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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 | |