Page 436 - Schaum's Outline of Theory and Problems of Advanced Calculus

INDEX 427

Degree, of a polynomial equation, 6 Differentiation (Cont.): Equations (Cont.):
of homogeneous fuctions, 122 rules for, 70, 78–80, 87 integral, 364, 369, 370
Del (r), 159 under the integral sign, 186, 194, polynomial, 6, 43
formulas involving, 159 203 Equipotential surfaces, 186
in curl, gradient, and divergence, Diffusivity, 357 Errors, applications to, 189, 200, 204
159 Directed line segments, 150 in computing sums of alternating
Deleted neighborhood, 6, 117 Directional derivatives, 186, 193, series, 266, 282
De Moivre’s theorem, 7, 15 202 mean square, 353
Dense, everywhere, 2 Dirichlet conditions, 337, 345 Essential singularity, 395
Denumerable set (see Countable integrals, 379, 385, 389 Eudoxus, 90
set) Dirichlet’s test: Euler, Leonhart, 336
Dependent variable, 39, 116 for integrals, 314 Euler’s, constant, 296, 378, 388
Derivatives, 65–89, 75–79 (See also for series, 270, 303 formulas or indentities, 8, 295
Differentiation) Discontinuities, 47, 119 theorem on homogeneous
chain rules for, 70, 124 removable, 56, 119 functions, 122
continuity and, 66, 72, 121, 130 Distance between points, 165 Even function, 338, 347–351
definition of, 65, 66 Distributive law, 2 Everywhere dense set, 2
directional, 186, 193, 202 for cross products, 154 Evolute, 185
evaluation of, 71, 75–89 for dot products, 153 Exact differentials, 122, 131, 132,
graphical interpretation of, 66 Divergence, 158, 159, 172–174 231 (See also Differentials)
higher order, 71, 120 in curvilinear coordinates, 161 Expansion of functions:
of functions of a complex in cylindrical coordinates, 161 in Fourier series (see Fourier
variable, 393, 400–403 of improper integrals, 306–309 series)
of infinite series of functions, 269, (See also Improper integrals) in power series, 272
400–403 of infinite series (see Infinite Expansions (see Series)
of elementary functions, 71, 78–80 series) Explicit functions, 123
of vector functions, 157, 171, 172 Divergence theorem, 236, 249–252, Exponential function, 42
partial (see Partial derivatives) 261 order, 334
right- and left-hand, 67, 77, 78, 86 proof of, 249, 250 Exponents, 3, 11
rules for finding, 70 Divergent integrals, 306–335
table of, 71 sequences, 23 (See also Sequences)
Determinant: series, 25 (See also Series) Factorial function (see Gamma
for cross product, 154 Division, 1 functions)
for curl, 159 by zero, 8 Fibonacci sequence, 35, 37
for scalar triple product, 155 of complex numbers, 6, 7 Field, 2
Jacobian (see Jacobian) Domain, of a function, 39, 116 conservative, 233
Dextral system, 152 of convergence, 272 scalar, 153
Difference equations, 65 Dot products, 153, 154, 165, 166 vector, 153
Differentiability, 66, 67, 121 commutative law for, 153 Fluid mechanics, 402
and continuity, 66, 72, 73 distributive law for, 153 Fourier coefficients, 337, 345
continuous, 66 laws for, 153, 154 expansion (see Fourier series)
piecewise, 66 Double series, 277 Fourier integrals, 363–374 (See also
Differential: Dummy variable, 94 Fourier transforms)
as a linear function, 68, 121 Duplication formula for gamma convergence of (see Fourier’s
elements of area, of volume, 160, function, 286, 378, 386 integral theorem)
163, 212, 213, 233 solution of boundary-value
Differential equation: problems by, 371
Gauss’, 276 e,4 Fourier, Joseph, 336
solution of, by Laplace Electric field vector, 181 Fourier series, 336 –362
transforms, 314, 330 Electromagnetic theory, 181 complex notation for, 339
Differential geometry, 158, 181 Elementary transcendental convergence of, 338, 354–356
Differentials, 67, 68, 69, 78, 120–122 functions, 43, 71, 95 differentiation and integration of,
approximations by use of, 78, 79, Elements, of a set, 1 339
120 Ellipse, 114 Dirichlet conditions for
exact, 122, 131, 132 area of, 114 convergence of, 337
geometric interpretation of, 68, Empty set, 1 half range, 338, 347–351
69, 121 Envelopes, 185, 186, 192 Parseval’s identity for, 310, 338,
of functions of several variables, Equality, of complex number, 6 351, 352
120, 130 of vectors, 158 solution of boundary-value
of vector functions, 156 Equations: problems by, 339, 356–358
total, 120 difference, 65 Fourier’s integral theorem, 363, 364
Differentiation (See also Derivatives) differential (see Differential heuristic demonstration of, 369
of Fourier series, 339, 353 equation) proof of, 369