360 INDEX

inverse function theorem, 166, 168inverse image, 51inverses and determinants, 42invertible, 23iterated integrals, 127

Jordan curve theorem, 322Jordan separation theorem, 323

kinetic energy, 346Kroneker delta, 334

Lagrange multipliers, 168, 169Lagrangian formalism, 347Laplace expansion, 40Laplacian

general curvilinear coordinates, 353least squares regression, 161Lebesgue

points, 239Lebesgue integral

computing them, 234desires to be linear, 206nonnegative function, 202other definitions, 205simple function, 203

Lebesgue measurable functionapproximation with Borel measurable,

237Lebesgue measure

approximation with Borel sets, 237properties, 237

Lebesgue number, 69, 89left inverse, 32, 33lim inf, 57

properties, 59lim sup, 57

properties, 59lim sup and lim inf, 211limit

continuity, 143infinite limits, 141point, 63

limit of a function, 141limit of a sequence, 64

well defined, 64limit point, 141limits

combinations of functions, 141

existence of limits, 58limits and continuity, 143Lindeloff property, 68linear combination, 24, 38, 92linear functional

positive, 193linear independence, 95linear map of measurable set, 242linear maps, 9linear relationship, 24linear relationships

row operations, 25linear space, 91linear transformation

defined on a basis, 113dimension of vector space, 113rank m, 172

linear transformationsa vector space, 113sum, 113

linearly dependent, 92linearly independent, 92linearly independent set

enlarging to a basis, 95Liouville, 296Lipschitz

continuous, 74Lipschitz functions, 262

of measurable sets, 241little o notation, 144local maximum, 171local minimum, 171locally finite, 252logarithm

branches, 299lower semicontinuous, 89Lyapunov Schmidt procedure, 177

manifolddifferentiable, 264measure, 266orientable, 264smooth, 264

manifold boundarywell defined, 262

manifolds, 261matrix

left inverse, 43