, 249

(See also Simple harmonie motion)

Arapere’s formula, 52ra.

Cylmdrical coordinates, 152, 185, 190, 191, 378, 386, 434, 438

simultaneous, 102-122, 139-141 solution of, 83-122 Systems of, 102-122

tests for convergence of, 9, 11, 12, 15, 20, 27, 31, 33

Mutually exclusive events, 497 N

fundam entai principle of, 6 limit of, 3

Coefficients, Fourier, 65 metric, 437 Cofactor, 111, 112 Combinatory analysis, fundamental principle of, 493

Direction angles, 146, 398 Direction components, 146 Direction cosines, 146, 147, 151, 188, 194, 398

Leibnitz’s test (see Test, for altemat-mg sériés)

linear, 357

Taylor’s formula, 35-46, 158 applications of, 41-46 Taylor’B sériés, 37, 464, 539 for functions of two variables, 155-158, 228

Viscous damping, 302, 366 Volume, as a triple intégral, 180 clement of, 185, 187, 190, 437 Volume intégral, 180, 415

Amplitude of coniplex number, 441 Amplitude function, 51 Analysis, harmonie, 545 Analytic functions, 451-491 Angle, as a line intégral, 195 direction, 146, 398 of lap, 240 of twist, 485 solid, 195

Direction ratios, 150, 151 Directional denvative, 143, 151, 219 (See also Gradient)

honïogeneous lmear, 119-122 non-homogeneous linear, 113-119

Gamma functions, 272-277 Gauss-Argand diagram, 440 Gaussian law of error, 520, 536 Gauss’s theorem, 193 General solution of differential équation, 230, 290, 292, 350, 358 Geomctnc sériés, 9 Gradient, V, 144, 152, 407, 410, 438 Graphical method, of curve fitting, 525

theorems on, 17, 21, 27, 28, 29, 31, 33, 34, 36, 38

N*bla, or del, V, 194, 195, 407, 414, 422

Sériés, asymptotic, 524 binomial, 40

Taylor’s theorem, 36 Tension, 239, 243, 244, 251, 362, 377 Test, Cauchy’s intégral, 12 comparison, 9 for alternating sériés, 15 for sériés, 9, 11, 12, 15, 20, 27, 31, 33

Commutative law, 394, 399, 400 Comparison test for senes, 9 Complementary function, 290, 292 Complété elliptic intégrais, 48 Compiex number, 440 absolute value of, 441 argument of, 441 conjugate of, 444, 488 vector représentation of, 440 Compiex roots of unity, 87

Length, of arc, 143 of ellipse, 47 of sine curve, 55 Level surface, 406 Limit, 2, 124, 454 Line, contour, 144 coordinate, 434

of elastic membrane, 377 of electric circuits, 386 of heat conduction, 367, 425 of vibratmg string, 361 Partial difîerentials, 128-143 Partial différentiation, 123-171 Partial fractions, method of, 297 Partial sum, 4

Angular veloeity, 61, 191, 236, 404,

Compiex variable, 440-491 functions of, 444-491 analytic, 451-491 dérivative of, 449 intégration of, 453 line intégral of, 454 Taylor’s expansion for, 464 Components of force, 217 Composite function, 134, 137 Condenser, 283, 299, 305, 308, 387 Conditionally convergent sériés, 16, 17, 21

Dam, gravity, 483 Dampmg, viscous, 30ü Dead-beat motion, 304 Décomposition of vectors, 396 Definite intégrais, 172

Dirichlet conditions, 65 Disoharge of condenser, 299 Discontinuity, fimte, 64 Discriminant of cubic, 89 Distance, élément of, 435 Distribution of charge, 487 Distribution charts, 506 Distribution turve, 504, 516 Distributive law, 399, 400 Divergence, of senes, 5, 8, 20 / of a vector, 411, 423, 438 Divergence theorem, 191, 415, 425, 428 dn u, 51

Error, Gaussian law of, 520, 536 mean, 516 mean absolute, 522 mean square, 522 of observation, 516 probable, 521 small, 56

of solution of équations, 83 Gravitational constant, 232 Gravitational law (see Attraction) Gravitational potential, 219, 408 Gravity, center of, 177, 182, 183, 187, 190, 191, 196, 522 Gravity dam, 483

umform convergence of, 23-30, 33 Inflection, point of, 159 Initial conditions, 235, 351 Intégral calculus, fundamental theorem of, 172, 457 Intégral curve, 226, 228, 279

Newtonian potential, 196 Newton’s law, of attraction, 218 of cooling, 254 of dynamics, first law, 231 second law, 231, 272, 363 third law, 231, 234 of gravitation, 232 Newton’s method of solution, 97

évaluation of intégrais by, 43—46 Fourier, 63-82 infinité, 1-62

ratio, 11, 20, 31 Weierstrass M, 27 Theory of lcast squares, 521 Thermodynamies, 222 Torque, 405

Intégral équation, 347 Intégral formula, Cauchy’s, 461 Intégral test for sériés, 12 Intégral theorem, Cauchy’s, 455 Intégrais, around closed curve, 201, 203, 206, 216, 421 change of variable in, 183-188 defimte, 172

direction cosines of, 146, 147, 151 normal, 144, 146-149 of equal potential, 277 of flow, 475 stream, 277, 432, 467 tangent, 143, 147, 151 vector équation of, 395 Line intégrais, 197-224, 410, 421, 454

Normal, to a curve, 144 to a plane, 146, 147 to a surface, 147, 188, 407 Normal approximation, 515 Normal denvative, 144, 146, 152 (See also Gradient)

Particular intégral, 290, 292, 297, 318, 359

Walhs’s formula, 45 Wave équation, 432 Wcdge, 473 Weierstrass test, 27 Work, 217, 404 Wronskian, 317

424

Conditions, Cauchy-Riemann, 221, 450, 455 Dirichlet, 65

change of variable in, 183-188 évaluation of, 172 mean-value theorem for, 210n. Deflection, 299

Dot product, 399

Error function, 516

Green’s theorem, for the plane, 202 in space, 191, 418 symmetric form of, 194, 418

Line intégrais, applications of, 217-224

intégration and différentiation of, 29, 33, 34

Total dérivatives, 130-143 Total differential, 127-143 Trajectories, orthogonal, 277-279 Transformation, by analytic fonctions, 467

Euler équation, 322 Euler formulas, 78, 251 Euler’s équations, 430 Euler’s theorem, 136 Evaluation of intégrais, by différentiation, 169 m sériés, 43-46 Even function, 68 Events, dépendent, 495 indépendant, 495 mutually exclusive, 497 Exact difïerential, 211, 212, 216, 222, 224,262,411,418,420 Exact difïerential équation, 262 Expansion, m Bessel functions, 339 m Fourier sériés, 65-82 m Legendre polynomials, 346 in Maclaurm’s sériés, 37 in power sériés, 37-46 in Taylor’s sériés, 37 m trigonométrie sériés, 65 uniqueness of, 38 Expectation, 500

double, 173, 192, 202, 275 elliptic, 47-55, 238 évaluation of, by means of sériés, 43-46

Normal distribution curve, 516 Normal équations, 537, 540 Normal form, 146

Particular solution, 230 Path, intégrais independent of, 208, 216, 452, 455

Sériés, of constants, 6-22 of functions, 23-62 power, 30-62

Applications, of conformai représentation, 479-491 of lme intégrais, 217-224 of scalar and vector produets, 404-406

for exact differential, 212, 216 Conductivity, 367, 426 Conductor, 486, 489 Conformai mappmg, 465, 471 Conformai représentation, applications of, 479-491 Conformai transformation, 467 Conjugate of a compiex number, 444, 488

Degree of differential équation, 225 Del, V (see Nabla)

Double intégrais, 173, 192, 202, 275 Drying of porous solids, 369 Dynamics, laws of, 231

around a closed curve, 202, 206, 216, 421

conformai, 467 Green’s, 191, 202, 418 of element of arc, 467 of intégrais, 202 Schwartz, 478, 485, 491 Stokes’s, 421 Trapézoïdal rule, 556 Trials, rcpeated and independent, 501

Expected number of successes, 508 Exponential form for trigonométrie functions, 78, 251, 446, 447 Exponential function, expansion for, 42, 446

iterated, 175, 180 line, 197-224, 410, 421, 454, 458 mean-value theorem for, 210n. multiple, 172-196 particular, 290, 292, 297, 318 surface, 188-196, 415, 421 transformation of (see Green’s theorem; Stokes’s theorem) triple, 177, 193 volume, 180, 415 witli a parameter, 47, 167 Integrating factor, 265 Intégration, by parts, 276 numerieal, 554—560 of complex functions, 453 of sériés, 29, 33, 34, 80 term by term, 33, 34, 80 Interpolation, method of, 101 Interpolation formulas, 550-554 Interval, 4

Normal law (see Gaussian law of error)

Pendulum, simple, 44, 234-238, 306 Periodic function, 64 Picard’s method, 347 Plane, équation of, 147 mclined, 280, 282, 306 normal form for, 146 tangent, 146-149 Point, of inflection, 159 smgular, 451 Poisson formula, 512 Polar coordinates, 183,184, 276, 279, 386, 438

solution of differential équations by, 228, 325-346

Zéro vector, 393

Approxnnate formula, for n\, 509 for probability of most probable number, 511

Conjugate functions, 468, 470 Conservation of matter, law of, 429 Conservative ficld of force, 219, 411 Consistent Systems of équations, 117-122

Delta amplitude, dn, 51 De Moivre’s theorem, 90, 442

Harmonie analysis, 545 Harmonie sériés, 8 Heat conduction, 367 Heat flow, 368-377 équation of, 368, 425 steady, 256, 368, 427 variable, 368, 373, 425 Hélix, 151, 152 Holomorphic functions, 451

définition of, 197, 454 évaluation of, 202-206, 458 for angle, 195 for area, 201 for work, 217 in space, 215, 410, 421 properties of, 206-217 transformation of, 202, 421 Linear dependence or independence, 116, 317

Trigonométrie functions, 78, 251, 446, 447

Dependence, functional, 2 linear, 116

Extremal values, 164 Extremum, 164

of convergence, 31, 33 of expansion, 38, 76 Inverse hyperbolic functions, 249, 255

Normal line, 144, 146-149 Normal orthogonal functions, 81 Numbers, complex, 440 measure, 397

Polygon, rectilinear, 478, 485 Polynomials, Legendre, 344, 384 Porous solids, drying of, 369 Potential, elcctrostatic, 487 gravitational, 219, 408 lines of equal, 277 Newtonian, 196

Taylor’s and Maclaurin’s, 37, 155, 228, 249, 464, 539 tests for convergence of, 9, 11, 12, 15, 20, 27, 31, 33

Zonal harmonies (see Legendre poly-nomials)

in applied mathematics, 55 Approximation, Laplace’s or normal,

Continuity, équations of, 221, 429, 481

e, 42 e**, 250

Linear differential équations, 288-349, 357

Trigonométrie sériés, 63-82 Triple intégrais, 177, 193

Dépendent events, 495 Dérivation of differential équations, 231-247 Dérivative, 125 directional, 143, 151, 219 normal, 144, 146, 152 of functions of a complex variable, 449, 452, 463

Inversions, 106

Numerical intégration, 554-560 Numerical solution of différentiel équations, 346

velocity, 221, 222, 277, 430, 432, 453, 467, 480

theorems on, 17, 21, 27, 28, 29, 31, 33, 34, 36, 38

515

of functions, 23, 28, 124, 448 Contour lino, 144

Effects, superposition of, 129, 223 E{k, <p), 48-51, 54 Elastic curve, 240, 307 Elasticity, 241, 422, 484-486 Electrodynamies, 422, 423n. Electron, 315

with constant coefficients, 287 357 with variable coefficients, 284, 315-349

Potential function, 219, 411 Power sériés, 30-62 différentiation of, 33, 34 évaluation of intégrais by, 43-46 expansion in, 35-46 fonctions defined by, 33 intégration of, 33, 34 interval of convergence of, 31, 33 operations on, 33-35 theorems on, 31-35 umform convergence of, 33 uniqueness of expansion in, 38 whose ternis are infinité sériés, 40 Power séries solutions of differential équations, 325-346 Précision constant, 520, 521 Pressure on dam, 484 Primitive, 458

of hyperbolic functions, 255 of senes, 29, 33 partial, 125-143, 153 total, 130-143 Descartes’s rule of signs, 94 Déterminants, 102-114 cofactors of, 111 expansion of, 106n., 111 functional or Jacobian, 183 Laplace development of, 111 minors of, 110 of matrix, 115 product of, 110 properties of, 107-112 solution of équations by, 102-114 Wronskian, 317 Déviation, standard, 523 Diagonal term of déterminant, 107 Diagram, pv, 223 Différences, 527

F(k, </>), 48-55 Factor, integrating, 265 Factor theorem, 92 Facto rial, n !, approximation for, 509 (See also Gamma functions) Falling body, 58, 232 Field, 406

Irrotational field, 418, 423, 430 Isolation of roots, 92 Isothermal process, 224 Iterated intégrais, 175, 180 Itération, method of, 297

uniform convergence of, 23-30 Shearing stresses, 485 Simple closed curve, 200 Simple harmonie motion, 233, 301, 314, 380 équation of, 234 penod of, 234

Approximations to binomial law, 512 Arc length, 143 of ellipse, 47

Convergence, absolute, 16, 17, 20, 21, 33

Electrostatie field, 475, 477, 479 Electrostatic force, 487 Electrostatie potential, 487 Electrostatics, 486-491 Elément, of arc, 467 of area, 184, 190, 437 of distance, 435 of volume, 185, 187, 190, 437 Elcmentary functions, 315 expansion of, 35-46, 65-82, 465 Ellipse, area of, 177, 202 center of gravity of, 177 length of arc of, 47 Ellipsoidal coordinates, 433 Elliptic functions, 51

Lmear differential operator, 287-299 Log z, 446

Principal part of incrément, 128 Probability, 492-524 Probability curvo, 521 Probable error, 521 Probable value, most, 505 probability of, 511 Product, of déterminants, 110 scalar, 399 vector, 400

Undetermined coefficients, 229 Uniform convergence, 23-30 test for, 27

Arc length, of sinusoid, 55 Area, 172

Differential, exact, 211, 212, 216, 222, 224, 262, 411, 418, 420 of area, 184, 190 of volume, 185, 187, 190 partial, 128-143 total, 127-143

conservative, 411 electrostatic, 475, 477, 479 irrotational, 418 Finite discontinuity, 64 Fittmg, curve, 525-560 Flexure, 298

Odd function, 68 Operator, 528

Simple pendulum, 44, 234-238, 306 Simply connected région, 205 Simpson’s rule, 556 Simultaneous differential équations, 312-315

conditional, 16, 17, 21 interval of, 31, 33 of sériés, 4, 7

Logarithmic paper, 526 M

Projection, Mercator’s, 479 stereographic, 479 Pulloy, slipping of belt on, 239 pv diagram, 223

Unit vectors, 394, 397 Unity, roots of, 87

as a double intégral, 178 as a lme intégral, 199-202 element of, 183, 184, 190, 437 positive and négative, 200 surface, 188-196

Differential équations, 225-391 Bernoulli’s, 286 Bessel’s, 332, 380 Cauchy-Riemann, 221, 450, 455 Cauchy’s, 322n. définition of, 225 degree of, 225 dérivation of, 231-247 Euler’s, 322, 430 exact, 262 first order, 256, 267 Fourier, 425

Flow, of a liquid, 220, 424, 428, 477,

Jn(x), 336

différentiel, 287-299, 357, 406 vector (see Curl; Divergence; Gradient; Nabla)

Simultaneous équations, 102-122, 139-141 sin x, 41, 250 siir ' x, 46

tests for, 9, 11, 12, 15, 20, 27, 31, 33

M test, 27

Argument of complex number, 441 Associative law, for sériés, 18 for vectors, 394

, 478,480-484

Jacobian, 141, 183, 190

Order of differential équation, 225 Ordinary differential équations, 225— • ' 349

Sine, hyperbolic, 247 length of curve, 55 power sériés for, 40, 41 Sme senes, 73 Singular point, 451 Smgular solution, 279 Singulanties of funetion, 222 sinh x, 247

radius of, 31, 33 umform, 23-30, 33 Cooling, law of, 254 Coordinate lines, 434 Coordinate surfaces, 434 Coordinates, curvilinear, 433-439 cylindrical, 152,185,190, 191, 378, 386, 434, 438

Maclaurin formula, 36 Maclaurin’s sériés, 37, 249 Magnitude of a vector, 393 Map, géographie, 479 of a curve, 466 Mapping functions, 467 Matrix, 114-122 augmented, 118 déterminants of, 115 rank of, 115

Quadrature, meehanical, 554 Quotient, of complex numbers, 444 of power sériés, 40

Variable, change of, 154, 183-188 complex, 440-491 dépendent, 2 independent, 1

Asymptotic formula for ■«', 509 Asymptotic séries, 524 Atmosphère, thickness of, 61 Attraction, law of, 218, 232 motion under, 58, 218 of cône, 196 of eyhnder, 196 of sphere, 196, 232 Augmented matrix, 118 Auxiliary équation, 292 Averages, method of, 534 Axes, right- or left-handed, 397

(See also Differential équations) Ordinary discontinuity, 64 Origin of a vector, 393 Orthogonal curves, 277, 468 Orthogonal functions, 81, 339, 345 Orthogonal Systems, 434 ^Orthogonal trajectories, 277-279 Orthogonal vectors, 398 Oscillation of a spring, 299 Oscillatory motion, 304 Overdamped, 303

Sink (see Source and sink) Six-ordmate scheme, 548 Slipping of belt on pulley, 239 Small numbers, law of, 512 sn u, 51

Maxima and minima, constramed, 163

Variable heat flow, 368, 373, 425 Variation, of constants (see Variation, of parameters)

Solenoidal field, 423 Solid angle, 195 Solids, drying of porous, 369 Solution, of cubic, 86-91

for functions of one variable, 158 for functions of several variables, 160

Radius of convergence, 31, 33 Radius vector, 195 Rank of matrix, 115 Ratio test, 11, 20, 31 Reaction, Chemical, 258 Rearrangement of sériés, 17 Rectilinear polygon, 478, 485 Recursion formula, 273, 328, 3

Base vectors, 396 Beam, 240-242, 307 Belt on pulley, slippmg of, 239 Bendmg moment, 241 Bernoulli-Euler law, 241, 307 Bernoulli’s équation, 286 Bessel functions, 273, 336, 381 expansion in, 339 Bessel’s équation, 332, 380 Beta function, 276 Binomial law, 502

p sériés, 10 Parabole, 244 Parabolic coordinates, 439 Paraboloid, hyperbolic, 162 Parachute, 253, 255

of differential équations, 226, 228, 325

Mean error, 516, 522 Mean-value theorems, 210n.

approximations to, 512 Binomial sériés, 40 Biot and Savart, law of, 52 Boundary conditions, 242, 351, 363, 370

general, 230, 290, 292, 350, 358 particular, 230

Measure numbers, 397 Mechanical quadrature, 554 Membrane, vibration of, 377 Mercator’s projection, 479 Metric coefficients, 437

Buckling, 299