15
	16
	17
	18
			295
75	equations with Separable Variables		. 257
77	Exact Differential Equations		262
74	First-order Differential Equations		256
	(See also Differential equations) 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
157.	* Precision Constant. Probable Error		521
	. 478, 480-484
	06a
	06b
	07a
	07b
	08a
	08b
	09a
	09b
105	105. Elimination of Arbitrary Functions.	.	....
	10a
	10b
	11a
	11b
129	129 Physical Interpretation of Divergence and Curl		423
	12a
	12b
	13a
	13b
	14a
	14b
96	96. Variation of Parameters	.	. 318
97	97. The Euler Equation		322
	A
168.	A More General Formula		558
86.	A Non-linear Equation Reducible to Linear Form (Bernoulli’s
86.	A Non-linear Equation Reducible to Linear Form (Bernoulli’s
	Absolute convergence of series, lb, 17, 20, 21
	Absolute value of complex number, 441
117.	Addition and Subtraction of Vectors		393
441	Addition, of series, 21 of vectors,
	Addition, of series, 21 of vectors, 393 parallelogram law of, 394 Adiabiftic process, 224 Aerodynamics, 133, 431 Algebra, fundamental theorem of, 92 Algebraic theorems, 92-94 Alternating series, 15 am i/, 51
394	Adiabiftic process, 224 Aerodynamics, 133, 431 Algebra, fundamental theorem of, 92 Algebraic theorems, 92-94 Alternating series, 15 am i/, 51
6	Algebra of Series	.	21
26.	Algebraic Solution of Cubic		86
4.	Alternating Series		15
	Amplitude of complex number, 441 Amplitude function, 51 Analysis, harmonic, 545 Analytic functions, 451-491 Angle, as a line integral, 195 direction, 146, 398 of lap, 240 of twist, 485 solid, 195
	Angular velocity, 61, 191, 236, 404, 424
	Answers	. .	561
12.	Application of Taylor’s Formula		41
	Applications of Conformal Representation.	. .	479
121.	Applications of Scalar and Vector Products		404
	Applications, of conformal representation, 479-491 of line integrals, 217-224 of scalar arid vector products, 404-406
	Approximate formula, for n!, 509 for probability of most probable number, 511
16	Approximate Formulas in Applied Mathematics		55
155.	Approximations to Binomial Law		512
	Approximations to binomial law, 512 Arc length, 143 of ellipse, 47
	approximations to, 512 Binomial series, 40 Biot and Savart, law of, 52 Boundary conditions, 242, 351, 363, 370
	Arc length, of sinusoid, 55 Area, 172
61.	Area of a Closed Curve		199
	Argument of complex number, 441 Associative law, for series, 18 for vectors, 394
	around a closed curve, 202, 206, 216, 421
	as a double integral, 178 as a line integral, 190-202 element of, 183, 184, 190, 437 positive and negative, 200 surface, 188-196
	Asymptotic formula for a1, 509 Asymptotic senes, 524 Atmosphere, thickness of, 61 Attraction, law of, 218, 232 motion under, 58, 218 of cone, 196 of cylinder, 196 of sphere, 196, 232 Augmented matrix, 118 Auxiliary equation, 292 Averages, method of, 534 Axes, right- or left-handed, 397
	B
	Base vectors, 396 Beam, 240-242, 307 Belt on pulley, slipping of, 239 Bending moment, 241 Bernoulli-Euler law, 241, 307 Bernoulli’s equation, 286 Bessel functions, 273, 336, 381 expansion in, 339 Bessel's equation, 332, 380 Beta function, 27*6 Binomial law, 502
100.	Bessel’s Equation		332
	Buckling, 299
	c, 42 c'xy 250
	Cable, flexible, 244 flow of electricity in, 386 supporting horizontal roadway, 242
	Cartography, 479 Catenary, 247, 252.
	Cauchy- Riemann equations, 221, 450, 455
	Cauchy’s equation, 322n.
140.	Cauchy’s Integral Formula . .		461
	Cauchy’s integral formula, 401 Cauchy’s integral test, 12 Cauchy’s integral theorem, 455 Center of gravity, 177, 182, 183, 187, 190, 191, 196, 522
137.	Cauchy’s Integral Theorem		455
137.	Cauchy’s Integral Theorem		455
	Change of variables, in derivatives, 154
	Chapter 1
	Chapter II FOURIER SERIES
	Chapter II FOURIER SERIES
	Chapter III
	Chapter III
	Chapter IV
	Chapter IV
	Chapter IX VECTOR ANALYSIS
	Chapter IX VECTOR ANALYSIS
	Chapter V
	Chapter V
	Chapter VI LINE INTEGRAL
	Chapter VI LINE INTEGRAL
	Chapter VII
	Chapter VII
	Chapter VIII
	Chapter VIII
	Chapter X
	Chapter X
	Chapter XI PROBABILITY
	Chapter XI PROBABILITY
	Chapter XI PROBABILITY
	Chapter XII
	Chapter XII
	Chapter XII
	Coefficients, Fourier, 65 metric, 437 Cofactor, 111, 112 Combinatory analysis, fundamental principle of, 493
	Commutative law, 394, 399, 400 Comparison test for series, 9 Complementary function, 290, 292 Complete elliptic integrals, 48 Complex number, 440 absolute value of, 441 argument of, 441 conjugate of, 444, 488 vector representation of, 440 Complex roots of unity, 87
22.	Complex Form of Fourier Senes		78
133.	Complex Numbers .		440
	COMPLEX VARIABLE
	COMPLEX VARIABLE
	Complex variable, 440-491 functions of, 444-491 analytic, 451-491 derivative of, 449 integration of, 453 line integral of, 454 Taylor’s expansion for, 464 Components of force, 217 Composite function, 134, 137 Condenser, 283, 299, 305, 308, 387 Conditionally convergent series, 16, 17, 21
	conditional, 16, 17, 21 interval of, 31, 33 of series, 4, 7
	Conditions, Cauchy-Riemann, 221, 450, 455 Dirichlet, 65
142.	Conformal Mapping ....		465
	Conjugate functions, 468, 470 Conservation of matter, law of, 429 Conservative field of force, 219, 411 Consistent systems of equations, 117-122
	conservative, 411 electrostatic, 475, 477, 479 irrotational, 418 Finite discontinuity, 64 Fitting, curve, 525-560 Flexure, 298
35	Consistent and Inconsistent Systems of Equations		117
	consistent, 117-122 dependent, 105 differential, 225-391 Euler’s, 430
161.	Constants Determined by Method of averages		534
50.	Constrained Maxima and Minima		163
7.	Continuity of Functions Uniform Convergence		23
	Continuity, equations of, 221, 429, 481
	Convergence, absolute, 16, 17, 20, 21, 33
	Coordinates, ellipsoidal, 433 parabolic, 439
	cos x, 46, 250 cosh, 247
	Cosine, hyperbolic, 247 power series for, 38, 40 Cosine series, 73
	Cosines, direction, 146, 147, 151, 188, 194, 398 coth, 249 Cramer’s rule, 113 Cross product, 400 Cubic equation, algebraic solution of,#86
126.	Curl of a Vector		418
132.	Curvilinear Coordinates		433
	Cylindrical coordinates, 152, 185, 190, 191, 378, 386, 434, 438
	D
	Dam, gravity, 483 Damping, viscous, 302*
	Dead-beat motion, 304 Decomposition of vectors, 396 Definite integrals, 172 change of variable m, 183-188 evaluation of, 172 mean-value theorem for, 21071. Deflection, 299
118.	Decomposition of Vectors. Base Vectors		396
52.	Definition and Evaluation of the Double Integral		173
60.	Definition of Line Integral		197
	definition of, 197, 454 evaluation 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
	Degree of differential equation, 225 Del, V (see Nabla)
	Delta amplitude, dn, 51 De Moivre’s theorem, 90, 442
	Dependence, functional, 2 linear, 116
	Dependent events, 495 Derivation of differential equations, 231-247 Derivative, 125 directional, 143, 151, 219 normal, 144, 146, 152 of functions of a complex variable, 449, 452, 463
	derivation of, 351
31.	Determinants of the nth Order.		100
31.	Determinants of the nth Order.		106
30.	Determinants of the Second and Third Order		102
159.	Differences		527
80.	Differential Equations of the Second Order		269
	Differential equations, 225-391 Bernoulli’s, 286 Bessel’s, 332, 380 Cauchy-Ricmann, 221, 450, 455 Cauchy’s, 322n. definition of, 225 degree of, 225 derivation of, 231-247 Euler’s, 322, 430 exact, 262 first order, 256, 267 Fourier, 425
	Differential equations, general solution of, 230, 290, 292, 350, 358 homogeneous, 259, 2G1 homogeneous linear, 290 integral curve of, 22(5, 228 integrating factor of, 205 Laplace’s, 309, 382, 385, 386, 439, 451, 470, 481 Legendre’s, 342, 384 linear, 226, 283-349, 357 numerical solution of, 346 of electric circuits, 301, 305, 386 of heat conduction, 367 of membrane, 377 of vibrating spring, 308 of vibrating string, 361 order of, 225 ordinary, 225-349 partial, 225, 350-391 particular integral of, 290, 292, 297, 318, 359 particular solution of, 230 second order, 269, 295 separation of variables m, 257 simultaneous, 312-315 singular solution of, 279 solution m series, 228, 325, 349, 364
	Differential expression, 225 Differential operators, 287-299, 357, 406
122.	Differential Operators		406
	differential, 287-299, 357, 406 vector (see Curl; Divergence;
	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
23.	Differentiation and Integration of Fourier Scries		80
41.	Differentiation of Implicit Functions .		137
51.	Differentiation under the Integral Sign		167
	Differentiation, of implicit functions, 132-142
18.	Dinchlct Conditions. Derivation of Fourier Coefficients .... 65
18	Dinchlet Conditions. Derivation fourier coeficients		65
	Direction angles, 146, 398 Direction components, 146 Direction cosines, 146, 147, 151, 188, 194, 398
	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 equation of, 395 Line integrals, 197-224, 410, 421, 454
	Direction ratios, 150, 151 Directional derivative, 143, 151, 219 (See also Gradient)
42.	Directional Derivatives		143
45.	Directional Derivatives in Space		151
	Dirichlet conditions, 65 Discharge of condenser, 299 Discontinuity, finite, 64 Discriminant of cubic, 89 Distance, element 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 uf 51
15	Discussion of Elliptic Integrals	.	48
15	Discussion of Elliptic Integrals	.	. . 48
152.	Distribution Curve		504
152.	Distribution Curve		504
124.	Divergence of a Vector		411
125.	Divergence Theorem		415
	Dot product, 399
	Double integrals, 173, 192, 202, 275 Drying of porous solids, 369 Dynamics, laws of, 231
	E
	Effects, superposition of, 129, 223 E{k, <p), 48-51, 54 Elastic curve, 240, 307 Elasticity, 241, 422, 484-486 Electrodynamics, 422, 423n. Electron, 315
	Electrostatic field, 475, 477, 479 Electrostatic force, 487 Electrostatic potential, 487 Electrostatics, 486-491 Element, of arc, 467 of area, 184, 190, 437 of distance, 435 of volume, 185, 187, 190, 437 Elementary 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
134.	Elementary Functions of a Complex Variable .		444
105.	Elimination of Arbitrary Functions.		.
	Elliptic integrals, 47-55 complete, 48
	EMPIRICAL FORMULAS AND CURVE FITTING
	EMPIRICAL FORMULAS AND CURVE FITTING
	EMPIRICAL FORMULAS AND CURVE FITTING
130.	Equation of Heat Flow		425
	Equation)	.	286
	Equation)	.	. 286
131.	Equations of Hydrodynamics		428
79.	Equations of the First Order in Which One of the Variables Does
79	Equations of the First Order m Which One of the Variables Does
	Equations with Separable Variables		257
	Equations, Cauchy-Uiemann, 221, 450, 455
	Error function, 516
	Error, Gaussian law of, 520, 536 mean, 516 mean absolute, 522 mean square, 522 of observation, 516 probable, 521 small, 56
	Euler equation, 322 Euler formulas, 78, 251 Euler’s equations, 430 Euler’s theorem, 136 Evaluation of integrals, by differentiation, 169 m series, 43-46 Even function, 68 Events, dependent, 495 independent, 495 mutually exclusive, 497 Exact differential, 211, 212, 216, 222, 224, 262, 411, 418, 420 Exact differential equation, 262 Expansion, in Bessel functions, 339 in Fourier series, 65-82 m Legendre polynomials, 346 in Maclaunil’s series, 37 in power series, 37-46 in Taylor’s scries, 37 m trigonometric series, 65 uniqueness of, 38 Expectation, 500
40.	Euler's Formula		130
13	Evaluation of Definite Integrals by Means of Power Series ...		43
	evaluation of integrals by, 43-46 Fourier, 63-82 infinite, 1-62
145	Examples of Conformal Maps .		471
99.	Existence of Power Series Solutions		329
101.	Expansion in Series of Bessel Functions		339
19	Expansion of Functions in Fourier series		67
11.	Expansion of Functions in Power Series		35
150.	Expectation. .		500
	Expected number of successes, 508 Exponential form for trigonometric functions, 78, 251, 446, 447 Exponential function, expansion for, 42, 446
138.	Extension of Cauchy’s Theorem		455
138.	Extension of Cauchy’s Theorem		455
21.	Extension of Interval of Expansion		76
	Extremal values, 164 Extremum, 164
	F
	F(k, <?), 48-55 Factor, integrating, 265 Factor theorem, 92 Factonal, n!, approximation for, 509 (See also Gamma functions) Falling body, 58, 232 Field, 406
	first kind, F(k, *>), 48-55, 238 second kind, E(k, <*>), 48-54 third kind, II(n, k} <p), 50 Empirical formulas, 525-500 Entropy, 224 Envelope, 279 Equation, auxiliary, 292 Bernoulli’s, 286 Bessel’s, 332, 380 characteristic, 292 cubic, 80 Euler, 322 Fourier, 425 indicia!, 334 integral, 347
115.	Flow of Electricity in a Cable . .		386
	Flow, of a liquid, 220, 424, 428, 477,
	for exact differential, 212, 216 Conductivity, 367, 426 Conductor, 486, 489 Conformal mapping, 465, 471 Conformal representation, applications of, 479-491 Conformal transformation, 467 Conjugate of a complex number, 444, 488
	for functions of one variable, 158 for functions of several variables, 160
92.	Forced Vibrations .		308
109.	Fourier Series Solution .		364
	Fourier, 425
36	Functions of Several Variables		123
1.	Fundamental Concepts		1
147.	Fundamental Notions		492
147	Fundamental Notions		492
	fundamental principle of, 6 limit of, 3
72	Further Examples of Derivation of Differential Equations		239
81.	Gamma Functions		272
	general, 230, 290, 292, 350, 358 particular, 230
53.	Geometric Interpretation of the Double Integral		177
	geometric interpretation of, 177 Multiplication, of complex numbers, 442
	Gradient; Nabla)
158.	Graphical Method .		525
	graphical solution of, 83 Curl, 418, 422, 423, 438 Current, 386, 427 Curve, distribution, 504, 516 clastic, 240, 307 map of, 466 Curve fitting, 525-560 Curves, integral, 226, 228, 279 orthogonal, 277, 468 Curvilinear coordinates, 433-439 Cylinder functions (see Bessel functions)
25.	Graphical Solutions		83
62.	Green’s Theorem for the Plane		202
58.	Green’s Theorem in Space		191
164.	Harmonic Analysis		545
110.	Heat Conduction		367
89.	'Hie Meaning of the Operator
46.	Higher Partial Derivatives		153
	hofnogeneous linear, 119-122 non-homogeneous linear, 113-119
	Homogeneous Differential equations		259
76	Homogeneous Differential Equations		. 259
28.	Horner’s Method		95
73	Hyperbolic Functions		247
66.	Illustrations of the Application of the Line Integrals
66.	Illustrations of the Application of the Line Integrals .		. 217
	in applied mathematics, 55 Approximation, Laplace’s or normal, 515
	in integrals, 183-188 Characteristic equation, 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 integral around, 201, 203, 206, 216, 421 simple, 200 cn w, 51
	inconsistent, 105, 117-122 normal, 537, 540
	Index	.	575
	Index		575
	INFINITE SERIES
78	Integrating Factors		265
	integration and differentiation of, 29, 33, 34
136.	Integration of Complex Functions ....		453
106.	Integration of Partial Differential Equations.		353
	integration of, 353
165.	Interpolation Formulas		550
13	J.3. Evaluation of Definite Integrals by Means of Power Series ... 43
55.	Jacobians. Change of Variable		183
166.	Lagrange’s Interpolation Formula		552
	Lagrange’s interpolation formula, 552
	Lagrange’s method of multipliers, 163-167
	Lamellar field, 423 Laplace’s approximation, 515 Laplace’s equation, 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
114.	Laplace’s Equation		382
	Laplace’s, 195, 309, 382, 385, 380, 439, 451, 470, 481 Legendre’s, 342, 384 of continuity, 221, 429, 481 of plane, 147 wave, 432
	Laplace's, 369, 382, 385, 386, 439
	Legendre polynomials, 344, 384 expansion in, 346 • Legendre’s equation, 342, 384 Leibnitz’s rule (see Differentiation, under integral sign)
102.	Legendre’s Equation		342
	Leibnitz’s test (see Test, for alternating series) length, of arc, 143 of ellipse, 47 of sine curve, 55 Level surface, 406 Limit, 2, 124, 454 Line, contour, 144 coordinate, 434
65.	Line Integrals in Space		215
	Line integrals, applic&ions of, 217-224
84.	Linear Differential equations of the first order		283
87	Linear Differential Equations of the nth Order		287
	Linear differential equations, 283-349, 357
	Linear differential operator, 287-299 Ivog z, 446
85.	Linear Equations of the First Order . .		284
95.	Linear Equations with Variable Coefficients		.... 315
95	linear Equations with Variable Coefficients .		315
107.	Linear Partial Differential Equations with Constant Coefficients		357
107.	Linear Partial Differential Equations with Constant coefficients
	linear, 357
	logarithmic paper, 526
	M
	M test, 27
	Maclaurin formula, 36 Maclaurin’s scries, 37, 249 Magnitude of a vector, 393 Map, geographic, 479 of a curve, 466 Mapping functions, 467 Matrix, 114-122 augmented, 118 determinants of, 115 rank of, 115
34.	Matrices and Linear Dependence		114
48.	Maxima and Minima of Functions of One Variable		158
49.	Maxima and Minima of Functions of Several variables		160
	Maxima and minima, constrained, 163
	Mean error, 516, 522 Mean-value theorems, 210n. Measure numbers, 397 Mechanical quadrature, 554 Membrane, vibration of, 377 Mercator’s projection, 479 Metric coefficients, 437
143.	Method of Conjugate Functions . .		467
162.	Method of Least Squares		536
163.	Method of Moments		544
	Minima (see Maxima and minima) Minimax, 162
33.	Minors .		110
	Modulus, of complex number, 441, 442
	Motion, dead-beat, 304 fluid, 220 1aws of, 231, 234 of a membrane, 377 oscijlatory, 304 pendulum, 48, 234 simple harmonic, 233, 301, 314, 380
	MULTIPLE INTEGRALS
119.	Multiplication of Vectors		399
	MULTIPLK INTEGRALS
64.	Multiply Connected Regions		212
149.	Mutually Exclusive Events		497
	Mutually exclusive events, 497 N
	Nabla, or del, V, 194, 195, 407, 414, 422
29.	Newton’s Method . .		97
69.	Newtonian Laws		231
	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
	Normal distribution curve, 516 Normal equations, 537, 540 Normal form, 146
	Normal law (see Gaussian law of error)
	Normal line, 144, 146-149 Normal orthogonal functions, 81 Numbers, complex, 440 measure, 397
	Normal, to a curve, 144 to a plane, 146, 147 to a surface, 147, 188, 407 Normal approximation, 515 Normal derivative, 144, 146, 152 (See also Gradient)
	Not Occur Explicitly		267
167.	Numerical Integration		554
	Numerical integration, 554-560 Numerical solution of differential equations, 346
103.	Numerical Solution of Differential Equations		346
	O
	Odd function, 68 Operator, 528
	of determinants, 110 of series, 21 of vectors, 399 Multiplicity of root, 93, 294 Multiplier, Lagrangian, 165 Multiply connected region, 205, 212, 455
	of differential equations, 226, 228, 325
	of elastic membrane, 377 of electric circuits, 386 of heat conduction, 367, 425 of vibrating string, 361 Partial differentials, 128-143 Partial differentiation, 123-171 Partial fractions, method of, 297 Partial sum, 4
	of elliptic function, fc, 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
	of functions, 23, 28, 124, 448 Contour line, 144
	of hyperbolic functions, 255 of series, 29, 33 partial, 125-143, 153 total, 130-143 Descartes’s rule of signs, 94 Determinants, 102-114 cofactors of, 111 expansion of, 106rc., Ill functional or Jacobian, 183 Laplace development of, 111 minors of, 110 of matrix, 115 product of, 110 properties of, 107-112 solution of equations by, 102-114 Wronskian, 317 Deviation, standard, 523 Diagonal term of determinant, 107 Diagram, pv, 223 Differences, 527
	of integration, 173 simply connected, 205 Regula falsi, 101 Regular functions, 451 Remainder in Taylor’s series, 36-37 Remainder theorem, 92 Repeated trials, 501 Representation, applications of conformal, 479-491 Residuals, 534, 537 Resonance, 310 Riemann surface, 473 Right-handed system of axes, 397 Rod, flow of heat in, 373 vibitltions of, 366, 367 Roots, of equations, 83-102 isolation of, 92 theorems on, 92-94 of unity, w, o>2, 87 Rot (see Curl)
	of series, 29, 33, 34, 80 partial, 123-171 term by term, 33, 34, 80 under integral sign, 167 Diffusion, 369, 427 Diffusivity, 368w.
	Order of differential equation, 225 Ordinary differential equations, 225-* ' 349
	ORDINARY DIFFERENTIAL EQUATIONS
	ORDINARY DIFFERENTIAL EQUATIONS
24.	Orthogonal Functions		81
82.	Orthogonal Trajectories		277
90.	Oscillation of a Spring and Discharge of a Condenser		299
	P
	p series, 10 Parabola, 244 Parabolic coordinates, 439 Paraboloid, hyperbolic, 162 Parachute, 253, 255
Section	Page
	Pao*
	Parallelogram law of addition, 394 Parameters, 277, 280 integrals containing, 167 variation of, 318
393	parallelogram law of,
	Parametric equations, 143, 149, 150, 199, 215, 247
	parametric, 143, 149, 150, 199, 215 representing special types of data, 528
37	Partial Derivatives		125
37	Partial Derivatives		. 125
	Partial derivatives, 125-143, 153 Partial differential equation, 350-391
	PARTIAL DIFFERENTIAL EQUATIONS
	PARTIAL DIFFERENTIAL EQUATIONS
	PARTIAL DIFFERENTIATION
	PARTIAL DIFFERENTIATION
	Particular integral, 290, 292, 297, 318, 359
	Particular solution, 230 Path, integrals independent of, 208, 216, 452, 455
	Pendulum, simple, 44, 234-238, 306 Periodic function, 64 Picard’s method, 347 Plane, equation of, 147 inclined, 280, 282, 306 normal form for, 146 tangent, 146-149 Point, of inflection, 159 singular, 451 Poisson formula, 512 Polar coordinates, 183,184, 276, 279, 386, 438
	polar, 183, 184, 276, 279, 386, 438 spherical, 152, 185, 382, 386, 434, 439
	Polygon, rectilinear, 478, 485 Polynomials, Legendre, 344, 384 Porous solids, drying of, 369 Potential, electrostatic, 487 gravitational, 219, 408 lines of equal, 277 Newtonian, 196
	Potential function, 219, 411 Power series, 30-62 differentiation of, 33, 34 evaluation of integrals by, 43-46 expansion in, 35-^16 functions defined by, 33 integration of, 33, 34 interval of convergence of, 31, 33 operations on, 33-35 theorems on, 31-35 uniform convergence of, 33 uniqueness of expansion in, 38 whose terms are infinite series, 40 Power series solutions of differential equations, 325-346 Precision constant, 520, 521 Pressure on dam, 484 Primitive, 458
9.	Power Series		30
	Preface		v
	Preface
17	Preliminary Remarks		63
67.	Preliminary Remarks		225
104.	Preliminary Remarks .		350
	Principal part of increment, 128 Probability, 492-524 Probability curve, 521 Probable error, 521 Probable value, most, 505 probability of, 511 Product, of determinants, 110 scalar, 399 vector, 400
154.	Probability of the Most Probable Number .		511
154.	Probability of the Most Probable Number .		511
144.	Problems Solvable by Conjugate Functions		470
	Projection, Mercator’s, 479 stereographic, 479 Pulley, slipping of belt on, 239 pv diagram, 223
32.	Properties of Determinants . . .		107
135.	Properties of Functions of a Complex Variable		448
63.	Properties of Line Integrals		206
10.	Properties of Power Series		33
8.	Properties of Uniformly Convergent Senes		28
	Q
	Quadrature, mechanical, 554 Quotient, of complex numbers, 444 of power series, 40
	R
	Radius of convergence, 31, 33 Radius vector, 195 Rank of matrix, 115 Ratio test, 11, 20, 31 Reaction, chemical, 258 Rearrangement of series, 17 Rectilinear polygon, 478, 485 Recursion formula, 273, 328, 331, 334
	radius of, 31, 33 uniform, 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
14	Rectification of Ellipse. Elliptic Integrals		.47
14	Rectification of Ellipse. Elliptic. Integrals		47
	Region, multiply connected, 205, 212, 455
120.	Relations between Scalar and Vector Products		402
68.	Remarks on Solutions		227
151.	Repeated and Independent Trials		501
151.	Repeated and Independent Trials		501
93.	Resonance		310
	Rotational field, 418 Rule, Cramer’s, 113 Simpson’s, 556 trapezoidal, J>56
	S
	Sbction
	Scalar field, 406, 408 Scalar point function, 406, 418 Scalar product, 399 application of, 404 Scalars, 392
116.	Scalars and Vectors		392
	Schwartz transformation, 478, 485, 491
	Section
	section
	Section
	Section
	Seepage flow, 483 Separation of variables, 257 Sequences, 2
2.	Series of Constants		6
5.	Series of Positive and Negative Terms		16
3.	Series of Positive Terms		9
	Series, asymptotic, 524 binomial, 40
	Series, of constants, 6-22 of functions, 23-62 power, 30-62
70.	Simple Harmonic Motion		233
71	Simple Pendulum	. .	234
	Simple pendulum, 44, 234-238, 306 Simply connected region, 205 Simpson’s rule, 556 Simultaneous differential equations, 312-315
94.	Simultaneous Differential equations		312
94.	Simultaneous Differential Equations		. . 312
	Simultaneous equations, 102-122, 139-141 sin xt 41, 250 sin-1 x, 46
	simultaneous, 102-122, 139-141 solution of, 83-122 systems of, 102-122
20	Sine and Cosine Series		73
	Sine, hyperbolic, 247 length of curve, 55 power series for, 40, 41 Sme series, 73 Singular point, 451 Singular solution, 279 Singularities of function, 222 smh x, 247
83	Singular Solutions	...	279
	Sink {see Source and sink) Six-ordinate scheme, 548 Slipping of belt on pulley, 239 Small numbers, law of, 512
	sn u, 51
	Solenoidal field, 423 Solid angle, 195 Solids, drying of porous, 369 Solution, of cubic, 86-91
98.	Solution in Series		325
	solution of differential equations by, 228, 325-346
	SOLUTION OF EQUATIONS
	SOLUTION OF EQUATIONS
	solution of, 226
27.	Some Algebraic Theorems		92
88	Some General Theorems		291
88	Some General Theorems	. .	.291
44.	Space Curves		149
56.	Spherical and Cylindrical coordinates		185
111.	Steady Heat Flow . .		369
127.	Stokes’s Theorem		421
57.	Surface Integrals		188
57.	Surface Integrals		.188
59.	Symmetrical Form of Green’s Theorem		194
43.	Tangent Plane and Normal Line to a Surface		146
	Taylor’s and Maclaurin’s, 37, 155, 228, 249, 464, 539 tests for convergence of, 9, 11, 12, 15, 20, 27, 31, 33
141.	Taylor’s Expansion. .		464
47.	Taylor’s Scries for Functions of Two Variables		155
	tests for, 9, 11, 12, 15, 20, 27, 31, 33
139.	The Fundamental Theorem of Integral Calculus		457
89	The Meaning of the Operator
	theorems on, 17, 21, 27, 28, 29, 31, 33, 34, 36, 38
156.	Tho Error Function ...		516
39	Total Derivatives		. 130
38	Total Differential		127
38	Total Differential		i27
108.	Transverse Vibration of Elastic String		361
108.	Transverse Vibration of Elastic String .		.
54.	Triple Integrals		179
128.	Two Important Theorems		422
	under gravity, 232 Multiple integrals, 172-196 definition and evaluation of, 173, 179
	uniform convergence of, 23-30 Shearing stresses, 485 Simple dosed curve, 200 Simple harmonic motion, 233, 301, 314, 380 equation of, 234 period of, 234
112.	Variable Heat Flow ...		373
112.	Variable Heat Flow ...		. . 373
96.	Variation of Parameters		318
123.	Vector Fields		409
	velocity, 221, 222, 277, 430, 432, 453, 467, 480
113.	Vibration of a Membrane		377
	vii
91.	Viscous Damping		302
	with constant coefficients, 287 367 with variable coefficients, 284, 315-349
	with constant coefficients, 287-315, 357
	with variable coefficients, 284, 315-349
	xi
			322
			357