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Mathematical Analysis : An Introduction to Functions of Several Variables - Mariano Giaquinta

Mathematical Analysis

An Introduction to Functions of Several Variables


Published: 28th April 2009
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This self-contained work is an introductory presentation of basic ideas, structures, and results of differential and integral calculus for functions of several variables.

The wide range of topics covered include: differential calculus of several variables, including differential calculus of Banach spaces, the relevant results of Lebesgue integration theory, differential forms on curves, a general introduction to holomorphic functions, including singularities and residues, surfaces and level sets, and systems and stability of ordinary differential equations. An appendix highlights important mathematicians and other scientists whose contributions have made a great impact on the development of theories in analysis.

Mathematical Analysis: An Introduction to Functions of Several Variables motivates the study of the analysis of several variables with examples, observations, exercises, and illustrations. It may be used in the classroom setting or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering.

Other books recently published by the authors include: Mathematical Analysis: Functions of One Variable, Mathematical Analysis: Approximation and Discrete Processes, and Mathematical Analysis: Linear and Metric Structures and Continuity, all of which provide the reader with a strong foundation in modern-day analysis.

From the reviews:

"This is a comprehensive introduction to the study of functions of several variables that includes several areas not commonly included in comparable textbooks. ... The Current book has a generally broader scope ... . There is a huge amount of mathematics here, presented carefully and with style. ... The treatment of holomorphic functions here is nicely done ... . In the end, I find that this text would be an agreeable source for most of its individual topics ... ." (William J. Satzer, The Mathematical Association of America, August, 2009)

"This is a classical textbook on functions of several variables. On 348 pages it covers the content of a graduate course of mathematical analysis devoted to the higher dimensional spaces. ... The textbook is suitable for students of mathematics, physics, engineering and technology, as well as for researchers." (Vladimir Janis, Zentralblatt MATH, Vol. 1177, 2010)

"This is a part of an ampler project of the authors ... . The applications and the examples included in the book make it more attractive. There are also exercises at the end of each chapter. ... will supply the reader with a fairly complete account of the fundamental results in mathematical analysis and applications, including Lebesgue integration in Rn and complex analysis of one variable. ... can be used for courses in real or complex analysis and their applications." (Tiberiu Trif, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010)

"This is a textbook on analysis of functions of several real variables and of functions of one complex variable. ... The book is concise and nicely written and may well serve as source for (graduate) courses in the areas covered as well as a textbook for students and as a reference book for the working mathematician." (R. Steinbauer, Monatshefte fur Mathematik, Vol. 165 (3-4), March, 2012)

Prefacep. v
Differential Calculusp. 1
Differential Calculus of Scalar Functionsp. 1
Directional and partial derivatives, and the differentialp. 1
Directional derivativesp. 1
The differentialp. 2
The gradient vectorp. 5
Direction of steepest ascentp. 6
Directional derivatives and differential in coordinatesp. 7
Partial derivativesp. 7
Jacobian matrixp. 8
The differential in the dual basisp. 8
The gradient vector in coordinatesp. 9
The tangent planep. 9
The orthogonal to the tangent spacep. 10
The tangent mapp. 11
Differentiability and blow-upp. 12
Differential Calculus for Vector-valued Functionsp. 12
Differentiabilityp. 14
Jacobian matrixp. 14
The tangent spacep. 16
The calculusp. 20
Differentiation of compositionsp. 21
Calculus for matrix-valued mapsp. 22
Theorems of Differential Calculusp. 24
Maps with continuous derivativesp. 24
Functions of class C1 (A)p. 24
Functions of class C1 (A)p. 25
Functions of class C2 (A)p. 26
Functions of classes Ck (A) and C&infinity; (A)p. 28
Mean value theoremp. 29
Scalar functionsp. 29
Vector-valued functionsp. 31
Taylor's formulap. 32
Taylor's formula of second orderp. 33
Taylor formulas of higher orderp. 34
Real analytic functionsp. 36
A converse of Taylor's theoremp. 37
Critical pointsp. 38
Some classical partial differential equationsp. 42
Invertibility of Maps Rn Rnp. 46
Banach's fixed point theoremp. 47
Local invertibilityp. 48
A few examplesp. 51
A variational proof of the inverse function theoremp. 55
Global invertibilityp. 56
Differential Calculus in Banach Spacesp. 57
G&ahat;teaux and Fréchet differentialsp. 57
Gradientp. 59
Mean value theoremp. 60
Higher order derivatives and Taylor's formulap. 61
Local invertibility in Banach spacesp. 62
Exercisesp. 62
Integral Calculusp. 67
Lebesgue's Integralp. 67
Definitions and properties: a short summaryp. 67
Lebesgue's measurep. 68
Measurable functionsp. 70
Lebesgue's integralp. 71
Basic properties of Lebesgue's Integralp. 72
The integral as area of the subgraphp. 74
Chebyshev's inequalityp. 74
Negligible sets and the integralp. 74
Riemann integrable functionsp. 75
Fubini's theorem and reduction to iterated integralsp. 76
Change of variablesp. 78
Differentiation and primitivesp. 78
Convergence Theoremsp. 81
Monotone convergencep. 81
Dominated convergencep. 83
Absolute continuity of the integralp. 86
Differentiation under the integral signp. 86
Mollifiers and Approximationsp. 89
C0-approximations and Lusin's theoremp. 89
Mollifying in Rnp. 91
Mollifying in ¿p. 94
Calculus of Integralsp. 96
Calculus of multiple integralsp. 96
Normal setsp. 97
Rotational figuresp. 99
Changes of coordinatesp. 100
Measure of the n-dimensional ballp. 104
Isodiametric inequalityp. 105
Euler's ¿ functionp. 106
Tetrahedronsp. 108
Monte Carlo methodp. 110
Differentiation under the integral signp. 111
Measure and Areap. 114
Hausdorff's measuresp. 114
Area formulap. 116
Calculus of the area of a surfacep. 119
The coarea formulap. 121
Gauss-Green Formulasp. 123
Two simple situationsp. 124
Admissible setsp. 126
Decomposition of unityp. 127
Gauss-Green formulasp. 128
Integration by partsp. 130
The divergence theoremp. 130
Geometrical meaning of the divergencep. 130
Divergence and transport of volumep. 131
Exercisesp. 132
Curves and Differential Formsp. 137
Differential Forms, Vector Fields, and Workp. 137
Vector fields and differential formsp. 137
Curvesp. 138
Integration along a curve and workp. 140
Conservative Fields and Potentialsp. 142
Exact differential formsp. 142
Closed Forms and Irrotational Fieldsp. 145
Closed formsp. 145
Poincaré lemmap. 147
Homotopic curves and workp. 148
Simply connected subsets and closed formsp. 150
Pull back of a differential formp. 151
Homotopy formulap. 153
Stokes's theorem in a squarep. 153
Homotopy formulap. 155
Stokes's Formula in the Planep. 156
Exercisesp. 158
Holomorphic Functionsp. 159
Functions from C to Cp. 159
Complex numbersp. 159
Complex derivativep. 159
Cauchy-Riemann equationsp. 160
The Fundamental Theorem of Calculus on Cp. 163
Line integralsp. 163
Holomorphic primitives and line integralsp. 164
Fundamental Theorems about Holomorphic Functionsp. 167
Goursat and Cauchy theoremsp. 167
Goursat lemmap. 167
Elementary domains and Goursat's theoremp. 168
Cauchy formula and power series developmentp. 171
Liouville's theoremp. 175
The unique continuation principlep. 176
Holomorphic differentialsp. 176
Winding numberp. 178
Stokes's formula and Cauchy's and Morera's theoremsp. 179
Examples of Holomorphic Functionsp. 180
Some simple functionsp. 180
Inverses of holomorphic functionsp. 182
Complex logarithmp. 183
Real powersp. 185
Singularitiesp. 187
Removable singularitiesp. 188
Polesp. 189
Essential singularitiesp. 190
Singularities at infinityp. 191
Singular points at boundary and radius of convergencep. 192
Laurent series developmentp. 192
Residuesp. 195
Calculus of residuesp. 196
Definite integrals by the residue methodp. 197
Sums of series by the residue methodp. 204
Z-transformp. 208
Z-transform of a sequence of vectorsp. 213
Systems of recurrences and Z-transformp. 215
Further Consequences of Cauchy's Formulap. 217
The argument principlep. 217
Rouché's theoremp. 218
Maximum principlep. 219
On the convergence of holomorphic functionsp. 220
Schwarz's lemmap. 221
Open mapping and the inverse theoremp. 221
Biholomorphismsp. 222
Riemann mapping theoremp. 223
Harmonic functions and Riemann's mapping theoremp. 225
Schwarz's and Poisson's formulasp. 226
Hilbert's transformp. 227
Exercisesp. 229
Surfaces and Level Setsp. 237
Immersed and Embedded Surfacesp. 237
Diffeomorphismsp. 237
Tangent vectorsp. 239
r-dimensional surfaces in Rnp. 241
Submanifoldsp. 241
Immersionsp. 241
Parameterizations of maximal rankp. 243
Implicit Function Theoremp. 247
Implicit functionsp. 247
The theoremp. 249
Foliationsp. 253
Submersionsp. 257
Irregular level setsp. 258
Some Applicationsp. 259
Small perturbationsp. 259
Quadratic systemsp. 260
Nonlinear Cauchy problemp. 260
A boundary value problemp. 261
C1-dependence on initial datap. 261
Rectifiability theorem for vector fieldsp. 262
Critical points and critical values: Sard's lemmap. 264
Morse lemmap. 266
Gradient flowp. 267
Constrained critical points: the multiplier rulep. 268
Some applicationsp. 270
Orthogonal projection and eigenvectorsp. 270
Inequalitiesp. 272
Lyapunov-Schmidt procedurep. 275
Maps with locally constant rank and functional dependencep. 279
Curvature of Curves and Surfacesp. 281
Curvature of a curve in Rnp. 281
Moving frame for a planar curvep. 284
Moving frame of a curve in R3p. 286
Curvature of a submanifold of Rnp. 288
First fundamental formp. 289
Second fundamental formp. 290
Curvature vectorp. 291
Mean curvature vectorp. 292
Curvature of surfaces of codimension onep. 293
Gradient and divergence on a surfacep. 296
First variation of the areap. 299
Laplace-Beltrami operator and the mean curvaturep. 300
Distance functionp. 301
Exercisesp. 303
Systems of Ordinary Differential Equationsp. 309
Linear Systemsp. 310
Linear systems of first-order ODEsp. 310
Linear systems with constant coefficientsp. 313
More about linear systemsp. 314
Higher-order equationsp. 318
Higher-order equations and first-order systemsp. 318
Homogeneous linear equations with constant coefficientsp. 319
Nonhomogeneous linear ODEsp. 321
Stabilityp. 322
Critical points and linearizationp. 322
Lyapunov's methodp. 325
Poincaré-Bendixson Theoremp. 328
Limit sets and invariant setsp. 329
Poincaré-Bendixson theoremp. 331
Systems on a torusp. 332
Exercisesp. 336
Mathematicians and Other Scientistsp. 339
Bibliographical Notesp. 341
Indexp. 343
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780817645090
ISBN-10: 0817645098
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 348
Published: 28th April 2009
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 1.51