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| Ordinary Differential Equations | |
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| First-Order Differential Equations | |
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| Preliminary Concepts | |
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| Separable Equations | |
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| Linear Differential Equations | |
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| Exact Differential Equations | |
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| Integrating Factors | |
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| Homogeneous, Bernoulli, and Riccati Equations | |
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| Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories | |
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| Existence and Uniqueness for Solutions of Initial Value Problems | |
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| Second-Order Differential Equations | |
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| Preliminary Concepts | |
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| Theory of Solutions y" + p(x)y' + q(x)y = f(x) | |
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| Reduction of Order | |
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| The Constant Coefficient Homogeneous Linear Equation | |
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| Euler's Equation | |
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| The Nonhomogeneous Equation y" + p(x)y' + q(x)y = f(x) | |
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| Application of Second-Order Differential Equations to a Mechanical System | |
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| The Laplace Transform | |
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| Definition and Basic Properties | |
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| Solution of Initial Value Problems Using the Laplace Transform | |
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| Shifting Theorems and the Heaviside Function | |
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| Convolution | |
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| Unit Impulses and the Dirac Delta Function | |
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| Laplace Transform Solution of Systems | |
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| Differential Equations with Polynomial Coefficients | |
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| Series Solutions | |
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| Power Series Solutions of Initial Value Problems | |
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| Power Series Solutions Using Recurrence Relations | |
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| Singular Points and the Method of Frobenius | |
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| Second Solutions and Logarithm Factors | |
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| Appendix on Power Series | |
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| Vectors and Linear Algebra | |
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| Vectors and Vector Spaces | |
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| The Algebra and Geometry of Vectors | |
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| The Dot Product | |
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| The Cross Product | |
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| The Vector Space R[superscript n] | |
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| Linear Independence, Spanning Sets, and Dimension in R[superscript n] | |
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| Abstract Vector Spaces | |
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| Matrices and Systems of Linear Equations | |
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| Matrices | |
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| Elementary Row Operations and Elementary Matrices | |
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| The Row Echelon Form of a Matrix | |
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| The Row and Column Spaces of a Matrix and Rank of a Matrix | |
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| Solution of Homogeneous Systems of Linear Equations | |
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| The Solution Space of AX = O | |
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| Nonhomogeneous Systems of Linear Equations | |
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| Summary for Linear Systems | |
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| Matrix Inverses | |
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| Determinants | |
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| Permutations | |
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| Definition of the Determinant | |
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| Properties of Determinants | |
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| Evaluation of Determinants by Elementary Row and Column Operations | |
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| Cofactor Expansions | |
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| Determinants of Triangular Matrices | |
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| A Determinant Formula for a Matrix Inverse | |
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| Cramer's Rule | |
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| The Matrix Tree Theorem | |
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| Eigenvalues, Diagonalization, and Special Matrices | |
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| Eigenvalues and Eigenvectors | |
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| Diagonalization of Matrices | |
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| Orthogonal and Symmetric Matrices | |
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| Quadratic Forms | |
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| Unitary, Hermitian, and Skew-Hermitian Matrices | |
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| Systems of Differential Equations and Qualitative Methods | |
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| Systems of Linear Differential Equations | |
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| Theory of Systems of Linear First-Order Differential Equations | |
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| Solution of X' = AX When A Is Constant | |
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| Solution of X' = AX + G | |
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| Qualitative Methods and Systems of Nonlinear Differential Equations | |
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| Nonlinear Systems and Existence of Solutions | |
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| The Phase Plane, Phase Portraits, and Direction Fields | |
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| Phase Portraits of Linear Systems | |
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| Critical Points and Stability | |
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| Almost Linear Systems | |
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| Predator/Prey Population Models | |
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| Competing Species Models | |
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| Lyapunov's Stability Criteria | |
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| Limit Cycles and Periodic Solutions | |
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| Vector Analysis | |
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| Vector Differential Calculus | |
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| Vector Functions of One Variable | |
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| Velocity, Acceleration, Curvature, and Torsion | |
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| Vector Fields and Streamlines | |
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| The Gradient Field and Directional Derivatives | |
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| Divergence and Curl | |
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| Vector Integral Calculus | |
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| Line Integrals | |
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| Green's Theorem | |
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| Independence of Path and Potential Theory in the Plane | |
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| Surfaces in 3-Space and Surface Integrals | |
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| Applications of Surface Integrals | |
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| Preparation for the Integral Theorems of Gauss and Stokes | |
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| The Divergence Theorem of Gauss | |
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| The Integral Theorem of Stokes | |
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| Fourier Analysis, Orthogonal Expansions, and Wavelets | |
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| Fourier Series | |
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| Why Fourier Series? | |
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| The Fourier Series of a Function | |
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| Convergence of Fourier Series | |
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| Fourier Cosine and Sine Series | |
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| Integration and Differentiation of Fourier Series | |
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| The Phase Angle Form of a Fourier Series | |
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| Complex Fourier Series and the Frequency Spectrum | |
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| The Fourier Integral and Fourier Transforms | |
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| The Fourier Integral | |
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| Fourier Cosine and Sine Integrals | |
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| The Complex Fourier Integral and the Fourier Transform | |
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| Additional Properties and Applications of the Fourier Transform | |
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| The Fourier Cosine and Sine Transforms | |
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| The Finite Fourier Cosine and Sine Transforms | |
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| The Discrete Fourier Transform | |
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| Sampled Fourier Series | |
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| The Fast Fourier Transform | |
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| Special Functions, Orthogonal Expansions, and Wavelets | |
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| Legendre Polynomials | |
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| Bessel Functions | |
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| Sturm-Liouville Theory and Eigenfunction Expansions | |
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| Orthogonal Polynomials | |
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| Wavelets | |
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| Partial Differential Equations | |
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| The Wave Equation | |
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| The Wave Equation and Initial and Boundary Conditions | |
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| Fourier Series Solutions of the Wave Equation | |
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| Wave Motion Along Infinite and Semi-infinite Strings | |
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| Characteristics and d'Alembert's Solution | |
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| Normal Modes of Vibration of a Circular Elastic Membrane | |
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| Vibrations of a Circular Elastic Membrane, Revisited | |
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| Vibrations of a Rectangular Membrane | |
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| The Heat Equation | |
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| The Heat Equation and Initial and Boundary Conditions | |
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| Fourier Series Solutions of the Heat Equation | |
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| Heat Conduction in Infinite Media | |
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| Heat Conduction in an Infinite Cylinder | |
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| Heat Conduction in a Rectangular Plate | |
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| The Potential Equation | |
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| Harmonic Functions and the Dirichlet Problem | |
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| Dirichlet Problem for a Rectangle | |
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| Dirichlet Problem for a Disk | |
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| Poisson's Integral Formula for the Disk | |
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| Dirichlet Problems in Unbounded Regions | |
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| A Dirichlet Problem for a Cube | |
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| The Steady-State Heat Equation for a Solid Sphere | |
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| The Neumann Problem | |
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| Canonical Forms, Existence and Uniqueness of Solutions, and Well-Posed Problems | |
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| Canonical Forms | |
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| Existence and Uniqueness of Solutions | |
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| Well-Posed Problems | |
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| Complex Analysis | |
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| Geometry and Arithmetic of Complex Numbers | |
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| Complex Numbers | |
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| Loci and Sets of Points in the Complex Plane | |
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| Complex Functions | |
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| Limits, Continuity, and Derivatives | |
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| Power Series | |
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| The Exponential and Trigonometric Functions | |
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| The Complex Logarithm | |
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| Powers | |
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| Complex Integration | |
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| Curves in the Plane | |
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| The Integral of a Complex Function | |
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| Cauchy's Theorem | |
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| Consequences of Cauchy's Theorem | |
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| Series Representations of Functions | |
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| Power Series Representations | |
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| The Laurent Expansion | |
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| Singularities and the Residue Theorem | |
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| Singularities | |
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| The Residue Theorem | |
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| Some Applications of the Residue Theorem | |
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| Conformal Mappings | |
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| Functions as Mappings | |
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| Conformal Mappings | |
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| Construction of Conformal Mappings Between Domains | |
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| Harmonic Functions and the Dirichlet Problem | |
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| Complex Function Models of Plane Fluid Flow | |
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| Historical Notes | |
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| Development of Areas of Mathematics | |
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| Ordinary Differential Equations | |
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| Matrices and Determinants | |
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| Vector Analysis | |
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| Fourier Analysis | |
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| Partial Differential Equations | |
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| Complex Function Theory | |
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| Biographical Sketches | |
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| Galileo Galilei (1564-1642) | |
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| Isaac Newton (1642-1727) | |
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| Gottfried Wilhelm Leibniz (1646-1716) | |
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| The Bernoulli Family | |
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| Leonhard Euler (1707-1783) | |
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| Carl Friedrich Gauss (1777-1855) | |
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| Joseph-Louis Lagrange (1736-1813) | |
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| Pierre-Simon de Laplace (1749-1827) | |
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| Augustin-Louis Cauchy (1789-1857) | |
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| Joseph Fourier (1768-1830) | |
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| Henri Poincare (1854-1912) | |
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| Answers and Solutions to Selected Odd-Numbered Problems | |
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| Index | |