INDEX 691

expansion along row (column), 450matrix inverse formula, 429, 453minor, 422product, 426, 449row operations, 425transpose, 447

determinant rankrow rank, 452

difference quotient, 340differentiable, 471, 474differential equations, 634differentiation rules, 129, 343dimension, 413directed line segment, 284direction vector, 284directional derivative, 462, 493Dirichlet function, 47Dirichlet integral, 684Dirichlet test, 170discriminant, 43distance, 54, 204distance formula, 285divergence, 581divergence, 640

general curvilinear coordinates, 641divergence theorem, 586domain, 47dominated convergence theorem, 663donut, 574dot product, 293

geometric description, 295double series

absolute convergence, 173interchange order of summation, 173

double suminterchange of order, 173

dual basis, 627dual basis, 632Dynkin’s lemma, 665

eigenvalue, 513eigenvalues, 454eigenvectors, 420Einstein summation convention, 310elementary matrices, 393elementary matrix, 415

inverse, 396properties, 396

epigraph, 119equal area rule, 367equality of mixed partial derivatives, 468Euclidean algorithm, 29Euler’s number, 72Eulerian coordinates, 595exponential growth, 240exponential growth and decay, 75extreme value theorem, 107

Fatou’s lemma, 661Fibonacci sequence, 81Fick’s law, 495, 605field axioms, 16, 38finite intersection property, 97

compact sets, 98first derivative test, 147focus, 290force

on a dam, 229force field, 366, 527Foucalt pendulum, 377Fourier law of heat conduction, 495Frenet Serret formulas, 359Frobinius norm, 442frustum of a cone, 225Fubini’s theorem, 667Fubini’s theorem., 246function

even, 141odd, 141uniformly continuous, 112

fundamental matrix, 625group property, 625

fundamental theorem line integrals, 619fundamental theorem of algebra, 42, 274,

333fundamental theorem of algebra

plausibility argument, 275future value of an annuity, 30

Gamma function, 682gamma function

existence and convergence, 238properties, 239

Gauss’s theorem, 586geometric mean, 514geometric series, 162