358 INDEX

completely separable, 67complex valued measurable functions, 207components of a vector, 93connected, 80

open balls, 82, 131connected component, 81

boundary, 302connected components, 81

equivalence class, 81equivalence relation, 81open sets, 81

connected setcontinuous function, 83, 132continuous image, 80

connected setsintersection, 80intervals, 81real line, 81union, 80

continuityalgebraic properties, 116bounded linear maps, 114coordinate maps, 115uniform, 74

continuous function, 72maximum and minimum, 74

continuous functionscompact support, 212equivalent conditions, 72

continuous image of compact set, 73contraction map, 74

fixed point theorem, 74convergence in measure, 232convex

sets, 132convex combination, 108convex hull, 108convolution, 251coordinate map, 102countable, 53counting zeros

analytic functions, 300cowlicks, 317Cramer’s rule, 43cross product, 329

area of parallelogram, 329coordinate description, 330geometric description, 329

curl, 336

definition of a Ck function, 152derivative

chain rule, 146continuity, 151continuity of Gateaux derivative, 151continuous, 145continuous Gateaux derivatives, 149Frechet, 144Gateaux, 147, 149generalized partial, 155higher order, 151, 152matrix, 147partial, 155second, 151well defined, 144

derivative of inverse, 229derivatives, 144determinant

definition, 35expansion along row, column, 41matrix inverse, 42permutation of rows, 36product, 39row, column operations, 37symmetric definition, 37transpose, 37

diameter of a set, 88differentiable, 144

continuous, 145continuous partials, 155

differentiable map of Lebesgue measurableset, 244

differential equations, 346dimension of a vector space, 94directional derivative, 149distance, 63distribution function, 217divergence, 336divergence, 352

general curvilinear coordinates, 353divergence theorem, 268, 270dominated convergence

generalization, 212dominated convergence theorem, 211dot product, 95dual basis, 339