Part I: ORDINARY DIFFERENTIAL EQUATIONS. 1. First Order Differential Equations. Preliminary Concepts. General and Particular Solutions. Implicitly Defined Solutions. Integral Curves. The Initial Value Problem. Direction Fields. Separable Equations. Some Applications of Separable Differential Equations. Linear Differential Equations. Exact Differential Equations. Integrating Factors. Separable Equations and Integrating Factors. Linear Equations and Integrating Factors. Homogeneous, Bernoulli and Riccati Equations. Homogeneous Differential Equations. The Bernoulli Equation. The Riccati Equation. Applications to Mechanics, Electrical Circuits and Orthogonal Trajectories. Mechanics. Electrical Circuits. Orthogonal Trajectories. Existence and Uniqueness for Solutions of Initial Value Problems. 2. Second Order Differential Equations. Preliminary Concepts. Theory of Solutions of y" + p(x)y' + q(x)y = f(x). The Homogeneous Equation y" + p(x)y' + q(x) = 0. The Nonhomogeneous Equation y" + p(x)y' + q(x)y = f(x). Reduction of Order. The Constant Coefficient Homogeneous Linear Equation. Case 1 A¦ û 4B > 0. Case 2 A¦ û 4B = 0. Case 3 A¦ û 4B < 0. An Alternative General Solution In the Complex Root Case. Euler's Equation. The Nonhomogeneous Equation y" + p(x)y' + q(x)y = f(x). The Method of Variation of Parameters. The Method of Undetermined Coefficients. The Principle of Superposition. Higher Order Differential Equations. Application of Second Order Differential Equations to a Mechanical System. Unforced Motion. Forced Motion. Resonance. Beats. Analogy With An Electrical Circuit. 3. The Laplace Transform. Definition and Basic Properties. Solution of Initial Value Problems Using the Laplace Transform. Shifting Theorems and the Heaviside Function. The First Shifting Theorem. The Heaviside Function and Pulses. The Second Shifting Theorem. Analysis of Electrical Circuits. Convolution. Unit Impulses and the Dirac Delta Function. Laplace Transform Solution of Systems. Differential Equations With Polynomial Coefficients. 4. Series Solutions. Power Series Solutions of Initial Value Problems. Power Series Solutions Using Recurrence Relations. Singular Points and the Method of Frobenius. Second Solutions and Logarithm Factors. Appendix on Power Series. Convergence of Power Series. Algebra and Calculus of Power Series. Taylor and Maclaurin Expansions. Shifting Indices.
Part II: VECTORS AND LINEAR ALGEBRA. 5. Vectors and Vector Spaces. The Algebra and Geometry of Vectors. The Dot Product. The Cross Product. The Vector Space Rn. Linear Independence, Spanning Sets and Dimension in Rn. Abstract Vector Spaces. 6. Matrices and Systems of Linear Equations. Matrices. Matrix Algebra. Matrix Notation for Systems of Linear Equations. Some Special Matrices. Another Rationale for the Definition of Matrix Multiplication. Random Walks in Crystals. Elementary Row Operations and Elementary Matrices. The Row Echelon Form of a Matrix. The Row and Column Spaces of a Matrix and Rank of a Matrix. Solution of Homogeneous Systems of Linear Equations. The Solution Space of AX = O. Nonhomogeneous Systems of Linear Equations. The Structure of Solutions of AX = B. Existence and Uniqueness of Solutions of AX = B. Summary for Linear Systems. Matrix Inverses. A Method for Finding A-1. 7. Determinants. Permutations. Definition of the Determinant. Properties of Determinants. Evaluation of Determinants by Elementary Row and Column Operations. Cofactor Expansions. Determinants of Triangular Matrices. A Determinant Formula for a Matrix Inverse. Cramer's Rule. The Matrix Tree Theorem. 8. Eigenvalues, Diagonalization, and Special Matrices. Eigenvalues and Eigenvectors. Gerschgorin's Theorem. Diagonalization of Matrices. Orthogonal and Symmetric Matrices. Quadratic Forms. Unitary, Hermitian and Skew Hermitian Matrices.
Part III: SYSTEMS OF DIFFERENTIAL EQUATIONS AND QUALITATIVE METHODS. 9. Systems of Linear Differential Equations. Theory of the Homogeneous System X' = AX. General Solution of the Nonhomogeneous System X' = AX + G. Solution of X' = AX When A Is Constant. Solution of X' = AX When A Has Complex Eigenvalues. Solution of X' = AX When A Does Not Have n Linearly Independent Eigenvectors. Solution of X' = AX By Diagonalizing A. Exponential Matrix Solutions of X' = AX. Solution of X' = AX + G. Variation of Parameters. Solution of X' = AX + G By Diagonalizing A. 10. Qualitative Methods and Systems of Nonlinear Differential Equations. Nonlinear Systems and Existence of Solutions. The Phase Plane, Phase Portraits and Direction Fields. Phase Portraits of Linear Systems. Critical Points and Stability. Almost Linear Systems. Predator/Prey Population Models. A Simple Predator/Prey Model. An Extended Predator/Prey Model. Competing Species Models. A Simple Competing Species Model. An Extended Competing Species Model. Lyapunov's Stability Criteria. Limit Cycles and Periodic Solutions.
Part IV: VECTOR ANALYSIS. 11. Vector Differential Calculus. Vector Functions of One Variable. Velocity, Acceleration, Curvature and Torsion. Tangential and Normal Components of Acceleration. Curvature As a Function of t. The Frenet Formulas. Vector Fields and Streamlines. The Gradient Field and Directional Derivatives. Level Surfaces, Tangent Planes and Normal Lines. Divergence and Curl. A Physical Interpretation of Divergence. A Physical Interpretation of Curl. 12. Vector Integral Calculus. Line Integrals. Line Integral With Respect to Arc Length. Green's Theorem. An Extension of Green's Theorem. Independence of Path and Potential Theory In the Plane. A More Critical Look at Theorem 12.5. Surfaces in 3- Space and Surface Integrals. Normal Vector to a Surface. The Tangent Plane to a Surface. Smooth and Piecewise Smooth Surfaces. Surface Integrals. Applications of Surface Integrals. Surface Area. Mass and Center of Mass of a Shell. Flux of a Vector Field Across a Surface. Preparation for the Integral Theorems of Gauss and Stokes. The Divergence Theorem of Gauss. Archimedes's Principle. The Heat Equation. The Divergence Theorem As A Conservation of Mass Principle. Green's Identities. The Integral Theorem of Stokes. An Interpretation of Curl. Potential Theory in 3- Space.
Part V: FOURIER ANALYSIS, ORTHOGONAL EXPANSIONS AND WAVELETS. 13. Fourier Series. Why Fourier Series? The Fourier Series of a Function. Even and Odd Functions. Convergence of Fourier Series. Convergence at the End Points. A Second Convergence Theorem. Partial Sums of Fourier Series. The Gibbs Phenomenon. Fourier Cosine and Sine Series. The Fourier Cosine Series of a Function. The Fourier Sine Series of a Function. Integration and Differentiation of Fourier Series. The Phase Angle Form of a Fourier Series. Complex Fourier Series and the Frequency Spectrum. Review of Complex Numbers. Complex Fourier Series. 14. The Fourier Integral and Fourier Transforms. The Fourier Integral. Fourier Cosine and Sine Integrals. The Complex Fourier Integral and the Fourier Transform. Additional Properties and Applications of the Fourier Transform. The Fourier Transform of a Derivative. Frequency Differentiation. The Fourier Transform of an Integral. Convolution. Filtering and the Dirac Delta Function. The Windowed Fourier Transform. The Shannon Sampling Theorem. Lowpass and Bandpass Filters. The Fourier Cosine and Sine Transforms. The Discrete Fourier Transform. Linearity and Periodicity. The Inverse N û Point DFT. DFT Approximation of Fourier Coefficients. Sampled Fourier Series. Approximation of a Fourier Transform by an N û Point DFT. Filtering. The Fast Fourier Transform. Computational Efficiency of the FFT. Use of the FFT in Analyzing Power Spectral Densities of Signals. Filtering Noise From a Signal. Analysis of the Tides in Morro Bay. 15. Special Functions, Orthogonal Expansions and Wavelets. Legendre Polynomials. A Generating Function for the Legendre Polynomials. A Recurrence Relation for the Legendre Polynomials. Orthogonality of the Legendre Polynomials. Fourier-Legendre Series. Computation of Fourier-Legendre Coefficients. Zeros of the Legendre Polynomials. Derivative and Integral Formulas for Pn(x). Bessel Functions. The Gamma Function. Bessel Functions of the First Kind and Solutions of Bessel's Equation. Bessel Functions of the Second Kind. Modified Bessel Functions. Some Applications of Bessel Functions. A Generating Function for Jn(x). An Integral Formula for Jn(x). A Recurrence Relation for Jv(x). Zeros of Jv(x). Fourier-Bessel Expansions. Fourier-Bessel Coefficients. Sturm-Liouville Theory and Eigenfunction Expansions. The Sturm-Liouville Problem. The Sturm-Liouville Theorem. Eigenfunction Expansions. Approximation In the Mean and Bessel's Inequality. Convergence in the Mean and Parseval's Theorem. Completeness of the Eigenfunctions. Orthogonal Polynomials. Chebyshev Polynomials. Laguerre Polynomials. Hermite Polynomials. Wavelets. The Idea Behind Wavelets. The Haar Wavelets. A Wavelet Expansion. Multiresolution Analysis With Haar Wavelets. General Construction of Wavelets and Multiresolution Analysis. Shannon Wavelets.
Part VI: PARTIAL DIFFERENTIAL EQUATIONS. 16. The Wave Equation. The Wave Equation and Initial and Boundary Condition. Fourier Series Solutions of the Wave Equation. Vibrating String with Given Initial Velocity and Zero Initial Displacement. Vibrating String With Initial Displacement and Velocity. Verification of Solutions. Transformation of Boundary Value Problems Involving the Wave Equation. Effects of Initial Condition and Constants on the Motion. Wave Motion Along Infinite and Semi-Infinite String. Fourier Transform Solution of Problems on Unbounded Domains. Characteristics and d'Alembert's Solution. A Nonhomogeneous Wave Equation. Forward and Backward Waves. Normal Modes of Vibration of a Circular Elastic Membrane. Vibrations of a Circular Elastic Membrane, Revisited. Vibrations of a Rectangular Membrane. 17. The Heat Equation. The Heat Equation and Initial and Boundary Conditions. Fourier Series Solutions of the Heat Equation. Ends of the Bar Kept at Temperature Zero. Temperature in a Bar With Insulated Ends. Temperature Distribution in a Bar With Radiating End. Transformations of Boundary Value Problems Involving the Heat Equation. A Nonhomogeneous Heat Equation. Effects of Boundary Conditions and Constants on Heat Conduction. Heat Conduction in Infinite Media. Heat Conduction in an Infinite Bar. Heat Conduction in a Semi-Infinite Bar. Integral Transform Methods for the Heat Equation in an Infinite Medium. Heat Conduction in an Infinite Cylinder. Heat Conduction in a Rectangular Plate. 18. The Potential Equation. Harmonic Functions and the Dirichlet Problem. Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. Poisson's Integral Formula for the Disk. Dirichlet Problems in Unbounded Regions. Dirichlet Problem for the Upper Half Plane. Dirichlet Problem for the Right Quarter Plane. An Electrostatic Potential Problem. A Dirichlet Problem for a Cube. The Steady-State Heat Equation for a Solid Sphere. The Neumann Problem. A Neumann Problem for a Rectangle. A Neumann Problem for a Disk. A Neumann Problem for the Upper Half Plane. 19. Canonical Forms, Existence and Uniqueness of Solutions, and Well-Posed Problems. Canonical Forms. Existence and Uniqueness of Solutions. Well-Posed Problems.
Part VII: COMPLEX ANALYSIS. 20. Geometry and Arithmetic of Complex Numbers. Complex Numbers. The Complex Plane. Magnitude and Conjugate. Complex Division. Inequalities. Argument and Polar Form of a Complex Number. Ordering. Binomial Expansion of (z = w) n. Loci and Sets of Points in the Complex Plane. Distance. Circles and Disks. The Equation |z-a| = |z-b|. Other Loci. Interior Points, Boundary Points, and Open and Closed Sets. Limit Points. Complex Sequences. Subsequences. Compactness and the Bolzano-Weierstrass Theorem. 21. Complex Functions. Limits, Continuity and Derivatives. Limits. Continuity. The Derivative of a Complex Function. The Cauchy-Riemann Equations. Power Series. Series of Complex Numbers. Power Series. The Exponential and Trigonometric Functions. The Complex Logarithm. Powers. Integer Powers. z1/n for Positive Integer n. Rational Powers. Powers zw. 22. Complex Integration. Curves in the Plane. The Integral of a Complex Function. The Complex Integral in Terms of Real Integrals. Properties of Complex Integrals. Integrals of Series of Functions. Cauchy's Theorem. Proof of Cauchy's Theorem for a Special Case. Proof of Cauchy's Theorem for a Rectangle. Consequences of Cauchy's Theorem. Independence of Path. The Deformation Theorem. Cauchy's Integral Formula. Cauchy's Integral Formula for Higher Derivatives. Bounds on Derivatives and Liouville's Theorem. An Extended Deformation Theorem. 23. Series Representations of Functions. Power Series Representations. Isolated Zeros and the Identity Theorem. The Maximum Modulus Theorem. The Laurent Expansion. 24. Singularities and the Residue Theorem. Singularities. The Residue Theorem. Some Applications of the Residue Theorem. The Argument Principle. RouchTs Theorem. Summation of Real Series. An Inversion Formula for the Laplace Transform. Evaluation of Real Integrals. 25. Conformal Mappings.
