396 Index

refinement, 103

path, 318

equivalence, 324

piecewise linear, 186

simple, 322

simple closed, 322

Peano’s axioms, 8

piecewise linear path, 186

piecewise smooth curve, 318

planes in Rp, 219

pointwise convergence, 74, 203

polar change of coordinates, 264

polynomial, 59

positive definite matrix, 254

2 × 2 case, 256

positively oriented, 349

2-cell, 341

3-cell, 360

power function, 60

power series, 148

coeﬃcients, 148, 154

continuity of sum, 150

derivative, 151

integration, 150

interval of convergence, 149

radius of convergence, 149

Taylor series, 154

prime numbers, 12

product of series, 144

proper subset, 2

properties of N, 10

radius of convergence, 149

range of a function, 5

rank

of a linear transformation, 216

of a matrix, 216

ratio for a geometric series, 131

ratio test, 137

rational function, 60

rational number field, 18

defects of, 19

rational number system, 17

real numbers, 21

rearrangement of a series, 141, 143

rectangle

aligned, 275

recursion relation, 9

reduced form, 366

of a 0-chain, 367

of a p-chain, 370

refinement of a partition, 103, 278

relatively

closed, 184

open, 184, 197

relatively prime, 20

remainder

Lagrange form, 157

Taylor’s formula, 155

Riemann integral, 104, 279

Riemann sum, 101, 277

root test, 136

row reduction, 304

Russell’s paradox, 382

saddle point, 256

sawtooth function, 60

scalar, 162

scalar multiplication, 162

scale matrices, 304

Schr¨ oder-Bernstein Theorem, 377

separated, 184

separation theorem, 182

sequence, 8

convergent, 36

in

Rd,

169

of functions, 73, 202

of partial sums, 130

of real numbers, 34

of statements, 8

pointwise convergent, 74

uniformly Cauchy, 77, 204

uniformly convergent, 74

series, 129

p-series, 135

absolute convergence, 132

alternating, 140

alternating series test, 140

comparison test, 132

conditional convergence, 140

convergence, 129

divergent, 129

Fourier, 148

geometric, 131

harmonic, 130, 136

integral test, 135

integration of, 147

of functions, 146

partial sum, 129, 206

power, 148

product, 144

ratio test, 137

rearrangement, 141

root test, 136