CONTENTS vii 4.6. LineandPlaneinSpace 124 4.6.1. PlaneinSpace 124 4.6.2. LineinSpace 131 4.6.3. MutualArrangementofPoints,Lines,andPlanes 135 4.7. Quadric Surfaces (Quadrics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.7.1. Quadrics (Canonical Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.7.2. Quadrics(GeneralTheory) 148 ReferencesforChapter4 153 5. Algebra 155 5.1. Polynomials and Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.1.1. Polynomials and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.1.2. LinearandQuadraticEquations 157 5.1.3. CubicEquations 158 5.1.4. Fourth-DegreeEquation 159 5.1.5. Algebraic Equations of Arbitrary Degree and Their Properties . . . . . . . . . . . . . . 161 5.2. MatricesandDeterminants 167 5.2.1. Matrices 167 5.2.2. Determinants 175 5.2.3. EquivalentMatrices.Eigenvalues 180 5.3. Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3.1. Concept of a Linear Space. Its Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . 187 5.3.2. Subspaces of Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.3.3. Coordinate Transformations Corresponding to Basis Transformations in a Linear Space 191 5.4. Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.4.1. RealEuclideanSpace 192 5.4.2. Complex Euclidean Space (Unitary Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.4.3. Banach Spaces and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.5. SystemsofLinearAlgebraicEquations 197 5.5.1. Consistency Condition for a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.5.2. Finding Solutions of a System of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 198 5.6. LinearOperators 204 5.6.1. NotionofaLinearOperator.ItsProperties 204 5.6.2. LinearOperatorsinMatrixForm 208 5.6.3. EigenvectorsandEigenvaluesofLinearOperators 209 5.7. Bilinear and Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.7.1. Linear and Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.7.2. Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5.7.3. QuadraticForms 216 5.7.4. Bilinear and Quadratic Forms in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . 219 5.7.5. Second-Order Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.8. SomeFactsfromGroupTheory 225 5.8.1. Groups and Their Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.8.2. Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5.8.3. GroupRepresentations 230 ReferencesforChapter5 233 viii CONTENTS 6. Limits and Derivatives 235 6.1. BasicConceptsofMathematicalAnalysis 235 6.1.1. NumberSets.FunctionsofRealVariable 235 6.1.2. LimitofaSequence 237 6.1.3. LimitofaFunction.Asymptotes 240 6.1.4. Infinitely Small and InfinitelyLargeFunctions 242 6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind . . . . . . . 243 6.1.6. ConvexandConcaveFunctions 245 6.1.7. Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 6.1.8. ConvergenceofFunctions 249 6.2. DifferentialCalculusforFunctionsofaSingleVariable 250 6.2.1. Derivative and Differential, Their Geometrical and Physical Meaning . . . . . . . . . 250 6.2.2. TableofDerivativesandDifferentiationRules 252 6.2.3. Theorems about Differentiable Functions. L’Hospital Rule . . . . . . . . . . . . . . . . . 254 6.2.4. Higher-Order Derivatives and Differentials. Taylor’s Formula . . . . . . . . . . . . . . . 255 6.2.5. Extremal Points. Points of Inflection 257 6.2.6. Qualitative Analysis of Functions and Construction of Graphs . . . . . . . . . . . . . . 259 6.2.7. Approximate Solution of Equations (Root-Finding Algorithms for Continuous Functions) 260 6.3. FunctionsofSeveralVariables.PartialDerivatives 263 6.3.1. PointSets.Functions.LimitsandContinuity 263 6.3.2. DifferentiationofFunctionsofSeveralVariables 264 6.3.3. Directional Derivative. Gradient. GeometricalApplications 267 6.3.4. ExtremalPointsofFunctionsofSeveralVariables 269 6.3.5. DifferentialOperatorsoftheFieldTheory 272 ReferencesforChapter6 272 7. Integrals 273 7.1. IndefiniteIntegral 273 7.1.1. Antiderivative. IndefiniteIntegralandItsProperties 273 7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral. Integration Examples 274 7.1.3. IntegrationofRationalFunctions 276 7.1.4. Integration of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 7.1.5. Integration of Exponential and Trigonometric Functions . . . . . . . . . . . . . . . . . . . 281 7.1.6. Integration of Polynomials Multiplied by Elementary Functions . . . . . . . . . . . . . 283 7.2. DefiniteIntegral 286 7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the DefiniteIntegral 286 7.2.2. Properties of DefiniteIntegralsandUsefulFormulas 287 7.2.3. General Reduction Formulas for the Evaluation of Integrals . . . . . . . . . . . . . . . . 289 7.2.4. General Asymptotic Formulas for the Calculation of Integrals . . . . . . . . . . . . . . . 290 7.2.5. Mean Value Theorems. Properties of Integrals in Terms of Inequalities. ArithmeticMeanandGeometricMeanofFunctions 295 7.2.6. Geometric and Physical Applications of the DefiniteIntegral 299 7.2.7. Improper Integrals with InfiniteIntegrationLimit 301 7.2.8. General Reduction Formulas for the Calculation of Improper Integrals . . . . . . . . 304 7.2.9. General Asymptotic Formulas for the Calculation of Improper Integrals . . . . . . . 307 7.2.10. Improper Integrals of Unbounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 7.2.11. Cauchy-Type Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 CONTENTS ix 7.2.12. Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 7.2.13. SquareIntegrableFunctions 314 7.2.14. Approximate (Numerical) Methods for Computation of Definite Integrals . . . . 315 7.3. Double and Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 7.3.1. Definition and Properties of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 317 7.3.2. Computation of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 7.3.3. Geometric and Physical Applications of the Double Integral . . . . . . . . . . . . . . . . 323 7.3.4. Definition and Properties of the Triple Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 7.3.5. Computation of the Triple Integral. Some Applications. Iterated Integrals and AsymptoticFormulas 325 7.4. LineandSurfaceIntegrals 329 7.4.1. LineIntegraloftheFirstKind 329 7.4.2. LineIntegraloftheSecondKind 330 7.4.3. SurfaceIntegraloftheFirstKind 332 7.4.4. SurfaceIntegraloftheSecondKind 333 7.4.5. IntegralFormulasofVectorCalculus 334 ReferencesforChapter7 335 8. Series 337 8.1. Numerical Series and Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 8.1.1. Convergent Numerical Series and Their Properties. Cauchy’s Criterion . . . . . . . 337 8.1.2. Convergence Criteria for Series with Positive (Nonnegative) Terms . . . . . . . . . . 338 8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional Convergence 341 8.1.4. Multiplication of Series. Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 8.1.5. Summation Methods. Convergence Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 344 8.1.6. Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.2. FunctionalSeries 348 8.2.1. Pointwise and Uniform Convergence of Functional Series . . . . . . . . . . . . . . . . . . 348 8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent Series 349 8.3. PowerSeries 350 8.3.1. Radius of Convergence of Power Series. Properties of Power Series . . . . . . . . . . 350 8.3.2. TaylorandMaclaurinPowerSeries 352 8.3.3. Operations with Power Series. Summation Formulas for Power Series . . . . . . . . 354 8.4. FourierSeries 357 8.4.1. Representation of 2π-Periodic Functions by Fourier Series. Main Results . . . . . 357 8.4.2. Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions . . . . . . . 359 8.4.3. Criteria of Uniform and Mean-Square Convergence of Fourier Series . . . . . . . . . 361 8.4.4. Summation Formulas for Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 8.5. AsymptoticSeries 363 8.5.1. Asymptotic Series of Poincar ´ e Type. Formulas for the Coefficients 363 8.5.2. OperationswithAsymptoticSeries 364 ReferencesforChapter8 366 9. Differential Geometry 367 9.1. TheoryofCurves 367 9.1.1. PlaneCurves 367 9.1.2. SpaceCurves 379 x CONTENTS 9.2. Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 9.2.1. Elementary Notions in Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 9.2.2. CurvatureofCurvesonSurface 392 9.2.3. IntrinsicGeometryofSurface 395 ReferencesforChapter9 397 10. Functions of Complex Variable 399 10.1. BasicNotions 399 10.1.1. ComplexNumbers.FunctionsofComplexVariable 399 10.1.2. FunctionsofComplexVariable 401 10.2. MainApplications 419 10.2.1. ConformalMappings 419 10.2.2. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 ReferencesforChapter10 433 11. Integral Transforms 435 11.1. GeneralFormofIntegralTransforms.SomeFormulas 435 11.1.1. IntegralTransformsandInversionFormulas 435 11.1.2. Residues.JordanLemma 435 11.2. LaplaceTransform 436 11.2.1. LaplaceTransformandtheInverseLaplaceTransform 436 11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some Functions 437 11.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series andAsymptoticExpansions 440 11.3. Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 11.3.1. Mellin Transform and the Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 441 11.3.2. Main Properties of the Mellin Transform. Relation Among the Mellin, Laplace, and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 11.4. VariousFormsoftheFourierTransform 443 11.4.1. FourierTransformandtheInverseFourierTransform 443 11.4.2. FourierCosineandSineTransforms 445 11.5. OtherIntegralTransforms 446 11.5.1. Integral Transforms Whose Kernels Contain Bessel Functions and Modified BesselFunctions 446 11.5.2. Summary Table of Integral Transforms. Areas of Application of Integral Transforms 448 ReferencesforChapter11 451 12. Ordinary Differential Equations 453 12.1. First-OrderDifferentialEquations 453 12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 453 12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration . . . . 456 12.1.3. ExactDifferentialEquations.IntegratingFactor 458 12.1.4. Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 12.1.5. AbelEquationsoftheFirstKind 462 12.1.6. Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 12.1.7. EquationsNotSolvedfortheDerivative 465 12.1.8. ContactTransformations 468 12.1.9. Approximate Analytic Methods for Solution of Equations . . . . . . . . . . . . . . . . 469 12.1.10. NumericalIntegrationofDifferentialEquations 471 CONTENTS xi 12.2. Second-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 12.2.1. Formulas for the General Solution. Some Transformations . . . . . . . . . . . . . . . 472 12.2.2. Representation of Solutions as a Series in the Independent Variable . . . . . . . . 475 12.2.3. AsymptoticSolutions 477 12.2.4. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 12.2.5. EigenvalueProblems 482 12.2.6. TheoremsonEstimatesandZerosofSolutions 487 12.3. Second-Order Nonlinear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 12.3.1. Form of the General Solution. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . 488 12.3.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 12.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable 492 12.3.4. Movable Singularities of Solutions of Ordinary Differential Equations. Painlev ´ eTranscendents 494 12.3.5. Perturbation Methods of Mechanics and Physics . . . . . . . . . . . . . . . . . . . . . . . 499 12.3.6. Galerkin Method and Its Modifications (Projection Methods) . . . . . . . . . . . . . 508 12.3.7. Iteration and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 12.4. LinearEquationsofArbitraryOrder 514 12.4.1. Linear Equations with Constant Coefficients 514 12.4.2. Linear Equations with Variable Coefficients 518 12.4.3. AsymptoticSolutionsofLinearEquations 522 12.4.4. CollocationMethodandItsConvergence 523 12.5. NonlinearEquationsofArbitraryOrder 524 12.5.1. StructureoftheGeneralSolution.CauchyProblem 524 12.5.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 12.6. LinearSystemsofOrdinaryDifferentialEquations 528 12.6.1. Systems of Linear Constant-CoefficientEquations 528 12.6.2. Systems of Linear Variable-CoefficientEquations 539 12.7. NonlinearSystemsofOrdinaryDifferentialEquations 542 12.7.1. Solutions and First Integrals. Uniqueness and Existence Theorems . . . . . . . . . 542 12.7.2. Integrable Combinations. Autonomous Systems of Equations . . . . . . . . . . . . . 545 12.7.3. Elements of Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 ReferencesforChapter12 550 13. First-Order Partial Differential Equations 553 13.1. Linear and Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 13.1.1. CharacteristicSystem.GeneralSolution 553 13.1.2. CauchyProblem.ExistenceandUniquenessTheorem 556 13.1.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations . . 558 13.1.4. Quasilinear Equations of General Form. Generalized Solution, Jump Condition, and Stability Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 13.2. NonlinearEquations 570 13.2.1. Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 13.2.2. CauchyProblem.ExistenceandUniquenessTheorem 576 13.2.3. Generalized Viscosity Solutions and Their Applications . . . . . . . . . . . . . . . . . 579 ReferencesforChapter13 584 xii CONTENTS 14. Linear Partial Differential Equations 585 14.1. Classification of Second-Order Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 585 14.1.1. EquationswithTwoIndependentVariables 585 14.1.2. EquationswithManyIndependentVariables 589 14.2. BasicProblemsofMathematicalPhysics 590 14.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems 590 14.2.2. First, Second, Third, and Mixed Boundary Value Problems . . . . . . . . . . . . . . . 593 14.3. Properties and Exact Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 14.3.1. Homogeneous Linear Equations and Their Particular Solutions . . . . . . . . . . . . 594 14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions . . . . . . . . . 598 14.3.3. General Solutions of Some Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . 600 14.4. Method of Separation of Variables (Fourier Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution 602 14.4.2. Problems for Parabolic Equations: Final Stage of Solution . . . . . . . . . . . . . . . 605 14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution . . . . . . . . . . . . . . 607 14.4.4. Solution of Boundary Value Problems for Elliptic Equations . . . . . . . . . . . . . . 609 14.5. IntegralTransformsMethod 611 14.5.1. Laplace Transform and Its Application in Mathematical Physics . . . . . . . . . . . 611 14.5.2. Fourier Transform and Its Application in Mathematical Physics . . . . . . . . . . . 614 14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution . . 615 14.6.1. CauchyProblemforParabolicEquations 615 14.6.2. CauchyProblemforHyperbolicEquations 617 14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green’s Function 618 14.7.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 618 14.7.2. Problems for Equation s(x) ∂w ∂t = ∂ ∂x p(x) ∂w ∂x –q(x)w + Φ(x, t) 620 14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s Function.GoursatProblem 623 14.8.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 623 14.8.2. Problems for Equation s(x) ∂ 2 w ∂t 2 = ∂ ∂x p(x) ∂w ∂x –q(x)w + Φ(x, t) 624 14.8.3. Problems for Equation ∂ 2 w ∂t 2 + a(t) ∂w ∂t = b(t) ∂ ∂x p(x) ∂w ∂x – q(x)w + Φ(x, t) 626 14.8.4. Generalized Cauchy Problem with Initial Conditions Set Along a Curve . . . . . 627 14.8.5. Goursat Problem (a Problem with Initial Data of Characteristics) . . . . . . . . . . 629 14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables . . . . . . . . . 631 14.9.1. Problems and the Green’s Functions for Equation a(x) ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 + b(x) ∂w ∂x + c(x)w =–Φ(x, y) 631 14.9.2. Representation of Solutions to Boundary Value Problems via the Green’s Functions 633 14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions viatheGreen’sFunction 634 14.10.1. ProblemsforParabolicEquations 634 14.10.2. Problems for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 14.10.3. Problems for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 14.10.4. Comparison of the Solution Structures for Boundary Value Problems for EquationsofVariousTypes 638 CONTENTS xiii 14.11. Construction of the Green’s Functions. General Formulas and Relations . . . . . . . . . . 639 14.11.1. Green’s Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 14.11.2. Green’s Functions Admitting Incomplete Separation of Variables . . . . . . . . 640 14.11.3. Construction of Green’s Functions via Fundamental Solutions . . . . . . . . . . 642 14.12. Duhamel’s Principles in Nonstationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 14.12.1. Problems for Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 646 14.12.2. Problems for Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . 648 14.13. Transformations Simplifying Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . 649 14.13.1. Transformations That Lead to Homogeneous Boundary Conditions . . . . . . 649 14.13.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions 650 ReferencesforChapter14 650 15. Nonlinear Partial Differential Equations 653 15.1. Classification of Second-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 15.1.1. Classification of Semilinear Equations in Two Independent Variables . . . . . . . 653 15.1.2. Classification of Nonlinear Equations in Two Independent Variables . . . . . . . . 653 15.2. Transformations of Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . 655 15.2.1. PointTransformations:OverviewandExamples 655 15.2.2. Hodograph Transformations (Special Point Transformations) . . . . . . . . . . . . . 657 15.2.3. Contact Transformations. Legendre and Euler Transformations . . . . . . . . . . . . 660 15.2.4. B ¨ acklund Transformations. Differential Substitutions . . . . . . . . . . . . . . . . . . . 663 15.2.5. Differential Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions. SimilarityMethod 667 15.3.1. PreliminaryRemarks 667 15.3.2. Traveling-Wave Solutions. Invariance of Equations Under Translations . . . . . 667 15.3.3. Self-Similar Solutions. Invariance of Equations Under Scaling Transformations 669 15.3.4. Equations Invariant Under Combinations of Translation and Scaling Transformations,andTheirSolutions 674 15.3.5. GeneralizedSelf-SimilarSolutions 677 15.4. ExactSolutionswithSimpleSeparationofVariables 678 15.4.1. Multiplicative and Additive Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . 678 15.4.2. Simple Separation of Variables in Nonlinear Partial Differential Equations . . . 678 15.4.3. Complex Separation of Variables in Nonlinear Partial Differential Equations . 679 15.5. MethodofGeneralizedSeparationofVariables 681 15.5.1. Structure of Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 681 15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System ofCoordinateFunctions 683 15.5.3. Solution of Functional Differential Equations by Differentiation . . . . . . . . . . . 684 15.5.4. Solution of Functional-Differential Equations by Splitting . . . . . . . . . . . . . . . . 688 15.5.5. Titov–GalaktionovMethod 693 15.6. MethodofFunctionalSeparationofVariables 697 15.6.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 15.6.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions 697 . . 28 3 7 .2. DefiniteIntegral 28 6 7 .2. 1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the DefiniteIntegral 28 6 7 .2. 2. Properties of DefiniteIntegralsandUsefulFormulas 28 7 7 .2. 3 GeometricalApplications 26 7 6.3.4. ExtremalPointsofFunctionsofSeveralVariables 26 9 6.3.5. DifferentialOperatorsoftheFieldTheory 27 2 ReferencesforChapter6 27 2 7. Integrals 27 3 7.1. IndefiniteIntegral 27 3 7.1.1 24 9 6 .2. DifferentialCalculusforFunctionsofaSingleVariable 25 0 6 .2. 1. Derivative and Differential, Their Geometrical and Physical Meaning . . . . . . . . . 25 0 6 .2. 2. TableofDerivativesandDifferentiationRules