676 INDEX

sum, 29geometric series, 154gradient, 452gradient

contravariant components, 625covariant components, 625

Gram Schmidt process, 422graph of function, 50greatest lower bound, 32Green’s theorem, 595, 604gronwall’s inequality, 185

half life, 76hanging chain, 370harmonic, 456heat equation, 456Heine Borel, 90Heine Borel theorem, 328Hessian matrix, 490, 504Holder continuous, 326Holder’s inequality, 187homogeneous coordinates, 397hyperbolic functions, 73, 74

image, 398implicit differentiation, 131implicit function theorem, 509, 512improper integral, 236improper Riemann integral, 230inconsistent, 380increment of volume

increment of area, 550independent, 400index

lowering, 615raising, 615

infinite seriesraised to a power, 256

infinite sumsproperties, 154

inner product, 285integral

continuous function, 175decreasing function, 643definition, 179uniform convergence, 177

integration by parts, 176, 195intercepts, 305

interestcompounded continuously, 150

interior point, 90, 310intermediate value theorem, 103, 104interval of convergence, 244inverse, 390

left inverse, 441right inverse, 441

inverse function theorem, 511, 512inverse image, 49inverses and determinants, 440invertible, 390iterated integral, 521, 660iterated integrals, 238, 660

Jacobian determinant, 550Jensens inequality, 668joule, 290

Kepler’s first law, 357Kepler’s laws, 357Kepler’s third law, 359kilogram, 298kinetic energy, 620Kroneker delta, 301

L’Hopital’s rule, 146L’Hopitals rule, 148Lagrange multipliers, 498, 513, 514Lagrange remainder, 143, 144, 503Lagrangian coordinates, 580Lagrangian formalism, 621Laplace expansion, 438Laplace transform, 233

obvious properties, 233Laplacian, 456

polar coordinates, 481Laplacian

general curvilinear coordinates, 627law of cosines, 58least squares regression, 457least upper bound, 32Lebesgue integral

desires to be linear, 647nonnegative function, 644simple function, 645

Lebesgue number, 328length of smooth curve, 338Leontief model, 397