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References 1. Apostol, T. Mathematical Analysis. Narosa Publishing House, New Delhi, 1974.
2. Artin, M. Algebra. Prentice-Hall, Englewood Cliffs, NJ, 1994.
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48, 241 Finney, 153 First partial derivative, 109 Fixed point, 117 Foote, 123 Fréchet differentiable at a point, 79 Fubini’s theorem, 200, 245 Functions, 1 Fundamental ttheorem of integral
Improper integral, 15 Increasing function, 11 Increasing sequence, 7 Induced mapping, 270 Infimum, 6 Infinite-dimensional space, 127 Injective, 3, 124, 131 Inner product in Rn, 25 Integrable, 13, 203 Integral test, 15, 74 Integration, 8 Integration by parts, 14, 278 Interior of a set, 42, 178 Interior point, 42 Intermediate value theorem, 117, 147 Intermediate value theorem (Bolzano),
9 Intersection, 2 Intersection of open sets, 40 Inverse, 3 Inverse function theorem, 121, 124 Inverse image, 3 Inverse of a differentiable map, 124 Inverse of a linear map, 34, 226 Inverse of a matrix, 21, 36 Inversion, 18 Invertible, 3 Invertible linear map, 34, 124, 232 Invertible matrix, 21, 36 Isolated point, 53 Iterated (or repeated) integral, 200, 213
J
Jacobian, 98, 232 Jacobian matrix, 98, 137
K Kantorovich, 130 k-chain, 275 k-index, 257
Index 392
k-surface, 251 Kumaresan, 138 Kunze, 21
L
L(Rn,Rm), 61, 64 Lagrange equations, 157 Lagrange multipliers, 138, 151, 157 Lagrangian, 157, 169 Lang, 127 Leading diagonal, 15 Least upper bound, 5 Least upper bound axiom, 4 Left distributive aw, 18 Leibnitz’s formula, 107 Limit of a function, 8, 54 Limit of a sequence, 7, 30 Limit point, 8, 53, 55, 56 Linear (or Fréchet) derivative, 79 Linear map (operator, transformation),
32 Linear ordering axioms, 4 Lipschitz condition, 115 Local inverse, 127 Local maximum, 11, 151 Local minimum, 11, 151 Local solution, 23, 137 Local strict maximum, 151, 161 Local strict minimum, 151, 161 Locally invertible, 127 Logarithm function (natural), 11 Loomis, 130 Lower bound, 5 Lower face, 179 Lower integral, 13 Lower Riemann integral, 188, 203, 204 Lower sum, 13, 187
M Main diagonal, 15 Map, 3 Mapping, 3
Matrices and determinants, 1 Matrix, 15 Matrix of a linear map, 34 Mean value theorem, 12, 110 Möbius band, 61, 108 Monotone function, 12 Monotone sequence, 7 Morrey, 238 Multi-index, 257
N Nanda, 130 Negative definite, 165 Negative semidefinite, 165 Newton’s method, 12, 129 Nonoverlapping, 180, 185, 231 Nonsingular matrix, 21 Norm of a linear map, 62 Null matrix, 17 n-vector, 24
O
Objective function, 151, 157 One-to-one, 3 One-to-one correspondence, 3 Onto, 3 Open ball, 42, 124, 126 Open cover (or covering), 10, 47, 48 Open cuboid, 177 Open cuboid formed by, 180 Open interval, 6 Open mapping, 131 Open set, 10, 40, 41, 67 Optimisation problem, 151 Ordered field, 5
P
Page, 127 Pairwise disjoint, 2 Partial derivative, 96 Partial sums of a double series, 71 Partition, 12
Index 393
Paving of a cuboid, 180 Polar coordinates, 124, 222, 239, 253 Positive definite, 165 Positive semidefinite, 165 Product of a scalar and a linear map, 35 Product of linear maps, 35 Product of matrices, 17 Product of scalar and vector, 24 Proper subset, 2 Protter, 238 Pugh, 14
Q Quadratic form, 165, 167, 168
R Radius of a ball in Rn, 40, 125 Range, 3 Range space, 3 Rational components, 203 Rational polar coordinates, 246 r-ball in Rn, 40 Rectangular coordinates, 124, 242 Refinement, 188, 190, 191, 195, 220 Repeated (or iterated) limit, 70 Restriction, 3 Riemann integrable, 13, 189 Riemann integral, 189, 203, 204 Right distributive law, 18 Row matrix, 16 Rudin, 26, 127
S Scalar, 24 Scalar product, 25 Schwarz, 109 Schwarz’s theorem, 110, 269 Second partial derivative, 109, 169 Self map, 117 Sequence, 6 Set, 1 Set algebra, 1
Set function, 229 Shirali, 53, 74 Shrinking lemma, 119 Shrinking map, 117 Simple differential form, 254 Singh, 21 Sohrab, 26 Spivak, 26 Square matrix, 15 Standard basis, 25 Standard representation, 258, 260 Stereographic projection, 39 Sternberg, 130 Stokes theorem, 250 Strictly decreasing function, 12 Strictly increasing function, 12 Subcover (or subcovering), 10, 47 Submatrix, 21 Subsequence, 7, 45, 46 Subset, 1 Substitution, 217 Substitution form of the FTC, 250 Substitution rule, 14, 249 Sum in Rn, 24 Sum of a double series, 71 Sum of linear maps, 35 Sum of two matrices, 16 Supremum, 5 Surjective, 3, 131 System of (or simultaneous) equations,
23
T Taylor’s theorem, 12, 109, 112 Term of a double series, 71 Terms of a double sequence, 69 Thomae function, 197 Thomas, 153 Transformation, 32 Transformation formula, 14, 217, 228,
231, 235, 250 Transformation of content, 222 Transpose, 16