INDEX 675

expansion along row (column), 438matrix inverse formula, 417, 441minor, 410product, 414, 437row operations, 413transpose, 435

determinant rankrow rank, 440

difference quotient, 332differentiable, 459, 462differential equations, 620differentiation rules, 123, 335dimension, 401directed line segment, 276direction vector, 276directional derivative, 450, 482Dirichlet function, 47Dirichlet integral, 670Dirichlet test, 162discriminant, 43distance, 54, 196distance formula, 277divergence, 567divergence, 626

general curvilinear coordinates, 627divergence theorem, 572domain, 47dominated convergence theorem, 649donut, 560dot product, 285

geometric description, 287double series

absolute convergence, 165interchange order of summation, 165

double suminterchange of order, 165

dual basis, 613dual basis, 618Dynkin’s lemma, 651

eigenvalue, 501eigenvalues, 442eigenvectors, 408Einstein summation convention, 302elementary matrices, 381elementary matrix, 403

inverse, 384properties, 384

epigraph, 114equal area rule, 357equality of mixed partial derivatives, 455Euclidean algorithm, 29Euler’s number, 72Eulerian coordinates, 580exponential growth, 232exponential growth and decay, 74extreme value theorem, 103

Fatou’s lemma, 647Fibonacci sequence, 79Fick’s law, 483, 591field axioms, 16, 38finite intersection property, 94

compact sets, 95first derivative test, 141focus, 282force

on a dam, 221force field, 356, 515Foucalt pendulum, 366Fourier law of heat conduction, 483Frenet Serret formulas, 350Frobinius norm, 430frustum of a cone, 217Fubini’s theorem, 653Fubini’s theorem., 238function

even, 135odd, 135uniformly continuous, 107

fundamental matrix, 611group property, 611

fundamental theorem line integrals, 605fundamental theorem of algebra, 42, 266,

325fundamental theorem of algebra

plausibility argument, 267future value of an annuity, 30

Gamma function, 668gamma function

existence and convergence, 230properties, 231

Gauss’s theorem, 572geometric mean, 502geometric series, 154