内容

目次

1. Vectors and Matrices. Introduction. Vectors in Space. LinearIndependence Lines and Planes. Determinants. Simultaneous LinearEquations. Matrices. Addition of Matrices Scalar Times Matrix.Multiplication of Matrices. Inverse of a Square Matrix. GaussianElimination. *Eigenvalues of a Square Matrix. *The Transpose.*Orthogonal Matrices. Analytic Geometry and Vectors n-Dimensional Space.*Axioms for Vn. Linear Mappings. *Subspaces Rank of a Matrix.*Other Vector Spaces. 2. Differential Calculus of Functions of SeveralVariables. Functions of Several Variables. Domains and Regions.Functional Notation Level Curves and Level Surfaces. Limits andContinuity. Partial Derivatives. Total Differential Fundamental Lemma.Differential of Functions of n Variables The Jacobian Matrix.Derivatives and Differentials of Composite Functions. The General ChainRule. Implicit Functions. *Proof of a Case of the Implicit FunctionTheorem. Inverse Functions Curvilinear Coordinates. GeometricalApplications. The Directional Derivative. Partial Derivatives of HigherOrder. Higher Derivatives of Composite Functions. The Laplacian inPolar, Cylindrical, and Spherical Coordinates. Higher Derivatives ofImplicit Functions. Mixima and Minima of Functions of Several Variables.*Extrema for Functions with Side Conditions Lagrange Multipliers.*Maxima and Minima of Quadratic Forms on the Unit Sphere. *FunctionalDependence. *Real Variable Theory Theorem on Maximum and Minimum. 3.Vector Differential Calculus. Introduction. Vector Fields and ScalarFields. The Gradient Field. The Divergence of a Vector Field. TheCurl of a Vector Field. Combined Operations. *Curvilinear Coordinatesin Space. Orthogonal Coordinates. *Vector Operations in OrthogonalCurvilinear Coordinates. *Tensors. *Tensors on a Surface orHypersurface. *Alternating Tensors. Exterior Product. 4. IntegralCalculus of Functions of Several Variables. The Definite Integral.Numerical Evaluation of Indefinite Integrals. Elliptic Integrals.Double Integrals. Triple Integrals and Multiple Integrals in General.Integrals of Vector Functions. Change of Variables in Integrals. ArcLength and Surface Area. Improper Multiple Integrals. Integralsdepending on a Parameter-Leibnitz's Rule. *Uniform Continuity.Existence of the Riemann Integral. *Theory of Double Integrals. 5.Vector Integral Calculus. Two-Dimensional Theory. Introduction.Line Integrals in the Plane. Integrals with Respect to Arc Length.Basic Properties of Line Integrals. Line Integrals as Integrals ofVectors. Green's Theorem. Independence of Path. Simply ConnectedDomains. Extension of Results to Multiply Connected Domains.Three-Dimensional Theory and Applications. Line Integrals in Space.Surfaces in Space. Orientability. Surface Integrals. The DivergenceTheorem. Stokes's Theorem. Integrals Independent of Path.Irrotational and Solenoidal Fields. *Change of Variables in a MultipleIntegral. *Physical Applications. *Potential Theory in the Plane.*Green's Third Identity. *Potential Theory in Space. *DifferentialForms. *Change of Variables in an m-Form and General Stokes's Theorem.*Tensor Aspects of Differential Forms. *Tensors and Differential Formswithout Coordinates. 6. Infinite Series. Introduction. InfiniteSequences. Upper and Lower Limits. Further Properties of Sequences.Infinite Series. Tests for Convergence and Divergence. Examples andApplications of Tests for Convergence and Divergence. *Extended Ratio Testand Root Test. *Computation with Series-Estimate of Error. Operationson Series. Sequences and Series of Functions. Uniform Convergence.Weierstrass M-Test for Uniform Convergence. Properties of UniformlyConvergent Series and Sequences. Power Series. Taylor and MacLaurinSeries. Taylor's Formula with Remainder. Further Operations on PowerSeries. *Sequences and Series of Complex Numbers. *Sequences and Seriesof Functions of Several Variables. *Taylor's Formula for Functions ofSeveral Variables. *Improper Integrals Versus Infinite Series.*Improper Integrals Depending on a Parameter-Uniform Convergence.*Principal Value of Improper Integrals. *Laplace Transformation.G-Function and B-Function. *Convergence of Improper Multiple Integrals.7. Fourier Series and Orthogonal Functions. Trigonometric Series.Fourier Series. Convergence of Fourier Series. Examples-Minimizing ofSquare Error. Generalizations. Fourier Cosine Series. Fourier SineSeries. Remarks on Applications of Fourier Series. Uniqueness Theorem.Proof of Fundamental Theorem for Continuous, Periodic, and Piecewise VerySmooth Functions. Proof of Fundamental Theorem. Orthogonal Functions.*Fourier Series of Orthogonal Functions. Completeness. *SufficientConditions for Completeness. *Integration and Differentiation of FourierSeries. *Fourier-Legendre Series. *Fourier-Bessel Series.*Orthogonal Systems of Functions of Several Variables. *Complex Form ofFourier Series. *Fourier Integral. *The Laplace Transform as a SpecialCase of the Fourier Transform. General Functions. 8. Functions of aComplex Variable. Complex Functions. Complex-Valued Functions of a RealVariable. Complex-Valued Functions of a Complex Variable. Limits andContinuity. Derivatives and Differentials. Integrals. AnalyticFunctions. Cauchy-Riemann Equations. The Functions log z, az, za, sin-1z, cos-1 z. Integrals of Analytic Functions. Cauchy Integral Theorem.Cauchy's Integral Formula. Power Series as Analytic Functions. PowerSeries Expansion of General Analytic Function. Power Series in Positiveand Negative Powers. Laurent Expansion. Isolated Singularities of anAnalytic Function. Zeros and Poles. The Complex Number a. Residues.Residue at Infinity. Logarithmic Residues. Argument Principle.Partial Fraction Expansion of Rational Functions. Application of Residuesto Evaluation of Real Integrals. Definition of Conformal Mapping.Examples of Conformal Mapping. Applications of Conformal Mapping. TheDirichlet Problem. Dirichlet Problem for the Half-Plane. ConformalMapping in Hydrodynamics. Applications of Conformal Mapping in the Theoryof Elasticity. Further Applications of Conformal Mapping. GeneralFormulas for One-to-One Mapping. Schwarz-Christoffel Transformation. 9.Ordinary Differential Equations. Differential Equations. Solutions.The Basic Problems. Existence Theorem. Linear Differential Equations.Systems of Differential Equations. Linear Systems. Linear Systems withConstant Coefficients. A Class of Vibration Problems. Solution ofDifferential Equations by Taylor Series. The Existence and UniquenessTheorem. 10. Partial Differential Equations. Introduction. Review ofEquation for Forced Vibrations of a Spring. Case of Two Particles. Caseof n Particles. Continuous Medium. Fundamental Partial DifferentialEquation. Classification of Partial Differential Equations. BasicProblems. The Wave Equation in One Dimension. Harmonic Motion.Properties of Solutions of the Wave Equation. The One-Dimensional HeatEquation. Exponential Decay. Properties of Solutions of the HeatEquation. Equilibrium and Approach to Equilibrium. Forced Motion.Equations with Variable Coefficients. Sturm-Liouville Problems.Equations in Two and Three Dimensions. Separation of Variables.Unbounded Regions. Continuous Spectrum. Numerical Methods.Variational Methods. Partial Differential Equations and Integral Equations.