692 INDEX

sum, 29geometric series, 162gradient, 464gradient

contravariant components, 639covariant components, 639

Gram Schmidt process, 434graph of function, 50greatest lower bound, 32Green’s theorem, 609, 618gronwall’s inequality, 193

half life, 76hanging chain, 380harmonic, 468heat equation, 468Heine Borel, 92Heine Borel theorem, 336Hessian matrix, 502, 516Holder continuous, 334Holder’s inequality, 195homogeneous coordinates, 409hyperbolic functions, 74

image, 410implicit differentiation, 137implicit function theorem, 521, 524improper integral, 244improper Riemann integral, 238inconsistent, 392increment of volume

increment of area, 564independent, 412index

lowering, 629raising, 629

infinite seriesraised to a power, 264

infinite sumsproperties, 162

inner product, 293integral

continuous function, 183decreasing function, 657definition, 187uniform convergence, 185

integration by parts, 184, 203intercepts, 313

interestcompounded continuously, 157

interior point, 92, 318intermediate value theorem, 108, 109interval of convergence, 252inverse, 402

left inverse, 453right inverse, 453

inverse function theorem, 523, 524inverse image, 49inverses and determinants, 452invertible, 402iterated integral, 533, 674iterated integrals, 246, 674

Jacobian determinant, 564Jensens inequality, 682joule, 298

Kepler’s first law, 368Kepler’s laws, 367Kepler’s third law, 370kilogram, 306kinetic energy, 634Kroneker delta, 309

L’Hopital’s rule, 153L’Hopitals rule, 155Lagrange multipliers, 511, 525, 526Lagrange remainder, 149, 151, 515Lagrangian coordinates, 594Lagrangian formalism, 635Laplace expansion, 450Laplace transform, 241

obvious properties, 241Laplacian, 468

polar coordinates, 493Laplacian

general curvilinear coordinates, 641law of cosines, 58least squares regression, 469least upper bound, 32Lebesgue integral

desires to be linear, 661nonnegative function, 658simple function, 659

Lebesgue number, 336length of smooth curve, 346Leontief model, 409