Index 395

in two variables, 256

local, 254

of a function, 30

Mean Value Theorem, 90

for the integral, 112

in several variables, 253

metric space, 161, 168, 169

complete, 172

M¨ obius band, 352

monotone

function, 67

sequence, 46

Monotone Convergence Theorem, 46

monotonicity and the derivative, 91

multiplication

by scalars, 162

in Q, 17

in R, 23

multiplicative

identity, 16

inverse, 17

natural domain, 59

natural logarithm, 121

natural numbers, 8

negatively oriented, 349

2-cell, 341

neighborhood, 174

nested sequence

of intervals, 51

of sets, 180

non-decreasing

function, 91

sequence, 46

non-differentiable function, 234

non-increasing

function, 91

sequence, 46

non-singular matrix, 213

norm

Euclidean, 164

general, 166

sup, 205

normal vector, 352

normed vector space, 166, 169

one-to-one, 5, 260

onto, 5

open

ball, 174

cover, 179

map, 262

relatively, 184, 197

set, 174

Open Mapping Theorem, 262

operations

on linear functions, 210

on matrices, 210

operator norm, 212

order

linear, 383

partial, 383

total, 383

well, 383

order relation

in N, 12

on Q, 18

on R, 23

ordered basis, 341

ordered field, 18

of rational numbers, 18

orientable surface, 352, 363

orientation, 350

negative, 351

of a smooth surface, 352

positive, 351

preserving, 323

reversing, 323

origin, 162

orthogonal vectors, 165

outer volume, 283

pairwise disjoint, 5

parallelogram law, 167

parameter

change, 323, 343, 349

independence, 324, 325, 349, 353

interval, 194, 317

parameterization

smooth, 246

parameterized

by arc length, 327

curve, 194

surface, 195, 246, 348

parametric equations, 218

partial derivative, 223

equality of mixed partials, 226

of order 2, 224

total degree, 225

partial sum of a series, 129, 206

partially ordered set, 383

partition

of a rectangle, 276

of an interval, 101