392 Index

curve

branch of, 245

closed, 194

degenerate, 195

derivative of, 244, 317

parameterized, 194

piecewise smooth, 318

smooth, 318

tangent line, 244

trace of, 318

cut number, 22

cycle

1-cycle, 367

p-cycle, 370

equivalence, 372

cylinder, 196

decreasing function, 67, 91

Dedekind cut, 21

degenerate

curve, 195

rectangle, 276

surface, 195

dependent variable notation, 240

derivative

and monotonicity, 91

and uniform continuity, 92

bounded, 92

definition, 84

directional, 243

of a power series, 151

of a smooth curve, 317

of an inverse function, 87

of curve, 244

theorems, 85

vanishing, 91

diameter of a set, 181

differentiable, 84

conditions for, 233

equivalent definition, 236

for vector functions, 230

differential

of a 1-form, 332

of a 2-form, 334

of the inner product, 239

differential form, 317

1-form, 320

2-form, 332

3-form, 334

p-form, 335

closed, 335

exact, 335

higher-order, 331

transformation laws, 335

differential matrix, 231, 232

direction, 243

directional derivative, 243

disjoint sets, 5

distance

in R, 33

in

Rd,

164

in a metric space, 168

distributive law, 16

for scalar multiplication, 162

divergent series, 129

division algorithm, 15

domain of a function, 5, 59, 191

dominated by a set, 376

elementary function, 60

elementary matrices, 304

elements of a set, 2

empty set, 3

endpoints of a curve, 194

equivalent

cycles, 372

paths, 324

Euclidean

inner product, 163

metric, 169

space, 161

exact differential form, 335

exponent laws, 123

exponential function, 60, 122

extended real number system, 27

exterior differential, 332, 334, 335

exterior product, 332

faces

of a p-cell, 370

of a cube, 359

factors, 12

field, 17

field of rational numbers, 17

finite subcover, 179

fixed point, 93

fluid flow, 363

flux, 354

form

1-form, 320

2-form, 332

3-form, 334

p-form, 335

Fourier coeﬃcients, 119