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Table of Contents
                            Advanced Calculus for Applications
Preface
Contents
CHAPTER 1 Ordinary Differential Equations
CHAPTER 2 The Laplace Transform
CHAPTER 3 Numerical Methods for Solving Ordinary Differential Equations
CHAPTER 4 Series Solutions of Differential Equations: Special Functions
CHAPTER 5 Boundary-Value Problems and Characteristic-Function Representations
CHAPTER 6 Vector Analysis
CHAPTER 7 Topics in Higher-Dimensional Calculus
CHAPTER 8 Partial Differential Equations
CHAPTER 9 Solutions of Partial Differential equations of Mathematical Physics
CHAPTER 10 Functions of a Complex Variable
Appendix A
Appendix B
Answers to Problems
Index
                        
Document Text Contents
Page 2

FRANCIS B. HILDEBRAND

Associate Professor of Mathematics
Massachusetts Institute of Technology

Advanced Calculus

for Applications

PRENTICE-HALL, INC.

Englewood Cliffs. New Jersey 1962

Page 328

Problems 317

20. Find the volume of the tetrahedronwith vertices at the points (0,0,0),
(1,1,1),(2,1,1),and(l ,2,1).

21. Determinea unit vector in the planeof the vectorsi + j and j + k and
perpendicularto thevectori + j + k.

22. If u is a unit vector,prove that u x (a x u) is the vectorprojectionof a on
a planeperpendicularto u.

23. Provethata x (b x c) + b x (c x a) + c x (a x b) = O.
24. (a) Provethat

I (a b d)c - (a b c)d,
(a x b) x (c x d) = t(a c d)b - (b c d)a.

(To obtain the first form, write temporarilyu = a x b.)
(b) Show that this vector is in the direction of the intersectionof a plane

including thevectorsa andb with oneincludingc andd.

Section6.S

25. If F is a function of I, find thederivativeof

dF d2F
F·-x-.

dl dt2

26. At time t, thevectorfrom the origin to a moving point is

r = a coswt + b sin wt,

wherea, b, and CJ) areconstants.
dr

(a) Find the velocity v = d and prove that r x v is constant,so that the
curvetracedout lies in a plane. t

(b) Show that the accelerationis directed toward the origin and is pro-
portional to thedistancefrom theorigin.

27. Let r representthevectorfrom a fixed origin 0 to a movingparticleof mass
m, subjectto a force F. dr

(a) If H denotesthemomentof themomentumvectormv = m- about0,
provethat dl

dH d
dt = m dt (r x v) = r x F = M,

whereM is themomentof the force F aboutO.
(b) If the force F alwayspassesthroughthe fixed point 0, show that

d
-(rxv)=O
dt '

andhencededucethat r x v = h, whereh is a constantvector. Deducealso that
themotion is in a plane.

Page 329

318 Vector IIlU11ys;s I chap. 6

(c) Showthat Ir x vI is twice theareaover which thevectorr would sweep
in unit time if v wereconstant,andhencededucethat, whena massat P is subject

to a Ucentral" force, which alwayspassesthrougha fixed point 0, the vectorOP
movesin a planeandsweepsoverequalareasin equaltimes.

28. If u is a unit vector originating at a fixed point 0, and rotating about a
fixed vectorC&) through0, with angularvelocity of constantmagnitude00, showthat

du
dt = C&) x u.

29. Let r = x i + Y j + z k representthepositionvectorfrom a fixed origin 0
to a point P, andsupposethat thexyz axis systemis rotatingabouta fixed vector
C&) through0, with angularvelocity of constantmagnitudeoo.

dr di
(a) By calculatingdt' andnoticingthatdt = C&) x i, andsoforth, obtainthe

velocity vectorin the form
v = vo + C&) x r,

wherethe vector
dx dy dz

vo = - i + - j + - k
dt dt dt

is thevelocity vectorwhich would beobtainedif theaxeswerefixed.
(b) Obtain the accelerationvectorin the form

8 = 8 0 + 2C&) x vo +C&) x (C&) x r),
whereVo is definedin part (a), and where

d2x . d2y . d2z
80 = dt2 I + dt2 J + dt2 k.

30. If the systemof Problem29 is rotating with angularvelocity of constant
magnitude00 about the z axis, the z axis being fixed, show that the equationsof
motion for a point massm areof the form

m(d2x_ 200 dy _ W 2x) - F
dt2 dt - z,

m(d2y ..L' 200 dx _ oo2y) - F
dt2 dt - y.

d2z
m dt2 '"- Fz

whereFx • Fy , and Fz are thecomponentsof the externalforce alongtherespective
rotatingaxes.(Notice that the masshencebehavesas though the axeswerefixed,

with an additional force 2moody + moo2x acting in the positivex direction andan
dx dt

additionalforce -2moodt + mw2y actingin the positivey direction.)

Page 656

Index

Semiconvergentseries, 171
Separabledifferential equation, 33, 36.

433
Separationconstant,433
Separationof variables, I, 428, 430
Series:

asymptotic,170
Fourier (see Fourier series)
Fourier-Bessel,226
Frobenius,129
geometric, 166
hypergeometric,165
power, 119
semiconvergent,171
Taylor, 94, 122, 348

Shaft, rotating, 193
Shearingforce, 194, 320 (38)
Shearlines, 412
Simple pole, 548
Simple region, 284
Simply connectedregion, 284
Simpson'sRule, 117 (25), 118 (26, 27)
Simultaneousdifferential equations,19

function, 32
Single-valuedfunction. 283, 521
Singular curve, on integral surface,

406
Singularities,of analytic functions:

branchpoint, 132. 516, 535, 540
essential,540, 541
at infinity. 542
isolated, 538
pole, 132. 538
removable,538
significanceof. 545

Singularity functions, 63
Singular point, of differential equation,

127, 168
Singularsolution, 5. 34
Sink, 291
Sliding clampedend support. 195
Smoothcurve, 281
Smoothsurface.288
Solution, of differential equation.

complementary.7, 388
complete.4
homogeneou'i,7
particular,6. 388
singular. 5. 34

Sonic velocity. 311

645

Source,291. 569
strengthof, 569

Sourcefunction, in heat flow. 50I (71),
503 (75)

Spacecurve, 271
Specific heat, 428
Speed,273
Spherical coordinates,304
Sphericalwaves,455
Springconstant,68
Square-wavefunction, 83 (9)
Stability, 199
Stagnationpoint, 490 (38), 572
Standardform, of linear ordinary differ-

ential equation:
flrst order, 4
second order, 127

Steadystate,429
Stirling's formula, 81, 90 (40)
Stodola-Vianello method, 200. 214
Stokes'stheorem,295
Straight line, equationsof, 315 (5)
Streamfunction, 307, 480 (7), 566
Streaming,572
Streamline,306, 480 (7), 566
Stressfunction, 427
Stretching,of plane. 605 (95)
String:

deflection of. 189
rotating, 189
vibrating, 244 (7)

Strip, 401
characteristic,404
proper. 403

Strip condition, 40I
Sturm-Liouvi1le problem. 208, 373 (40)

proper,209
Substantialderivative, 310
Superpositionintegral, 451
Supersonicfluid flow. 312. 475
Surface:

conical, 384
cylindrical, 383
equationof, 287
equipotential,286
normal to, 287
Riemann.537

Surfacearea.288, 300, 370 (21)
Surfacecharge.584. 615 (/21)
Surfaceintegral, 287

Page 657

646

TABLE:

of Bessel functions of order n + lh.
77

of Gamma function, 619
of Laplacetransforms.74
of zerosof Bessel functions, 619

Tangentialacceleration,274
Tangentvector, 272
Taylor series,94, 122, 348, 528

with remainder, 123. 349
uniquenessof, 590 (34)

Telegraphequations.427, 464
Temperature,steadystate:

inside circle, 436
outsidecircle. 437
in circular annulus,433
in circular cylinder. 487 (30-32), 488

(34)

in half-plane, 460
in rectangle,430
in rectangularparallelepiped.441, 488

(33)
in rod. 450
in sphere,439

Tension, 189, 320 (38). 446
Thermalconductivity, 428
Thermal diffusivity. 429
Torsion. 274

radiusof, 274
Torsional stiffness, 320 (38)
Total differential. 335
Transform:

Fourier, 240
Fourier-Be'\l\el, 242
Laplace. 51

Transientsolution, heat flow. 450
Translation.of axes. 329 (86). 605 (95)
Translation properties,of Laplace tranl\-

forms, 59
Traveling waves.454
Triangular-wavefunction. 83 (10)
Trigonometricfunctionl\. 512
Triple scalar product. 268
Twisting moment. 120 (38)

Index

UNDE1ERMINED coefficients, method of,
11, 19

Unit doublet function. 65
Unit impulsefunction, 64
Unit step function, 87 (27)
Unit vector. 263

VARIATION, 356
of parameters,25

Vectors, 262
algebraof. 262
differentiation of. 270, 278
productsof, 265

Velocity. 273
Velocity deviation potential, 31 t
Velocity potential, 305, 566
Vianello (see Stodola-Vianello method)
Vibrating beam, 244 (10)
Vibrating membrane,446
Vibrating string, 244 (7)
Vibration. 68

natural modesof. 70
Viscosity. coefficient of. 334 (105)
Viscous flu id. 334 (105)
Volume, elementof. 300, 346
Vortex. 305, 570

strengthof, 570

WAVE equation.327 (78), 391. 427. 446.
464

Wavel\. 454
standing, 496 (58)

Weber'l\ function. 146
Weighting function. 206
Work, 281
Wronskiandeterminant.4,27,29.31,41

(7), 179 (29), 370 (20)

YOUNG'S modulu'i. 194

ZfROS. of Bes'iel functions. 154. 6 t7
table of, 619

7erevector. 261

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