Hardcover
Published: 28th April 2009
ISBN: 9780817645090
Number Of Pages: 348
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 R^{n} 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)
Preface | p. v |
Differential Calculus | p. 1 |
Differential Calculus of Scalar Functions | p. 1 |
Directional and partial derivatives, and the differential | p. 1 |
Directional derivatives | p. 1 |
The differential | p. 2 |
The gradient vector | p. 5 |
Direction of steepest ascent | p. 6 |
Directional derivatives and differential in coordinates | p. 7 |
Partial derivatives | p. 7 |
Jacobian matrix | p. 8 |
The differential in the dual basis | p. 8 |
The gradient vector in coordinates | p. 9 |
The tangent plane | p. 9 |
The orthogonal to the tangent space | p. 10 |
The tangent map | p. 11 |
Differentiability and blow-up | p. 12 |
Differential Calculus for Vector-valued Functions | p. 12 |
Differentiability | p. 14 |
Jacobian matrix | p. 14 |
The tangent space | p. 16 |
The calculus | p. 20 |
Differentiation of compositions | p. 21 |
Calculus for matrix-valued maps | p. 22 |
Theorems of Differential Calculus | p. 24 |
Maps with continuous derivatives | p. 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 theorem | p. 29 |
Scalar functions | p. 29 |
Vector-valued functions | p. 31 |
Taylor's formula | p. 32 |
Taylor's formula of second order | p. 33 |
Taylor formulas of higher order | p. 34 |
Real analytic functions | p. 36 |
A converse of Taylor's theorem | p. 37 |
Critical points | p. 38 |
Some classical partial differential equations | p. 42 |
Invertibility of Maps Rn Rn | p. 46 |
Banach's fixed point theorem | p. 47 |
Local invertibility | p. 48 |
A few examples | p. 51 |
A variational proof of the inverse function theorem | p. 55 |
Global invertibility | p. 56 |
Differential Calculus in Banach Spaces | p. 57 |
G&ahat;teaux and FrÃ©chet differentials | p. 57 |
Gradient | p. 59 |
Mean value theorem | p. 60 |
Higher order derivatives and Taylor's formula | p. 61 |
Local invertibility in Banach spaces | p. 62 |
Exercises | p. 62 |
Integral Calculus | p. 67 |
Lebesgue's Integral | p. 67 |
Definitions and properties: a short summary | p. 67 |
Lebesgue's measure | p. 68 |
Measurable functions | p. 70 |
Lebesgue's integral | p. 71 |
Basic properties of Lebesgue's Integral | p. 72 |
The integral as area of the subgraph | p. 74 |
Chebyshev's inequality | p. 74 |
Negligible sets and the integral | p. 74 |
Riemann integrable functions | p. 75 |
Fubini's theorem and reduction to iterated integrals | p. 76 |
Change of variables | p. 78 |
Differentiation and primitives | p. 78 |
Convergence Theorems | p. 81 |
Monotone convergence | p. 81 |
Dominated convergence | p. 83 |
Absolute continuity of the integral | p. 86 |
Differentiation under the integral sign | p. 86 |
Mollifiers and Approximations | p. 89 |
C0-approximations and Lusin's theorem | p. 89 |
Mollifying in Rn | p. 91 |
Mollifying in Â¿ | p. 94 |
Calculus of Integrals | p. 96 |
Calculus of multiple integrals | p. 96 |
Normal sets | p. 97 |
Rotational figures | p. 99 |
Changes of coordinates | p. 100 |
Measure of the n-dimensional ball | p. 104 |
Isodiametric inequality | p. 105 |
Euler's Â¿ function | p. 106 |
Tetrahedrons | p. 108 |
Monte Carlo method | p. 110 |
Differentiation under the integral sign | p. 111 |
Measure and Area | p. 114 |
Hausdorff's measures | p. 114 |
Area formula | p. 116 |
Calculus of the area of a surface | p. 119 |
The coarea formula | p. 121 |
Gauss-Green Formulas | p. 123 |
Two simple situations | p. 124 |
Admissible sets | p. 126 |
Decomposition of unity | p. 127 |
Gauss-Green formulas | p. 128 |
Integration by parts | p. 130 |
The divergence theorem | p. 130 |
Geometrical meaning of the divergence | p. 130 |
Divergence and transport of volume | p. 131 |
Exercises | p. 132 |
Curves and Differential Forms | p. 137 |
Differential Forms, Vector Fields, and Work | p. 137 |
Vector fields and differential forms | p. 137 |
Curves | p. 138 |
Integration along a curve and work | p. 140 |
Conservative Fields and Potentials | p. 142 |
Exact differential forms | p. 142 |
Closed Forms and Irrotational Fields | p. 145 |
Closed forms | p. 145 |
PoincarÃ© lemma | p. 147 |
Homotopic curves and work | p. 148 |
Simply connected subsets and closed forms | p. 150 |
Pull back of a differential form | p. 151 |
Homotopy formula | p. 153 |
Stokes's theorem in a square | p. 153 |
Homotopy formula | p. 155 |
Stokes's Formula in the Plane | p. 156 |
Exercises | p. 158 |
Holomorphic Functions | p. 159 |
Functions from C to C | p. 159 |
Complex numbers | p. 159 |
Complex derivative | p. 159 |
Cauchy-Riemann equations | p. 160 |
The Fundamental Theorem of Calculus on C | p. 163 |
Line integrals | p. 163 |
Holomorphic primitives and line integrals | p. 164 |
Fundamental Theorems about Holomorphic Functions | p. 167 |
Goursat and Cauchy theorems | p. 167 |
Goursat lemma | p. 167 |
Elementary domains and Goursat's theorem | p. 168 |
Cauchy formula and power series development | p. 171 |
Liouville's theorem | p. 175 |
The unique continuation principle | p. 176 |
Holomorphic differentials | p. 176 |
Winding number | p. 178 |
Stokes's formula and Cauchy's and Morera's theorems | p. 179 |
Examples of Holomorphic Functions | p. 180 |
Some simple functions | p. 180 |
Inverses of holomorphic functions | p. 182 |
Complex logarithm | p. 183 |
Real powers | p. 185 |
Singularities | p. 187 |
Removable singularities | p. 188 |
Poles | p. 189 |
Essential singularities | p. 190 |
Singularities at infinity | p. 191 |
Singular points at boundary and radius of convergence | p. 192 |
Laurent series development | p. 192 |
Residues | p. 195 |
Calculus of residues | p. 196 |
Definite integrals by the residue method | p. 197 |
Sums of series by the residue method | p. 204 |
Z-transform | p. 208 |
Z-transform of a sequence of vectors | p. 213 |
Systems of recurrences and Z-transform | p. 215 |
Further Consequences of Cauchy's Formula | p. 217 |
The argument principle | p. 217 |
RouchÃ©'s theorem | p. 218 |
Maximum principle | p. 219 |
On the convergence of holomorphic functions | p. 220 |
Schwarz's lemma | p. 221 |
Open mapping and the inverse theorem | p. 221 |
Biholomorphisms | p. 222 |
Riemann mapping theorem | p. 223 |
Harmonic functions and Riemann's mapping theorem | p. 225 |
Schwarz's and Poisson's formulas | p. 226 |
Hilbert's transform | p. 227 |
Exercises | p. 229 |
Surfaces and Level Sets | p. 237 |
Immersed and Embedded Surfaces | p. 237 |
Diffeomorphisms | p. 237 |
Tangent vectors | p. 239 |
r-dimensional surfaces in Rn | p. 241 |
Submanifolds | p. 241 |
Immersions | p. 241 |
Parameterizations of maximal rank | p. 243 |
Implicit Function Theorem | p. 247 |
Implicit functions | p. 247 |
The theorem | p. 249 |
Foliations | p. 253 |
Submersions | p. 257 |
Irregular level sets | p. 258 |
Some Applications | p. 259 |
Small perturbations | p. 259 |
Quadratic systems | p. 260 |
Nonlinear Cauchy problem | p. 260 |
A boundary value problem | p. 261 |
C1-dependence on initial data | p. 261 |
Rectifiability theorem for vector fields | p. 262 |
Critical points and critical values: Sard's lemma | p. 264 |
Morse lemma | p. 266 |
Gradient flow | p. 267 |
Constrained critical points: the multiplier rule | p. 268 |
Some applications | p. 270 |
Orthogonal projection and eigenvectors | p. 270 |
Inequalities | p. 272 |
Lyapunov-Schmidt procedure | p. 275 |
Maps with locally constant rank and functional dependence | p. 279 |
Curvature of Curves and Surfaces | p. 281 |
Curvature of a curve in Rn | p. 281 |
Moving frame for a planar curve | p. 284 |
Moving frame of a curve in R3 | p. 286 |
Curvature of a submanifold of Rn | p. 288 |
First fundamental form | p. 289 |
Second fundamental form | p. 290 |
Curvature vector | p. 291 |
Mean curvature vector | p. 292 |
Curvature of surfaces of codimension one | p. 293 |
Gradient and divergence on a surface | p. 296 |
First variation of the area | p. 299 |
Laplace-Beltrami operator and the mean curvature | p. 300 |
Distance function | p. 301 |
Exercises | p. 303 |
Systems of Ordinary Differential Equations | p. 309 |
Linear Systems | p. 310 |
Linear systems of first-order ODEs | p. 310 |
Linear systems with constant coefficients | p. 313 |
More about linear systems | p. 314 |
Higher-order equations | p. 318 |
Higher-order equations and first-order systems | p. 318 |
Homogeneous linear equations with constant coefficients | p. 319 |
Nonhomogeneous linear ODEs | p. 321 |
Stability | p. 322 |
Critical points and linearization | p. 322 |
Lyapunov's method | p. 325 |
PoincarÃ©-Bendixson Theorem | p. 328 |
Limit sets and invariant sets | p. 329 |
PoincarÃ©-Bendixson theorem | p. 331 |
Systems on a torus | p. 332 |
Exercises | p. 336 |
Mathematicians and Other Scientists | p. 339 |
Bibliographical Notes | p. 341 |
Index | p. 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
Publisher: BIRKHAUSER BOSTON INC
Country of Publication: US
Dimensions (cm): 23.5 x 15.5
x 2.54
Weight (kg): 0.69