Enhanced descriptions from Syndetics:

This book provides a general introduction to applied analysis; vector analysis with physical motivation, calculus of variation, Fourier analysis, eigenfunction expansion, distribution, and so forth, including a catalogue of mathematical theories, such as basic analysis, topological spaces, complex function theory, real analysis, and abstract analysis. This book also uses fundamental ideas of applied mathematics to discuss recent developments in nonlinear science, such as mathematical modeling of reinforced random motion of particles, semiconductor device equation in applied physics, and chemotaxis in biology. Several tools in linear PDE theory, such as fundamental solutions, Perron's method, layer potentials, and iteration scheme, are described, as well as systematic descriptions on the recent study of the blowup of the solution.

Table of contents provided by Syndetics

• Preface to the Second Edition (p. v)
• Preface to the First Edition (p. vii)
• Chapter 1 Field Formation (p. 1)
• 1.1 Mathematical Modeling (p. 1)
• 1.1.1 Chemical Reaction (p. 1)
• 1.1.2 Population Dynamics (p. 5)
• 1.1.3 Dynamical Systems (p. 8)
• 1.2 System of Differential Equations (p. 14)
• 1.2.1 Outer Product (p. 14)
• 1.2.2 Particle Motion (p. 16)
• 1.2.3 Lotka-Volterra System (p. 23)
• 1.2.4 Gradient System (p. 26)
• 1.2.5 Hamilton System (p. 29)
• Chapter 2 Geometric Objects (p. 33)
• 2.1 Basic Notions of Vector Analysis (p. 33)
• 2.1.1 Gradient (p. 33)
• 2.1.2 Divergence (p. 38)
• 2.1.3 Rotation (p. 44)
• 2.1.4 Fluid Motion (p. 48)
• 2.1.5 Tensor (p. 53)
• 2.1.6 Solid Deformation (p. 57)
• 2.2 Curvature (p. 58)
• 2.2.1 Surface (p. 58)
• 2.2.2 Curve (p. 62)
• 2.2.3 Curve on Surface (p. 67)
• 2.2.4 Integration (p. 73)
• 2.2.5 Differential Form (p. 78)
• 2.3 Extremal (p. 82)
• 2.3.1 Lagrange Multiplier (p. 82)
• 2.3.2 Implicit Function Theorem (p. 85)
• 2.3.3 Convex Function (p. 87)
• 2.3.4 Kuhn-Tucker Duality (p. 93)
• 2.3.5 Linear Programming (p. 97)
• Chapter 3 Calculus of Variations (p. 101)
• 3.1 Isoperimetric Inequality (p. 101)
• 3.1.1 Analytic Proof (p. 101)
• 3.1.2 Geometric Proof (p. 105)
• 3.2 Indirect Method (p. 109)
• 3.2.1 Euler Equation (p. 109)
• 3.2.2 Minimal Surface (p. 112)
• 3.2.3 Analytic Mechanics (p. 115)
• 3.2.4 Quantum Mechanics (p. 117)
• 3.3 Direct Method (p. 123)
• 3.3.1 Vibrating String (p. 123)
• 3.3.2 Minimizing Sequence (p. 125)
• 3.3.3 Sobolev Space (p. 127)
• 3.3.4 Lower Semi-Continuity (p. 130)
• 3.4 Numerical Scheme (p. 134)
• 3.4.1 Finite Difference Method (p. 134)
• 3.4.2 Finite Element Method (p. 135)
• Chapter 4 Infinite-Dimensional Analysis (p. 139)
• 4.1 Hilbert Space (p. 139)
• 4.1.1 Bounded Linear Operator (p. 139)
• 4.1.2 Representation Theorem of Riesz (p. 141)
• 4.1.3 Complete Orthonormal System (p. 144)
• 4.2 Fourier Analysis (p. 147)
• 4.2.1 Historical Note (p. 147)
• 4.2.2 Completeness (p. 150)
• 4.2.3 Uniform Convergence (p. 155)
• 4.2.4 Pointwise Convergence (p. 157)
• 4.2.5 Fourier Transformation (p. 160)
• 4.3 Eigenvalue Problem (p. 166)
• 4.3.1 Vibrating Membrane (p. 166)
• 4.3.2 Gel'fand Triple (p. 171)
• 4.3.3 Self-Adjoint Operator (p. 175)
• 4.3.4 Symmetric Form (p. 177)
• 4.3.5 Compact Operator (p. 179)
• 4.3.6 Eigenfunction Expansion (p. 182)
• 4.3.7 Minimax Principle (p. 185)
• 4.3.8 Hilbert-Schmidt Operator (p. 188)
• 4.4 Distribution (p. 189)
• 4.4.1 Delta Function (p. 189)
• 4.4.2 Locally Convex Space (p. 193)
• 4.4.3 Fréchet Space (p. 195)
• 4.4.4 Inductive Limit (p. 196)
• 4.4.5 Bounded Set (p. 199)
• 4.4.6 Definition and Examples (p. 200)
• 4.4.7 Fundamental Properties (p. 204)
• 4.4.8 Support (p. 208)
• 4.4.9 Convergence (p. 209)
• 4.4.10 Fourier Transformation (revisited) (p. 211)
• Chapter 5 Scattering (p. 215)
• 5.1 Direct Theory (p. 215)
• 5.1.1 Jost Solution (p. 215)
• 5.1.2 S-Matrix (p. 226)
• 5.1.3 GLM Equation (p. 233)
• 5.2 Reverse Theory (p. 239)
• 5.2.1 GLM Equation (continued) (p. 239)
• 5.2.2 Reconstruction (p. 247)
• 5.2.3 Consistency (p. 253)
• Chapter 6 Random Motion of Particles (p. 261)
• 6.1 Mean Field Limit (p. 261)
• 6.1.1 Master Equation (p. 261)
• 6.1.2 Einstein's Formula (p. 266)
• 6.1.3 Local Information Model (p. 270)
• 6.1.4 Smoluchowski Equation (p. 272)
• 6.1.5 Multiscale Model (p. 277)
• 6.2 Kinetic Model (p. 279)
• 6.2.1 Transport Equation (p. 279)
• 6.2.2 Boltzmann Equation (p. 282)
• 6.2.3 Semiconductor Device Equation (p. 286)
• 6.2.4 Drift-Diffusion Model (p. 289)
• Chapter 7 Linear PDE (p. 295)
• 7.1 Well-Posedness (p. 295)
• 7.1.1 Heat Equation (p. 295)
• 7.1.2 Uniqueness (p. 297)
• 7.1.3 Existence (p. 299)
• 7.2 Fundamental Solution (p. 301)
• 7.2.1 Cauchy Problem (p. 301)
• 7.2.2 Gaussian Kernel (p. 304)
• 7.2.3 Semigroup (p. 306)
• 7.3 Potential (p. 311)
• 7.3.1 Harmonic Function (p. 311)
• 7.3.2 Poisson Integral (p. 313)
• 7.3.3 Perron Solution (p. 318)
• 7.3.4 Boundary Regularity (p. 321)
• 7.3.5 Green's Function (p. 323)
• 7.3.6 Newton Potential (p. 325)
• 7.3.7 Layer Potential (p. 330)
• 7.3.8 Fredholm Theory (p. 336)
• 7.4 Regularity (p. 337)
• 7.4.1 Poisson Equation (p. 337)
• 7.4.2 Schauder Estimate (p. 339)
• 7.4.3 Dirichlet Principle (p. 346)
• 7.4.4 Moser's Iteration Scheme (p. 348)
• 7.4.5 Local Minimum Principle (p. 353)
• 7.4.6 BMO (p. 360)
• Chapter 8 Non-linear PDE (p. 371)
• 8.1 Perturbation (p. 371)
• 8.1.1 Duhamel Principle (p. 371)
• 8.1.2 Semilinear Heat Equation (p. 373)
• 8.1.3 Global Existence (p. 377)
• 8.1.4 Blowup (p. 381)
• 8.2 Energy (p. 384)
• 8.2.1 Lyapunov Function (p. 384)
• 8.2.2 Global-in-Time Solution (p. 388)
• 8.2.3 Unbounded Solution (p. 394)
• 8.2.4 Stable and Unstable Sets (p. 399)
• 8.3 Rescaling (p. 403)
• 8.3.1 ODE Part (p. 403)
• 8.3.2 Variational Structure (p. 406)
• 8.3.3 Scaling Invariance (p. 409)
• 8.3.4 Forward Self-Similar Transformation (p. 414)
• 8.3.5 Backward Self-Similar Transformation (p. 418)
• 8.4 Chemotaxis (p. 421)
• 8.4.1 Cellular Slime Molds (p. 421)
• 8.4.2 Michaelis Menten Reduction (p. 422)
• 8.4.3 Free Energy (p. 425)
• 8.4.4 Smoluchowski-Poisson Equation (p. 429)
• 8.4.5 Mass Quantization (p. 433)
• Appendix A Catalogue of Mathematical Theories (p. 445)
• A.1 Basic Analysis (p. 445)
• A.2 Ordinary Differential Equations (p. 449)
• A.3 Topological Spaces (p. 451)
• A.4 Complex Analysis (p. 454)
• A.5 Real Analysis (p. 460)
• A.6 Abstract Analysis (p. 465)
• Appendix B An Elliptic-Parabolic System (p. 469)
• B.1 The System (p. 469)
• B.2 Linearized System (p. 472)
• B.3 Mapping F (p. 479)
• B.4 Solvability (p. 488)
• Appendix C Commentary (p. 491)
• C.1 Elliptic Equations (p. 491)
• C.2 Parabolic Equations (p. 492)
• C.3 Self-Interacting Particles (p. 493)
• C.4 Cell Dynamics (p. 494)
• Bibliography (p. 497)
• Index (p. 505)

Reviews provided by Syndetics

CHOICE Review

Senba (Miyazaki Univ., Japan) and Suzuki (Osaka Univ., Japan) offer a theoretical and very thorough introduction to some topics in applied analysis. In particular, they treat basic vector analysis, calculus of variations, infinite dimensional vector analysis including Hilbert spaces, Fourier series and eigenvalue problems, several models of random motion, and linear and nonlinear partial differential equations in some detail. The Keller-Segel system of partial differential equations from mathematical biology is given special study. These results are presented in a theoretical manner, the level of rigor is high, and important results are either proved or referenced. Exercises, mainly theoretical, are included and there is an extensive bibliography. The level suits a beginning graduate student. A serious drawback to the attractively printed text is the style of English used. Sentences and phrases are often awkward, sometimes unclear, and occasionally unintelligible. This is unfortunate because the text is thorough and covers some valuable topics. In its current form, the book cannot be recommended. Perhaps with the work of an editor, the English usage could be revised so the book can be made acceptable. ^BSumming Up: Not recommended. J. H. Ellison Grove City College