Index 393

Fourier series, 148

fraction, 17

Fubini’s Theorem

first version, 296

second version, 297

third version, 300

function, 5

n-homogeneous, 241

aﬃne, 217

domain, 5

graph, 7

image, 5

inverse image, 5

linear, 208

local inverse, 260

max/min, 30

non-differentiable, 234

one-to-one, 5, 260

onto, 5

smooth, 228

with singularities, 124

Fundamental Theorem of Calculus

first form, 115

for curves, 322

second form, 116

Gauss elimination, 304

Gauss’s Theorem

classical form, 363

for a 3-cell, 361

on a cube, 360

geometric series, 131

convergence, 131

gradient, 243

graph of a function, 7

gravitational force field, 365

greatest lower bound, 26

Green’s Theorem

classical version, 346

on a cell, 344

on a rectangle, 338, 339

group homomorphism, 367

harmonic series, 130, 136

Heine-Borel Theorem, 180

homogeneous function, 241

homomorphism, 367

homotopic paths, 347

identity

additive, 16

multiplicative, 16

image

of a linear transformation, 216

under a function, 5

image under a function, 5

Implicit Function Theorem, 269

improper integral, 123

increasing function, 67, 91

indeterminate forms, 93

index of summation, 131

induction

axiom, 8

for definitions, 9

for propositions, 9

step, 9

uses, 12

inductive definition

of addition, 10

of product, 14

inductive definitions, 9

inductive proof

of associative law, 11

of the commutative law, 11

infimum, 27

infinite limits, 83

of sequences, 48

infinite series, 129

infinite-dimensional vector space

basis, 385, 386

injection, 375

injective, 375

inner product, 163

space, 163, 169

inner volume, 283

integers, 17

integrable, 104, 279, 288

over a Jordan region, 290

integral, 104, 279

definition, 101

existence, 108

improper, 123

interval additivity, 112

linearity, 109

Mean Value Theorem, 112

multi-iterated, 300

of a 2-form, 339

of a 3-form, 358

of a power series, 150

of a series, 147

of a uniformly convergent sequence,

114, 291

order preserving, 111