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Vectors and Matrices | |

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Introduction | |

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Vectors in Space | |

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Linear Independence | |

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Lines and Planes | |

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Determinants | |

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Simultaneous Linear Equations | |

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Matrices | |

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Addition of Matrices | |

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Scalar Times Matrix | |

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Multiplication of Matrices | |

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Inverse of a Square Matrix | |

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Gaussian Elimination | |

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Eigenvalues of a Square Matrix | |

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The Transpose | |

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Orthogonal Matrices | |

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Analytic Geometry and Vectors n-Dimensional Space | |

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Axioms for Vn | |

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Linear Mappings | |

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Subspaces | |

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Rank of a Matrix | |

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Other Vector Spaces | |

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Differential Calculus of Functions of Several Variables | |

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Functions of Several Variables | |

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Domains and Regions | |

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Functional Notation Level Curves and Level Surfaces | |

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Limits and Continuity | |

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Partial Derivatives | |

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Total Differential Fundamental Lemma | |

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Differential of Functions of n Variables | |

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The Jacobian Matrix | |

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Derivatives and Differentials of Composite Functions | |

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The General Chain Rule | |

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Implicit Functions | |

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Proof of a Case of the Implicit Function Theorem | |

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Inverse Functions | |

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Curvilinear Coordinates | |

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Geometrical Applications | |

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The Directional Derivative | |

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Partial Derivatives of Higher Order | |

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Higher Derivatives of Composite Functions | |

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The Laplacian in Polar, Cylindrical, and Spherical Coordinates | |

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Higher Derivatives of Implicit Functions | |

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Mixima and Minima of Functions of Several Variables | |

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Extrema for Functions with Side Conditions | |

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Lagrange Multipliers | |

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Maxima and Minima of Quadratic Forms on the Unit Sphere | |

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Functional Dependence | |

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Real Variable Theory | |

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Theorem on Maximum and Minimum | |

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Vector Differential Calculus | |

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Introduction | |

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Vector Fields and Scalar Fields | |

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The Gradient Field | |

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The Divergence of a Vector Field | |

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The Curl of a Vector Field | |

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Combined Operations | |

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Curvilinear Coordinates in Space | |

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Orthogonal Coordinates | |

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Vector Operations in Orthogonal Curvilinear Coordinates | |

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Tensors | |

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Tensors on a Surface or Hypersurface | |

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Alternating Tensors | |

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Exterior Product | |

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Integral Calculus of Functions of Several Variables | |

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The Definite Integral | |

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Numerical Evaluation of Indefinite Integrals | |

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Elliptic Integrals | |

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Double Integrals | |

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Triple Integrals and Multiple Integrals in General | |

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Integrals of Vector Functions | |

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Change of Variables in Integrals | |

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Arc Length and Surface Area | |

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Improper Multiple Integrals | |

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Integrals depending on a Parameter Leibnitz's Rule | |

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Uniform Continuity | |

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Existence of the Riemann Integral | |

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Theory of Double Integrals | |

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Vector Integral Calculus | |

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Two-Dimensional Theory | |

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Introduction | |

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Line Integrals in the Plane | |

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Integrals with Respect to Arc Length | |

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Basic Properties of Line Integrals | |

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Line Integrals as Integrals of Vectors | |

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Green's Theorem | |

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Independence of Path | |

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Simply Connected Domains | |

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Extension of Results to Multiply Connected Domains | |

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Three-Dimensional Theory and Applications | |

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Line Integrals in Space | |

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Surfaces in Space | |

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Orientability | |

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Surface Integrals | |

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The Divergence Theorem | |

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Stokes's Theorem | |

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Integrals Independent of Path | |

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Irrotational and Solenoidal Fields | |

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Change of Variables in a Multiple Integral | |

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Physical Applications | |

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Potential Theory in the Plane | |

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Green's Third Identity | |

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Potential Theory in Space | |

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Differential Forms | |

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Change of Variables in an m-Form and General Stokes's Theorem | |

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Tensor Aspects of Differential Forms | |

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Tensors and Differential Forms without Coordinates | |

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Infinite Series | |

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Introduction | |

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Infinite Sequences | |

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Upper and Lower Limits | |

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Further Properties of Sequences | |

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Infinite Series | |

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Tests for Convergence and Divergence | |

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Examples and Applications of Tests for Convergence and Divergence | |

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Extended Ratio Test and Root Test | |

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Computation with Series Estimate of Error | |

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Operations on Series | |

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Sequences and Series of Functions | |

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Uniform Convergence | |

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Weierstrass M-Test for Uniform Convergence | |

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Properties of Uniformly Convergent Series and Sequences | |

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Power Series | |

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Taylor and MacLaurin Series | |

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Taylor's Formula with Remainder | |

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Further Operations on Power Series | |

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Sequences and Series of Complex Numbers | |

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Sequences and Series of Functions of Several Variables | |

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Taylor's Formula for Functions of Several Variables | |

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Improper Integrals Versus Infinite Series | |

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Improper Integrals Depending on a Parameter Uniform Convergence | |

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Principal Value of Improper Integrals | |

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Laplace Transformation | |

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<F128>G-Function and B-Function | |

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Convergence of Improper Multiple Integrals | |

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Fourier Series and Orthogonal Functions | |

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Trigonometric Series | |

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Fourier Series | |

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Convergence of Fourier Series | |

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Examples Minimizing of Square Error | |

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Generalizations | |

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Fourier Cosine Series | |

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Fourier Sine Series | |

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Remarks on Applications of Fourier Series | |

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Uniqueness Theorem | |

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Proof of Fundamental Theorem for Continuous, Periodic, and Piecewise Very Smooth Functions | |

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Proof of Fundamental Theorem | |

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Orthogonal Functions | |

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Fourier Series of Orthogonal Functions | |

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Completeness | |

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Sufficient Conditions for Completeness | |

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Integration and Differentiation of Fourier Series | |

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Fourier-Legendre Series | |

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Fourier-Bessel Series | |

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Orthogonal Systems of Functions of Several Variables | |

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Complex Form of Fourier Series | |

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Fourier Integral | |

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The Laplace Transform as a Special Case of the Fourier Transform | |

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General Functions | |

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Functions of a Complex Variable | |

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Complex Functions | |

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Complex-Valued Functions of a Real Variable | |

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Complex-Valued Functions of a Complex Variable | |

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Limits and Continuity | |

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Derivatives and Differentials | |

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Integrals | |

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Analytic Functions | |

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Cauchy-Riemann Equations | |

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The Functions log z, az, za, sin-1 z, cos-1 z | |

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Integrals of Analytic Functions | |

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Cauchy Integral Theorem | |

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Cauchy's Integral Formula | |

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Power Series as Analytic Functions | |

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Power Series Expansion of General Analytic Function | |

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Power Series in Positive and Negative Powers | |

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Laurent Expansion | |

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Isolated Singularities of an Analytic Function | |

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Zeros and Poles | |

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The Complex Number <F128>Ã | |

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Residues | |

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Residue at Infinity | |

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Logarithmic Residues | |

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Argument Principle | |

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Partial Fraction Expansion of Rational Functions | |

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Application of Residues to Evaluation of Real Integrals | |

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Definition of Conformal Mapping | |

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Examples of Conformal Mapping | |

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Applications of Conformal Mapping | |

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The Dirichlet Problem | |

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Dirichlet Problem for the Half-Plane | |

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Conformal Mapping in Hydrodynamics | |

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Applications of Conformal Mapping in the Theory of Elasticity | |

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Further Applications of Conformal Mapping | |

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General Formulas for One-to-One Mapping | |

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Schwarz-Christoffel Transformation | |

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Ordinary Differential Equations | |

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Differential Equations | |

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Solutions | |

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The Basic Problems | |

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Existence Theorem | |

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Linear Differential Equations | |

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Systems of Differential Equations | |

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Linear Systems | |

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Linear Systems with Constant Coefficients | |

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A Class of Vibration Problems | |

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Solution of Differential Equations by Taylor Series | |

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The Existence and Uniqueness Theorem | |

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Partial Differential Equations | |

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Introduction | |

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Review of Equation for Forced Vibrations of a Spring | |

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Case of Two Particles | |

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Case of n Particles | |

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Continuous Medium | |

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Fundamental Partial Differential Equation | |

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Classification of Partial Differential Equations | |

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Basic Problems | |

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The Wave Equation in One Dimension | |

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Harmonic Motion | |

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Properties of Solutions of the Wave Equation | |

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The One-Dimensional Heat Equation | |

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Exponential Decay | |

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Properties of Solutions of the Heat Equation | |

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Equilibrium and Approach to Equilibrium | |

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Forced Motion | |

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Equations with Variable Coefficients | |

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Sturm-Liouville Problems | |

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Equations in Two and Three Dimensions | |

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Separation of Variables | |

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Unbounded Regions | |

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Continuous Spectrum | |

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Numerical Methods | |

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Variational Methods | |

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Partial Differential Equations and Integral Equations | |