Transcription
TABLE OF CONTENTS
viTABLE OF CONTENTSPreface . . . iiCourse Objectives . . . .ivTable of Contents . . vMathematical Notations and Symbols . xiCHAPTER IAA BBMATH FUNDAMENTALS11.1 Sets, Numbers, Coordinates, Functions . .1. Sets2. Operations with sets3. Proofs4. Numbers5. Constants and variables6. Coordinates7. Functions8. Review questions and exercises41.2 Complex Numbers . .1. Definition of the Field of Complex Numbers:Standard formExponential formTrigonometric formEuler’s Formula2. Operations with Complex Numbers:De Moivre’s Formula and nth rootProperties of absolute value and complex conjugate3. Examples4. Review Questions and Exercises5. Complex Numbers with Maple171.3 Real Functions . . 271. Constant Function2. Absolute Value3. Linear Function4. Quadratic Function5. Polynomials6. Rational Function7. Irrational Functions8. Exponential Function9. Logarithmic Function10. Trigonometric Functions11. Inverse Trigonometric Functions12. Hyperbolic Function14. Review Questions and Exercises1.4 Mathematical Induction . 491. Definition2. Example3. Review questions and exercises
viiCHAPTER IICALCULUS532.1 Limits and Continuity . . 551. Limitslimit of a functionone-sided limitslimits at infinityinfinite limits2. Limit Theorems3. Examples4. Continuity5. Continuity Theorems6. Examples7. Review Questions and Exercises8. Limits and Continuity of Functions with Maple2.2 Differentiation . . 731. Definitions:derivative at the pointthe derivative functionhigher order derivativesgeometrical sensephysical sense2. Differentiation Rules3. Theorems4. Examples5. Review Questions and Exercises6. Differentiation Calculus with Maple2.3 Integration . . 871. Indefinite Integral and Antiderivative:properties of indefinite integralmethods of integration2. Definite Integral:Riemann’s SumFundamental Theorem of CalculusProperties of definite integral3. Improper Integrals4. Numerical Integration5. Application of Definite Integral6. Examples7. Theorems8. Review Questions and Exercises9. Integral Calculus with Maple2.4 Sequences and Series . . 1211. Sequences2. Infinite Series3. Power Series4. Taylor Series5. Review Questions and Exercises6. Sequences and Series with Maple2.5 Graphing Functions .1391. Definitions2. Graphing differentiable functions3. The general approach to plotting graph of a function4. Examples5. Review Questions and Exercises6. Graphing with Maple
viiiCHAPTER IIILINEAR ALGEBRAlinear transformation Ax bn x1 xn b1 bm CHAPTER IVm1.2.3.4.5.6.7.8.9.10.11.12.159Vector SpacesLinear Combination, Linear Independence, BasisVectorsMatricesLinear Transformations with the Help of MatricesDeterminantsMatrix InverseRow ReductionLinear SystemsEigenvalue ProblemLinear Algebra with MapleReview questions and exercisesVECTOR AND TENSOR ANALYSIS2034.1 Vectors and Tensors . . 2051. Introduction1. Euclidean Space E32. Geometric Vectors3. Vector Spaces4. Dot Product5. Cross Product6. Examples7. Tensors8. Tensor Algebra9. Summary of Tensors7. Review Questions and Exercises4.2 Vector and Tensor Analysis . . 2401. Tensor Function2. Tensor Field3. Space Curves4. Level curves and surfaces5. Operator “nabla”, gradient, directional derivative6. FluxFlux of the 2nd order tensor7. DivergenceDivergence of 2nd order tensor field8. Curl9. Operator “nabla” and related differential operators10. Line Integral11. Volume Integral12. The Divergence Theorem4.2 Vector Analysis – Revisited and Enhanced . 2581. Vector Functions3. Differentiation and integration of vector functions4. Scalar and vector fields5. Double integral6. Surface integral of the scalar functions7. Surface integral of the vector functions – flux of a vector field8. Divergence9. Curl10. Line integral11. Volume integral12. Integral theorems
ixCHAPTER VORDINARY DIFFERENTIAL EQUATIONS3155.1 Ordinary Differential Equations – Basics . 3175.2 1st order ODEs . . 3241. Exact ODEs2. Equation Reducible to Exact – Integrating Factor3. Separable Equations4. Homogeneous Equations5. Linear 1st Order ODEs6. Special Equations: Bernoulli, Ricatti, Clairaut, Lagrange7. Approximate and Numerical Methods for 1st Order ODEs8. Equations of Reducible Order9. Orthogonal Trajectories10. Exercises5.3 Linear ODEs . 3091. Linear ODE2. Homogeneous Linear ODE3. Non-Homogeneous Linear ODE4. Fundamental Set of Linear ODE with Constant Coefficients5. Particular sSolution of Linear ODE6. Euler-Cauchy equation7. Review Questions and Exercises8. Linear ODE with Maple5.4 Power Series Solution . . 3291. Definitions2. Power Series Solution3. The Method of Frobenius4. Review Questions and Exercises5. Power Series Solution with Maple5.5 Systems of ODEs . 3451. Definitions and Notations2. Theory of Linear Systems of ODEs3. The Fundamental Set of a Linear System with Constant Coefficients4. Autonomous Systems5. Examples6. Review Questions and Exercises7. Systems of ODEs with MapleCHAPTER VI :V vuNORMED VECTOR SPACES1.2.3.4.u vu v u v5.Normed Vector Spaces – Banach and Hilbet spacesThe Generalized Fourier seriesSturm-Liouville TheoremSturm-Liouville problem for equation X"-μX 0Summary TableRoots of transcendental equationReview questions, examples and exercises429
xCHAPTER VIIJ0 ( x )J1 ( x )J2 ( x)J3 ( x)CHAPTER VIIISPECIAL FUNCTIONS4731.Heaviside step functionH ( x)2.Dirac delta functionδ ( x)3.Sine integral functionSi ( x )4.Error functionerf ( x ) ,erfc ( x )5.Gamma functionΓ ( x)6.Bessel functionsJν ( x ) ,Yν ( x ) ,Iν ( x ) ,Kν ( x )7.Legendre functionsPn ( x )PARTIAL DIFFERENTIAL EQUATIONS5338.1 Fundamental Principles of Science and Engineering . . . 5351. Conservation of Mass2. Conservation of Energy3. Momentum Principle4. Entropy Principle5. Principles of State and Properties6. Fundamental Phenomena in Engineering7. Basics of Thermodynamics and Heat and Mass Transfer8. Fundamental phenomena (empirical laws) of heat transfer9. Vector analysis in heat and mass transfer10. Domain11. Derivation of the governing equation12. Modelling of boundary conditions13. Classical Initial-Boundary Value problems (IBVP)14. Thermodynamical properties15. Mathematical dimension in modelling heat transfer8.2 The Method of Separation of Variables –Stationary Boundary Value Problems . 5731. The Concept of Separation of Variables2. The Laplace Equation:basic case – 3 homogeneous boundary conditionsnon-homogeneous boundary conditions – superposition principlenon-homogeneous equation (the Poisson Equation)Laplace equation in cylindrical coordinatesLaplace equation in spherical coordinatesReview questions and exercises8.3 The Method of Separation of Variables –Transient Initial-Boundary Value Problems . . 6051. The Heat Equation – 1-D Plane Wall:Basic case: homogeneous equation and boundary conditionsGeneral case: non-homogeneous equation and boundary conditions,steady state solution – reduction to basic case2. The Heat Equation 2-D and 3-D ectangular domainThe Helmholz Equation – 3-D rectangular domainSolid cylinderHollow cylinderSolid sphereLaplace’s equation in spherical coordinates4. The Wave Equation:homogeneous equation with homogeneous boundary conditionsnormal modes of string vibration
xi4.5.CHAPTER IXWave Equation in polar coordinates with angular symmetrySingular Sturm-Liouville Problem – vibrations of ring stringReview questions and exercisesTHE INTEGRAL TRANSFORM METHODS6979.1 Laplace Transform . 6991. Definition2. Properties3. Examples4. Solution of IVP for ODEs5. Solution of PDE in Semi-Infinite Regions6. Wave Equation – semi-infinite string7. Solution of Volterra integral equation of the 2nd kindAbel’s integral equation8. Review Questions and Exercises9. Laplace Transform with Maple9.2 Fourier Transform . . . 7211. Definition2. Properties3. Examples4. Solution of ODE in the infinite domain5. Solution of PDEs in the infinite domain:1. The Heat Equation – Gauss’s Kernel – Green’s Function2. The Wave Equation – D’Alambert Solution3. The Laplace Equation – Poisson’s Integral Formula6. Fourier Integrals – Fourier integral representations7. Integral Fourier transform in the semi-infinite regions1. Fourier integral transform kernel2. Heat Equation in the semi-infinite region3. Laplace’s Equation in the semi-infinite strip4. Laplace’s equation in the 1st quadrant8. Review Questions and Exercises9.3 Finite Fourier Transform . . . . 7531. Introduction – integral transform over finite intervalTable – Finite Fourier Transform – kernel and operation properties2. Heat Equation in the finite layer3. Conduction and advection in the rectangular duct4. Heat equation in the sphere – roasting a turkey5. Examples and exercises9.4 Hankel Transform . . . 8031. Definition and operational properties2. Example – cooling of the space with initial bell-shaped temperature profile3. Instantaneous energy sources4. Heat Equation with instantaneous line source, cylindrical source, point source5. Wave Equation – axisymmetric infinite membrane6. Solution of PDE by application of two integral transforms (Fourier and Hankel)7. Finite Hankel Transform – definition and operational properties8. Development of velocity profile in a pipe under a pressure gradient9. Viscous flow inside of the rotating cylinder10. Annular cylindrical domain – Finite Hankel Transform-2 (FHT-2)11. Viscous flow between two cylinders12. Heat transfer through the cylindrical wall – Thermal shock13. Simple example of the transient heat transfer14. Some additional notes
xii9.5 Generalization of the Integral Transform Method . . . 8551. Introduction2. Supplemental eigenvalue problem3. Finite Integral Transform4. Operational property of the Finite Integral Transfer5. Transient heat transfer in the fin9.6 Conjugate Integral Transform . . . . 8691. Conjugate problem2. Supplemental Sturm-Liouville problem3. Two-layer unsteady Couette flow4. Flow over a heated block5. Exercises9.7 Additional Integral Transforms . . . . 8971. Solution of 3-D heat equation in Cylindrical coordinates2. Mellin Transform3. Legendre Transform4. Jacobi and Gegenbauer Transform5. Laguerre Transform6. Hermit Transform7. Hilbert and Stiltjes transform8. Z Transform9. Analytical solution of the P-1 models of radiative transferCHAPTER XPHASE-CHANGE PROBLEMS1.Introduction to Classical Stefan Problem . . . . 9181.2.APPENDIX917Phase-change problem in the semi-infinite regionBoundary and interface conditions2.Solidification in half space (Neumann’s solution) . . . . 9203.Exercise – Melting in half space . . . . 926SUMMARY .19.Greek LettersComplex NumbersLimitsDifferentiationIntegrationInfinite SeriesVectors1st order ODELinear ODEPower Series SolutionSystems of ODEHyperbolic functionsSturm-Liouville problemCoordinate SystemsLaplace TransformTable of Laplace TransformsMaple tutorialBessel FunctionsFourier Seris929
Table – Finite Fourier Transform – kernel and operation properties 2. Heat Equation in the finite layer . 3. Conduction and advection in the rectangular duct . . Mellin Transform . 3. Legendre Transform . 4. Jacobi and Gegenbauer Transform . 5. Laguerre Transform . 6. Hermit Transform
