533

An n following a page reference indicates the information is found in a note.

2 × 2 matrix, 31, 63

3 × 3 matrix, 31–32, 63

ε’s and δ’s limits, 99–102

I_n, 66

n-dimensional Euclidian space, 60

n-space vectors, 60–62

ℝⁿ, 2

x axis, 1

x coordinate, 1

x-simple regions, 283, 287, 428, 430

y axis, 1

y coordinate, 2

y-simple domain, 340

y-simple regions, 283, 286, 287, 428–439

z axis, 1

z coordinate, 2

0-form, 477

1-form, 478

2-form, 478–479

3-form, 479–480

symbols, xviii

absolute maximum, 180, 192, 193

absolute minimum, 180, 192, 193

absolute value, xxiii

acceleration, 217–228

action, principle of, 166–168

action at a distance, 243, 419

additive inverse, 3

adiabatic process, 375

affine approximation, 108–109

Alexandov, 417

algebra of forms, 483–488

al-Khuwarizmi, xviii

Ampère’s law, 372, 408, 452, 472

analytic function, 166

Andromeda galaxy, 419

angle between two vectors, 22–23

angular momentum, conservation of, 450

angular velocity vector, 250

anticommutativity, 483

Apollonius of Perga, xv

approximations, 158

Arabian mathematics, xvii–xviii

Archimedes, xvi, xix, 266, 333, 389

arch length

definition, 228, 231

differential, 230–232

formula justification, 232–234

function, 232

reparametrization, 234

area

curl as circulation per unit area, 445–448

Green’s theorem, 433–434

surfaces, 383–392

Argand, 46

Aristarchus of Samus, xix

Aristotle, xvi

Ars Magna [the Great Art] (Cardano), 44

associativity, 3, 46n1, 67, 483

average value, 357

average value of a function, 329–330

Babylonian mathematics, xiii–xvii

ball, volume of, 326

basic 1-form, 478

basic 2-form, 478–479

basic 3-form, 479–480

bearing, 30

Bentley, Richard, 419

Bernoulli, Jacob, 52

Bernoulli, Johann II, 155, 167, 358, 419

best linear approximation, 110

binormal vector, 235

bonded function

definition, 271

integratability, 274

bordered Hessian determinant, 197, 198, 199

bound vectors, 6

boundaries, 90–91

boundary curve, 440

boundary points, 90, 91

boundary regions, 283

bounded set, 180

brachistrochrone, 358

Brahe, Tycho, xx

Bunyakovskii, 61n4

Buys-Ballot’s law, 262

C¹, 114

calculus of variations, 358

capped cylinder, 451

Cardano, Gerolamo, 44, 45

Cartesian coordinates, 1, 2

Cartesian product, 263

Catoptrica (Euclid), xv

534

Cauchy, Augustin-Louis, 35, 45, 61n4, 281, 390

Cauchy–Bunyakovskii–Schwarz (CBS) inequality, 61n4

Cauchy–Riemann equations, 399

Cauchy–Schwarz inequality, 23–25, 61

Cavalieri, Bonaventura, 266

Cavalieri’s principle, 265–266

CBS (Cauchy–Bunyakovskii–Schwarz) inequality, 61n4

center of gravity, 399

center of mass, 330–333

centripetal force, 221

chain rule

described, 124, 126–127

as differentiation rules, 218

example, 153, 156

first special case, 127–128

implicit function theorem and surface, 206

Lagrange multiplier method, 186

second special case, 128–132

Stokes’ theorems and, 441

vector quantities and, 448

change of variables formula

applications, 329–338

cylindrical coordinates, 324

described, 307–308, 318–320

double integrals, 319

Gaussian integral, 322–323

polar coordinates, 320–323

spherical coordinates, 325–326

triple integrals, 323–324

change of variables theorem, 314–328

changing the order of integration, 289–294

charge density, 472

chemical equation, 4

circular orbit, 220–222

circulation, 373, 446

circulation and curl, 445–448

class C¹ functions, 114

class C² functions, 150

class C^k, 237

Clifford, W. K., 351, 419

closed curve, 368

closed interval, xxiii

closed set, 180

closed surface, 452

Cobb–Douglas production function, 203

coefficients, matrix of, 195n12

commutative, 66

complex numbers, 45–48

component curves, 370

component functions, 117

components, 1, 4

component scalar fields, 237

composition, 99

conductivity, 238

cone, 379, 385, 392

conformal parametrization, 399, 423

conical refraction, 46

conic sections, xiv

conservation of angular momentum, 450

conservation of energy, 240

conservative fields

definition, 453

physical interpretation, 455–457

planar case, 458–459

conservative vector field, 453

constant multiple rule, 125

constant vector field, 491

constrained extrema, 185–203

LaGrange multiplier method for several constraints, 191–193

second derivative test, 197–201

continuity, 88–105

of compositions, 99

definition, 97

open sets and, 88–90

theorems, 113–114

continuous functions, 95–98

conversion of energy, 240

coordinates, 1–2, 4

Copernicus, Nicolaus, xvi, xix

Coulomb’s law, 239, 243, 409

Cramer, 34

Cramer’s rule, 35

Crick, Francis, 418

critical points, 168, 177, 181, 182, 186, 198

cross product, 31, 35–39, 44, 47–48

cross product rule, 218

cross-sectional area, 265–266

cross section of a torus, 391

cubic equations, 44

curl

as circulation per unit area, 445–448

definition, 249–250

divergence, 253

gradients, 252

rotational flow, 251

rotations and, 250–251

scalar curl, 252–253

curvature

definition, 355

hemisphere, 415–417

on a path, 235

planes, 415

of surfaces, 414–417

surfaces of constant, 417, 418

total curvature, 414

535

curves, 116–124

components, 370

integral of 1-forms over, 480–481

knotted, 355

line integrals over, 368–371

piecewise, 229

planar, 353–355

total curvature, 355

cyclicly permuting, 36

cyclist, 374

cycloid, 119

cycloidal path, 121

cylinder, 270, 451

cylindrical coordinates

change of variables, 324

described, 52–54

Stokes’ theorems, 448

cylindrical hole, 392

d’Alembert, Le Rond, 155

DaVinci, Leonardo, 333

definite integral, xxv

degenerate critical point, 177

degenerate type, 176

Delaunay, 417

del Ferro, Scipione, xix, 44

del operator, 245, 256

density

charge, 472

current, 472

derivative of a function, xxiv

derivative of a k-form, 484–485

derivative operator, 245

derivatives

directional, 136–137

gradients, 112–113

partial, 105–108, 111

properties of, 124–134

Descartes, René, xix, 68

determinants

geometry of, 39–41

matrix, 31–32

properties of, 32–35, 66–67

determinant test for positive definiteness, 175

Dido, Queen of Carthage, 190

Dieterici’s equation, 133

differentiability

functions of two variables, 109

general case, 110–112

tangent plane, 110

theorems, 113–114

differential equations, 154

differential forms, 476–491

0-form, 477

1-form, 478

2-form, 478–479

3-form, 479–480

algebra of forms, 483–488

cross product and, 48n3

definition, 360

Gauss’, 487

Gauss’ theorem, 487, 488

Green’s theorem, 487

integral of 1-forms over curves, 480–481

integral of 2-forms over surfaces, 481–482

integral of 3-forms over regions, 483

Stokes’ theorems, 488

differentiation, 105–116

differentiation of paths, 217–219

Dirac, Paul, xviii

directed simple curve, 368

directional derivatives, 136–137

directions of fastest increase, 137–138

Dirichlet’s functional, 399

discontinuous functions, 97

discriminant of the Hessian, 176

displacement

infinitesimal, 231

vector, 27–29

distance

definition, xxiii

from point to plane, 43–44

between vector endpoints, 21–22, 62

distributivity, 3, 483

divergence

curls, 253

cylindrical coordinates, 448

definition, 245

Gauss’ theorem, 463–466, 467, 468, 470

Green’s theorem, 436–437

Laplace operator, 254

physical interpretation, 246

spherical coordinates, 448, 470–471

divergence-free, 468

domain, xxiv

dot product, 19–20, 24, 38, 44, 48, 55, 60, 245

dot product rule, 218

double helix, 418

double integrals

bonded function, 274

Cavalieri’s principle, 265–266

change of variables formula, 319

changing the order of integration, 289–294

Fubini’s theorem, 276–280

mean value equality, 292–293

mean value inequality, 292

over a rectangle, 271–283

over elementary regions, 283–289

reduction to iterated integrals, 267–269, 285–288

536

as volumes, 263–265

doughnut surface, 375

dr notation for line integral, 371

economics, 196–197

Egyptian mathematics, xiii–xvii

eigenvalue, 203

eigenvector, 203

Einstein, Albert, 243, 418–420

Einstein’s field equations, 420, 422

Einstein’s general theory of relativity, 418

elasticity, 155, 348

electric field, 472

Electromagnetic Theory (Heaviside), 49

elementary regions

described, 297–298

double integrals over, 283–289

Gauss’ theorem, 461–463

Green’s theorem, 428

other types of, 299

symmetric, 300

triple integrals over, 298–302

Elementary Treatises on Quaternions (Tait), 47

Elements (Euclid), xv

Elements of Vector Analysis (Gibbs), 49

ellipsoid, 391, 413

elliptic, 213

endpoints, 368

energy, conversion of, 240

energy vector field, 238

epicycles, xv, 119

epicycloid, 119

equality of mixed partials, 151, 152

Equilibrium (Archimedes), 333

equipotential surfaces, 141, 239

escape velocity, 240–241

Escher, M. C., 402

Euclid, xv, xix, 236, 333

Euclidian n-space

matrices, 63–73

vectors in, 60–62

Eudoxus, xv

Euler, Leonhard, 45, 48, 76n1, 149, 152, 155, 187, 222, 390

Euler equations, 152

Euler’s theorems, 146

European mathematics, xviii–xxi

exceptional points, 453

exhausting regions, 340

extrema of real-valued functions, 166–185

extreme points, 168

extremum, 168

Faraday’s law, 407, 449–450, 472

Fary–Milnor theorems, 356

fence, Tom Sawyer’s, 354

Feynman, Richard, 222, 223–224, 427

Feynman integrals, 223

field concept, 242–243

Fields medal, 356

Fior, Antonio, 44

first-order Taylor formula, 159, 160, 164

flexural rigidity, 348

flow lines, 241–242

flux, 407, 408, 467–468

flux per unit volume, 467–468

Fontana, Nicolo, 44

force fields

gravitational, 238, 239, 240, 459

work done by, 358–359

force vectors, 29

Fourier, Joseph, 154

Fourier series, 154, 281

free vectors, 6

Frenet formulas, 235

frequency, orbit, 221

frustum, 392

Fubini, Guido, 281

Fubini’s theorem, 268, 271, 276–280, 342–344

functions

analytic, 166

arch length, 232

average value, 329–330

class C¹, 114

class C², 150

Cobb–Douglas production function, 203

component, 117

continuous, 95–98

definition, xxiv

differentiability, 109

graphs, 77

Green’s, 475

harmonic, 157

mappings and, 76–77

one-to-one, 310–311

onto, 311–313

potential, 455, 458, 475

quadratic, 172

of several variables, 76

smooth, 193

functions unbounded at isolated points, 344–345

fundamental solution, 156

fundamental theorem of algebra, 45

fundamental theorem of calculus, 159, 232, 280, 430–431, 476

fundamental theorem of integral calculus, 276

537

Galileo, 153, 358

gauge freedom, 472

Gauss, Karl Friedrich, 45, 46, 408, 413, 418, 420

Gauss–Bonnet theorem, 420–422

Gauss curvature, 414, 416, 417, 418, 420

Gaussian integral, 322–323

Gauss’ law, 408, 468–470, 472

Gauss divergence theorem, 256

Gauss’ theorem

divergence as flux per unit volume, 467–468

divergence theorem, 463–466

elementary regions and boundaries, 461–463

generalizing, 466–467

general implicit function theorem, 207–208

general second-derivative test, 176

general vector field, 236

geodesics, 420

geometric example, 195–196

La Geometrie (Descartes), 66

geometry

of determinants, 39–41

real-valued functions, 76–87

scalar multiplication, 3, 6, 42

theorems by vector methods, 11–12

vector addition, 2–4

vector subtraction, 7

geometry theorems by vector methods, 11–12

geosynchronous orbit, 222

Gibbs, Josiah Willard, 48, 49, 256–257, 258

global maximum, 180–182, 193–195

global minimum, 180–182, 193–195

gradients, 112–113, 135, 138

gradient vector field

conservative fields, 453–461

described, 140–141, 238–240

line integrals over, 366–368

graphs

cylindrical coordinates, 448

definition, xxiv

orientation, 404

real-valued functions, 77

smooth vs. nonsmooth, 105

spherical coordinates, 448

Stokes’ theorems for, 439–443

surface area, 387

surface integrals over, 394–396, 409–410, 411

gravitational constant, 453

gravitational field, escaping earth’s, 240

gravitational force fields, 238, 239, 240, 459

gravitational potential, 155, 238, 334–337

gravitational potential energy, 457

Greek mathematics, xiii–xvii

Green, George, 431

Green’s identities, 475

Green’s theorem

area of region bounded by curve, 433–434

correct orientation for boundary curves, 432

differential forms, 487

divergence theorem in the plane, 436–437

generalizing, 432–433

lemmas, 429–431

overview, 427

simple and elementary regions and boundaries, 428

vector form, 434–437

Gregory, James, 76n1

Halley, Edmund, xxi, xxii

halo orbits, 226

Hamilton, Sir William Rowan, xxii, 46–47, 48, 222, 256, 476

Hamilton’s principle, 222, 223–224

harmonic functions, 157

heat equation, 154, 155, 156

heat flux vector field, 238

Heaviside, Oliver, 48, 49

helicoid, 386, 394, 397, 417

heliocentric theory, xix

helix, 121, 352, 418

hemisphere, curvature, 415–417

Hessian, 172–175, 176

Hessian matrix, 175–176, 179, 197

higher-order approximations, 158

higher-order derivatives, 149

Hilbert, David, 422

Hilbert’s action principle, 422

Hipparchus, xv

Hölder-continuous, 105

homogeneity with respect to functions, 483

homogeneous of degree, 146

Hooke, Robert, xxi

hot-air balloon, 451

Huygens, Christian, 45, 68, 119, 390

hydrodynamic equation, 258

hyperbolic paraboloid, 80

hyperboloid, 381

hypocycloid, 119, 219, 374, 433

ideal gas law, 147

imaginary numbers, xix, 44–46

implicit function theorem, 203–215

538

improper integrals

exhausting regions, 340

as limits, 340–341

as limits of iterated integrals, 341–342

one-variable, 339, 340

in plane, 339–340

incompressible fluid, 468

Indian mathematics, xvii–xviii

induced orientation, 440, 445, 469

inequality

Cauchy–Schwartz, 23–25, 61

mean value, 292

triangle, 26–27, 62, 126, 236

infinitesimal displacement, 231

inhomogeneouswave equation, 473

inner product, 3, 19–20, 21, 24–25, 60

integer, xxiii

integral

double, 271–283

Feynman, 223

Gaussian, 322–323

improper, 339–347

iterated, 267–269, 272–276, 286, 341–342

line, 358–375, 442–443

oriented, 366

path, 351–358

Riemann, 281

scalar functions over surfaces, 393

surface, 394–396, 400–401, 406–411

topological invariant, 421

triple, 294–305, 323–324

integral curves, 241, 242

integratability, 272

integration

double integral reduction, 267–269, 285–288

triple integral reduction, 295, 296

integration by parts, 159, 160

intersection, xxiv

inverse function theorem, 208–209

invertible matrices, 66

irrational number, xxiii

irrotational vector field, 251, 455, 457

isobar, 262

isoquant, 196

isotherms, 238

iterated integrals

Fubini’s theorem for, 342–344

improper integrals as limits of iterated integrals, 341–342

properties of, 272–276

reduction of double integrals, 267–269, 285–288

iterated partial derivatives, 150–158

Jacobi, 35

Jacobian determinant, 209, 315–318, 323

Kelvin’s circulation theorem, 407

Kepler, Johannes, xx, 222

Kepler’s laws of celestial motion, xx, 153, 221

kernel, 214

knotted curve, 355

Lagrange, Joseph Louis, 35, 56

Lagrange multiplier method

constrained extrema and, 185–190

global maximum, 193–195

global minimum, 193–195

for several constraints, 191–193

Laplace, Pierre-Simon de, 35, 155

Laplace operator, 254

Laplace’s equation, 154, 155

law of cosines, 22, 61

law of planetary motion, 222

Lebesgue, Henri, 281

left-hand limit, 104

Leibniz, Gottfried Wilhelm, xx, xxi, 34, 45, 68, 167–168, 225–226, 266, 281, 419, 476

lemniscate, 327

length, vectors, 20, 21, 60

level contours, 78

level curves, 78–85

level sets, 78–85

level surface, 79, 138–139

L’Hôpital’s rule, 100

limits

boundaries, 90–91

concept of, 91–94

definition, 92, 93, 99

open sets and, 88–90

properties of, 94–95

in terms of ε’s and δ’s, 99–102

uniqueness of, 94

linear approximation, 108–109, 158, 164, 165

line integrals

definition, 359–363

differential forms, 360

dr notation, 371

of gradient field, 366–368

over curves, 368–371

reparametrization, 363–366

Stokes’ theorems, 442–443

work done by force fields, 358–359, 362

lines

dimensionality, 17

equations of, 12–17

parametrical expression, 12–17

539

passing through endpoints of two vectors, 14

point-direction form, 12

point–point form of parametric equations, 15

segment description, 16

Lipschitz-continuous, 105

Listing, J. B., 402

local extrema, 168

local maximum

definition, 168

first derivative test, 169–171

second derivative test, 171–175

second-derivative test for two variables, 176–180

local minimum

definition, 168

first derivative test, 169–171

second-derivative test, 171–175

second-derivative test for two variables, 176–180

Maclaurin, 34

magnetic field, 472

mappings, functions and, 76–77

maps

from ℝ² to ℝ², 308–314

definition, xxiv

images of, 310

Jacobian determinant, 315–318

one-to-one, 310–311

onto, 311–313

parametrized surfaces as, 376–378

Marcellus, 389

mass

center of, 330–333

density, 337

mathematics, xiii

matrices

2 × 2 matrix, 31, 63

3 × 3 matrix, 31–32, 63

coefficients, 195n12

determinants, 31–35, 39–41, 66–67

general matrices, 63–66

Hessian, 175–176, 179, 197

invertible, 66

partial derivatives, 111, 130

properties of, 66–68

triple product, 36, 46, 67

Maupertuis, Pierre-Louis de, 166, 167, 168

Maupertuis’ principle, 166–168, 222

maximum

absolute, 180–182, 192, 193

global, 180–182, 193–195

Maxwell, James Clerk, 48, 49, 256, 258

Maxwell field equations, 243, 452, 471–474

Maxwell’s equations, 155

mean curvature, 415, 417

mean-value equality, 292–293

mean-value inequality, 292

mean-value theorem, 353

Menaechmus, xiv

The Method of Fuxions and Infinite Series (Newton), 52

method of least squares, 214, 215n16

method of sections, 80–85

method of substitution, 318

Milky Way, 419

Milnor, John, 356

minimal surfaces, 423

minimum

absolute, 180–182, 192, 193

global, 180–182, 193–195

local, 168

mixed partial derivatives, 150–156

Möbius, A. F., 402

Möbius strip, 402

moment of a force, 51

moments of inertia, 333–334

momentum, 72

Muir, T., 35

multiplication, 3, 6, 44, 46, 65n5

negative, 3

negative-definite quadratic function, 173, 174, 175

negative pressure gradient, 262

neighborhood, 90, 91, 98, 113, 205

Newton, Sir Isaac, xxi–xxii, 46, 52, 222, 266, 335, 358, 413, 418, 419, 476

Newton’s law of gravitation, 141, 153, 168, 220, 238, 239, 243, 419

Newton’s mechanics, 222

Newton’s potential, 155, 158

Newton’s second law, 217–228, 240

nondegenerate critical point, 177, 179

nonsmooth graph, 105

norm of a vector, 60

normalized vectors, 21

notations, 76n1

octonians, 48n3

Oersted Hans Christian 372n6

one-dimensional wave equation, 155

one-to-one maps, 310–311

one-variable implicit function theorem, 203

On Floating Bodies (Archimedes), 333

On Growth and Form (D’Arcy), 418

On the Equilibrium and Centers of Mass of Plane Figures (Archimedes), 333

540

onto maps, xxiv, 311–313

open ball, 88

open disk, 88

open interval, xxiii

open sets, 88–90

opposite path, 363

Optics (Euclid), xv

orbit

circular, 220–222

geosynchronous, 222

halo, 226

order of integration, 289–294

ordinary differential equations, 154

orientation

graphs, 404

surfaces, 401–404

vector surface element of a sphere, 404

orientation-preserving parametrization, 403, 404

orientation-preserving reparametrization, 363, 364

orientation-reversing parametrization, 403, 404

orientation-reversing reparametrization, 363, 364

oriented integral, 366

oriented simple curve, 368

oriented surface, 401–402, 403

origin, 1

orthogonal projections, 25–26

orthogonal vectors, 24, 36

orthonormal, 58

orthonormal vectors, 24

paddle wheel, 445, 455

Pappus of Alexandria, 333

Pappus’ theorem, 392

paraboloid, 300, 302

paraboloid of revolution, 79

parallelepiped, 40–41

parallelogram

area of, 385

change of variables, 320

cross product calculation, 38

parametric description, 16, 17

parallelogram law, 69

parallel planes, 42–43

parametrized by arc length, 235

parametrized surface

definition, 377

as mappings, 376–378

regular surface, 378–379

tangent plane to, 379–381

tangent vectors, 378

parametrized surfaces

graph restrictions, 375–376

Stokes’ theorems, 444–445

surface integrals, 410–411

parametrization, 117, 309, 362, 368, 372, 378–381, 405

partial derivatives

described, 105–108

equality of mixed partials, 151–156

iterated partial, 150–158

matrix of, 111

mixed partial, 150

partial differential equations, 153–155

Pascal, Blaise, 119

path, 116–124

differentiation, 217–219

integration of secular functions over, 351–358

piecewise smooth, 229, 230

path-connected region, 305

path-independent integral, 453

path integral, 351–358

Peano, Giuseppe, 183

perpendicular vectors, 24

Philosophiae Naturalis Principia Mathematica (Newton), 335

physical applications of vectors, 27–29

piecewise curve, 229

Pierce, J. M., 49

planar curves, 353–355

Planck, Max, 222, 225

planes

curvature, 415

dimensionality, 17

equations of, 41–43

parallel, 42–43

parametric description, 17

parametrization, 375, 376

path in, 117

three coordinate planes, 17

Plato, xiv–xv, xviii

Poincaré, 226

point to plane, distance from, 43–44

Poisson’s equation, 155, 475

polar coordinates, 52, 53, 131, 320–323

polarization identity, 69

Pope, Alexander, xxi

position vector, 371

positive-definite quadratic function, 173

positive orientation, 440

potential, 475

potential equation, 155

potential function, 455, 458, 475

potential temperature, 147

Poynting vector, 475

principal normal vector, 235

541

Principia (Newton), xxi, xxii

principle of least action, 167–168, 223, 225–226

“Principles of the Motions of Fluids” (Euler), 155

product rule, 125

properties

continuous functions, 98

derivatives, 124–134

determinants, 32–35, 66–67

iterated integrals, 272–276

of limits, 94–95

triple integrals, 295–297

proper time of a path, 235

property of the unit element, 3

property of zero, 3

Ptolemaic model of planetary motion, 166

Ptolemaic theory, xv

Ptolemy of Alexandria, xv, xvi

Pythagorean theorem, 20

quadratic approximations, 158, 164, 165

quadratic equations, 44–45

quadratic functions, 172

quaternions, 46–48, 256

quotient rule, 125

radio waves, 471–474

range, xxiv

rational number, xxiii

real-valued functions

extrema, 166–185

geometry, 76–87

regular differentiable path, 219

regular partition, 271

regular surface, 378–379

relative extrema, 168, 188

relativistic triangle inequality, 236

remarkable theorems, 418

reparametrization, 363–366

restaurant plans, 412

Riemann, Bernhard, 45, 281, 399n11, 418, 420, 476

Riemann integral, 281

Riemann sum, 263–265, 269, 272, 278, 353, 385

right-hand limit, 104

right-hand rule, 37

Rodrigues, Olinde, 49

rotary vector field, 237

saddle, 80, 83

saddle point, 168, 170, 177

saddle-type critical point, 175, 176, 200

scalar curl, 252–253, 458

scalar field, 236

scalar multiplication, 3, 6, 42, 44

scalar multiplication rule, 218

scalar part, 47, 48

scalar quantity, 46

scalar-valued function, 76

Schwarz, 61n4

second-order Taylor formula, 159–160, 163, 164

sections, method of, 80–85

semimajor axis, 195

semiminor axis, 195

sets

bounded, 180

closed, 180

level, 78–85

open, 88–90

simple closed curve, 368, 369

simple curve, 368

simple regions, 283, 287

single-sheeted hyperboloid of revolution, 83

single-variable Taylor theorem, 158–160

sink, 468

slice method—Cavalieri’s principle, 266

smooth function, 193

smooth graph, 105

smooth path, 181

smooth surface, 378

Snell’s law, 51

soap bubble, 390–391, 417

soap film surfaces, 417, 423

solid ellipsoid, 347

solid of revolution, 270

solutions, existence of, 190–191

space, path in, 117

space analysis, 68

special implicit function theorem, 203–206

speed, 120, 220, 230

sphere, 411, 445

spherical coordinates

change of variables, 325–326

described, 54–58

divergence, 470–471

Stokes’ theorems, 448

standard basis vectors, 8–10, 60

steady flow, 237

Stokes’ theorems

conservative fields, 453, 455

curl as circulation per unit area, 445–448

differential forms, 488

for graphs, 439–443

parametrized surfaces, 444–445

streamlines, 241

strictly subharmonic relative, 213

542

Stokes’ theorem

Faraday’s law, 449–450

reorientation applications, 449

strong maximum principle, 439

strong minimum principle, 439

subharmonic function, 439

subset, xxiii

sum, 2

sum, Riemann, 263–265, 269, 272, 278, 353, 385

sum rule, 125, 218

superharmonic function, 439

surface area

definition, 384–385

graph surface area, 387

surfaces of revolution, 387–389

surface integrals

independence of parametrization, 405

over graphs, 394–396, 409–410

physical interpretation, 406–411

scalar integral relationship, 405–406

summary of formulas, 410–411

of vector fields, 400–413

surfaces

curvature of, 414–417

described, 78–85

implicit function theorem and, 205–206

integral of 2-forms over, 481–482

integrals of scalar functions over, 393–400

symmetric elementary regions, 300

Tait, Peter Guthrie, 47, 48

tangent line to a path, 122–123

tangent plane, 110, 139, 379–381

tangent vectors, 120, 129, 378

target, xxiv

Tartaglia, Niccolo, xix, 44, 45, 333

Tartaglia–Cardano solution, 45

Taylor series, 164

Taylor’s theorem, 158–166, 173

temperature, 147, 154, 155, 375, 412

tetrahedron, 286, 287

Thales of Miletus, xiv

theorema egregium (remarkable theorem), 418

theorems

change of variables, 314–328

Euler’s, 146

Fary–Milnor, 356

Fubini’s, 268, 271, 276–280, 342–344

fundamental theorem of algebra, 45

fundamental theorem of calculus, 159, 232, 280, 430–431, 476

fundamental theorem of integral calculus, 276

Gauss,’ 461, 463–466, 488

Gauss–Bonnet, 420–422

general implicit function, 207–208

Green’s, 428–439

implicit function, 203–215

inverse function, 208–209

Kelvin’s circulation, 407

mean-value, 353

Pappus,’ 392

Pythagorean, 20

remarkable, 418

special implicit function, 203–205

Stokes,’ 250, 407

and surfaces, 205–206

Taylor’s, 158–166, 173

The Theory of Determinants in the Historical Order of Development (Muir), 35

theory of mirrors, xv

thermodynamic path, 375

The Theodicy (Leibniz), 167

third-order Taylor formula, 163

Thomae, Karl J., 281

Thompson, D’Arcy, 418

three-body problem, 226

Tom Sawyer’s fence, 354

topological invariant, 421

torsion, 235

torus, 375, 391, 421, 476

total curvature, 355, 414

traces out, 117

trajectories, 26

transformations, xxiv

Treatise on Electricity and Magnetism (Maxwell), 48, 256

triangle inequality, 26–27, 62, 126, 236

triple, 2, 3

triple integral

change of variables formula for, 323–324

definition, 294–295

over elementary regions, 298–302

properties, 295–297

reduction to integrated integrals, 295, 296

triple product, 36, 46, 67

twice continuously differentiable, 150

unbound regions, 345

unicellular organisms, 418

union, xxiii

unit ball, 298, 328

unit cube, 462

unit disk, 283, 284

unit speed, 235

unit sphere, 392, 403

unit tangent, 235

unit vectors, 21, 58

543

Vandermonde, 35

van der Waals gas, 375

vector analysis, 68

Vector Analysis (Wilson), 49

vector fields

basic identities of vector analysis, 254–256

concept of, 236–238

conservative fields, 453–461

curl, 249–253

definition, 236

divergence, 245–248

flow lines, 241–242

general, 236

gradient, 140–141, 238–240

integral curves, 241, 242

integration of over paths, 358–375

Laplace operator, 254

rotary, 237

surface integrals, 400–413

types of, 236, 237

vector joining two points, 10–11

vector methods, geometry theorems by, 11–12

vector moment, 51

vector operations, geometry of, 4–8

vector part, 47, 48

vector product, 35, 44

vectors

addition, 2–4

bound, 6

definition, 4, 46

displacement, 27–29

force, 29

free, 6

length, 20, 21

normalized, 21

orthogonal, 24, 36

orthonormal, 24

perpendicular, 24

physical interpretation, 5

scalar multiplication, 3, 6, 42

subtraction, 7

unit, 21, 58

velocity, 28

zero, 29

vector standard basis, 8–10

vector-valued functions, 76, 217–262

velocity field V, 237

velocity vector, 28, 120, 122, 123, 129

Watson, James, 418

wave equation, 155

wedge product, 483

Weierstrass, 190–191

Wente, Henry, 417

Wessel, 46

Wiener, Norbert, 215n16

Wilson, E. B., 49, 257

Wimsey, Peter, 190

work, 30, 358–359, 362

Wren, Sir Christopher, xxi

Young’s modulus of elasticity, 348

zero element, 3

zero vector, 29

544