Calculus and linear algebra in recipes : terms, phrases and numerous examples in short learning units / / Christian Karpfinger

• Autore: Karpfinger Christian
• Pubblicazione: Berlin, Germany : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (1015 pages)
• Disciplina: 512.5
• Soggetto topico: Algebras, Linear; Calculus; Differential equations; Àlgebra lineal; Càlcul; Equacions diferencials
• Soggetto genere / forma: Llibres electrònics
• Note generali: Includes index.
• Nota di contenuto: Intro -- Foreword to the Third Edition -- Preface to the Second Edition -- Preface to the First Edition -- Contents -- 1 Speech, Symbols and Sets -- 1.1 Speech Patterns and Symbols in Mathematics -- 1.1.1 Junctors -- 1.1.2 Quantifiers -- 1.2 Summation and Product Symbol -- 1.3 Powers and Roots -- 1.4 Symbols of Set Theory -- 1.5 Exercises -- 2 The Natural Numbers, Integers and Rational Numbers -- 2.1 The Natural Numbers -- 2.2 The Integers -- 2.3 The Rational Numbers -- 2.4 Exercises -- 3 The Real Numbers -- 3.1 Basics -- 3.2 Real Intervals -- 3.3 The Absolute Value of a Real Number -- 3.4 n-th Roots -- 3.5 Solving Equations and Inequalities -- 3.6 Maximum, Minimum, Supremum and Infimum -- 3.7 Exercises -- 4 Machine Numbers -- 4.1 b-adic Representation of Real Numbers -- 4.2 Floating Point Numbers -- 4.2.1 Machine Numbers -- 4.2.2 Machine Epsilon, Rounding and Floating Point Arithmetic -- 4.2.3 Loss of Significance -- 4.3 Exercises -- 5 Polynomials -- 5.1 Polynomials: Multiplication and Division -- 5.2 Factorization of Polynomials -- 5.3 Evaluating Polynomials -- 5.4 Partial Fraction Decomposition -- 5.5 Exercises -- 6 Trigonometric Functions -- 6.1 Sine and Cosine -- 6.2 Tangent and Cotangent -- 6.3 The Inverse Functions of the Trigonometric Functions -- 6.4 Exercises -- 7 Complex Numbers: Cartesian Coordinates -- 7.1 Construction of C -- 7.2 The Imaginary Unit and Other Terms -- 7.3 The Fundamental Theorem of Algebra -- 7.4 Exercises -- 8 Complex Numbers: Polar Coordinates -- 8.1 The Polar Representation -- 8.2 Applications of the Polar Representation -- 8.3 Exercises -- 9 Linear Systems of Equations -- 9.1 The Gaussian Elimination Method -- 9.2 The Rank of a Matrix -- 9.3 Homogeneous Linear Systems of Equations -- 9.4 Exercises -- 10 Calculating with Matrices -- 10.1 Definition of Matrices and Some Special Matrices. ; 10.2 Arithmetic Operations -- 10.3 Inverting Matrices -- 10.4 Calculation Rules -- 10.5 Exercises -- 11 LR-Decomposition of a Matrix -- 11.1 Motivation -- 11.2 The LR-Decomposition: Simplified Variant -- 11.3 The LR-Decomposition: General Variant -- 11.4 The LR-Decomposition-with Column Pivot Search -- 11.5 Exercises -- 12 The Determinant -- 12.1 Definition of the Determinant -- 12.2 Calculation of the Determinant -- 12.3 Applications of the Determinant -- 12.4 Exercises -- 13 Vector Spaces -- 13.1 Definition and Important Examples -- 13.2 Subspaces -- 13.3 Exercises -- 14 Generating Systems and Linear (In)Dependence -- 14.1 Linear Combinations -- 14.2 The Span of X -- 14.3 Linear (In)Dependence -- 14.4 Exercises -- 15 Bases of Vector Spaces -- 15.1 Bases -- 15.2 Applications to Matrices and Systems of Linear Equations -- 15.3 Exercises -- 16 Orthogonality I -- 16.1 Scalar Products -- 16.2 Length, Distance, Angle and Orthogonality -- 16.3 Orthonormal Bases -- 16.4 Orthogonal Decomposition and Linear Combination with Respect to an ONB -- 16.5 Orthogonal Matrices -- 16.6 Exercises -- 17 Orthogonality II -- 17.1 The Orthonormalization Method of Gram and Schmidt -- 17.2 The Vector Product and the (Scalar) Triple Product -- 17.3 The Orthogonal Projection -- 17.4 Exercises -- 18 The Linear Equalization Problem -- 18.1 The Linear Equalization Problem and Its Solution -- 18.2 The Orthogonal Projection -- 18.3 Solution of an Over-Determined Linear System of Equations -- 18.4 The Method of Least Squares -- 18.5 Exercises -- 19 The QR-Decomposition of a Matrix -- 19.1 Full and Reduced QR-Decomposition -- 19.2 Construction of the QR-Decomposition -- 19.3 Applications of the QR-Decomposition -- 19.3.1 Solving a System of Linear Equations -- 19.3.2 Solving the Linear Equalization Problem -- 19.4 Exercises -- 20 Sequences -- 20.1 Terms. ; 20.2 Convergence and Divergence of Sequences -- 20.3 Exercises -- 21 Calculation of Limits of Sequences -- 21.1 Determining Limits of Explicit Sequences -- 21.2 Determining Limits of Recursive Sequences -- 21.3 Exercises -- 22 Series -- 22.1 Definition and Examples -- 22.2 Convergence Criteria -- 22.3 Exercises -- 23 Mappings -- 23.1 Terms and Examples -- 23.2 Composition, Injective, Surjective, Bijective -- 23.3 The Inverse Mapping -- 23.4 Bounded and Monotone Functions -- 23.5 Exercises -- 24 Power Series -- 24.1 The Domain of Convergence of Real Power Series -- 24.2 The Domain of Convergence of Complex Power Series -- 24.3 The Exponential and the Logarithmic Function -- 24.4 The Hyperbolic Functions -- 24.5 Exercises -- 25 Limits and Continuity -- 25.1 Limits of Functions -- 25.2 Asymptotes of Functions -- 25.3 Continuity -- 25.4 Important Theorems about Continuous Functions -- 25.5 The Bisection Method -- 25.6 Exercises -- 26 Differentiation -- 26.1 The Derivative and the Derivative Function -- 26.2 Derivation Rules -- 26.3 Numerical Differentiation -- 26.4 Exercises -- 27 Applications of Differential Calculus I -- 27.1 Monotonicity -- 27.2 Local and Global Extrema -- 27.3 Determination of Extrema and Extremal Points -- 27.4 Convexity -- 27.5 The Rule of L'Hospital -- 27.6 Exercises -- 28 Applications of Differential Calculus II -- 28.1 The Newton Method -- 28.2 Taylor Expansion -- 28.3 Remainder Estimates -- 28.4 Determination of Taylor Series -- 28.5 Exercises -- 29 Polynomial and Spline Interpolation -- 29.1 Polynomial Interpolation -- 29.2 Construction of Cubic Splines -- 29.3 Exercises -- 30 Integration I -- 30.1 The Definite Integral -- 30.2 The Indefinite Integral -- 30.3 Exercises -- 31 Integration II -- 31.1 Integration of Rational Functions -- 31.2 Rational Functions in Sine and Cosine -- 31.3 Numerical Integration. ; 31.4 Volumes and Surfaces of Solids of Revolution -- 31.5 Exercises -- 32 Improper Integrals -- 32.1 Calculation of Improper Integrals -- 32.2 The Comparison Test for Improper Integrals -- 32.3 Exercises -- 33 Separable and Linear Differential Equations of First Order -- 33.1 First Differential Equations -- 33.2 Separable Differential Equations -- 33.2.1 The Procedure for Solving a Separable Differential Equation -- 33.2.2 Initial Value Problems -- 33.3 The Linear Differential Equation of First Order -- 33.4 Exercises -- 34 Linear Differential Equations with Constant Coefficients -- 34.1 Homogeneous Linear Differential Equations with Constant Coefficients -- 34.2 Inhomogeneous Linear Differential Equations with Constant Coefficients -- 34.2.1 Variation of Parameters -- 34.2.2 Approach of the Right-Hand Side Type -- 34.3 Exercises -- 35 Some Special Types of Differential Equations -- 35.1 The Homogeneous Differential Equation -- 35.2 The Euler Differential Equation -- 35.3 Bernoulli's Differential Equation -- 35.4 The Riccati Differential Equation -- 35.5 The Power Series Approach -- 35.6 Exercises -- 36 Numerics of Ordinary Differential Equations I -- 36.1 First Procedure -- 36.2 Runge-Kutta Method -- 36.3 Multistep Methods -- 36.4 Exercises -- 37 Linear Mappings and Transformation Matrices -- 37.1 Definitions and Examples -- 37.2 Image, Kernel and the Dimensional Formula -- 37.3 Coordinate Vectors -- 37.4 Transformation Matrices -- 37.5 Exercises -- 38 Base Transformation -- 38.1 The Tansformation Matrix of the Composition of Linear Mappings -- 38.2 Base Transformation -- 38.3 The Two Methods for Determining Transformation Matrices -- 38.4 Exercises -- 39 Diagonalization: Eigenvalues and Eigenvectors -- 39.1 Eigenvalues and Eigenvectors of Matrices -- 39.2 Diagonalizing Matrices -- 39.3 The Characteristic Polynomial of a Matrix. ; 39.4 Diagonalization of Real Symmetric Matrices -- 39.5 Exercises -- 40 Numerical Calculation of Eigenvalues and Eigenvectors -- 40.1 Gerschgorin Circles -- 40.2 Vector Iteration -- 40.3 The Jacobian Method -- 40.4 The QR-Method -- 40.5 Exercises -- 41 Quadrics -- 41.1 Terms and First Examples -- 41.2 Transformation to Normal Form -- 41.3 Exercises -- 42 Schur Decomposition and Singular Value Decomposition -- 42.1 The Schur Decomposition -- 42.2 Calculation of the Schur Decomposition -- 42.3 Singular Value Decomposition -- 42.4 Determination of the Singular Value Decomposition -- 42.5 Exercises -- 43 The Jordan Normal Form I -- 43.1 Existence of the Jordan Normal Form -- 43.2 Generalized Eigenspaces -- 43.3 Exercises -- 44 The Jordan Normal Form II -- 44.1 Construction of a Jordan Base -- 44.2 Number and Size of Jordan Boxes -- 44.3 Exercises -- 45 Definiteness and Matrix Norms -- 45.1 Definiteness of Matrices -- 45.2 Matrix Norms -- 45.2.1 Norms -- 45.2.2 Induced Matrix Norm -- 45.3 Exercises -- 46 Functions of Several Variables -- 46.1 The Functions and Their Representations -- 46.2 Some Topological Terms -- 46.3 Consequences, Limits, Continuity -- 46.4 Exercises -- 47 Partial Differentiation: Gradient, Hessian Matrix, Jacobian Matrix -- 47.1 The Gradient -- 47.2 The Hessian Matrix -- 47.3 The Jacobian Matrix -- 47.4 Exercises -- 48 Applications of Partial Derivatives -- 48.1 The (Multidimensional) Newton Method -- 48.2 Taylor Development -- 48.2.1 The Zeroth, First and Second Taylor Polynomial -- 48.2.2 The General Taylor polynomial -- 48.3 Exercises -- 49 Extreme Value Determination -- 49.1 Local and Global Extrema -- 49.2 Determination of Extrema and Extremal Points -- 49.3 Exercises -- 50 Extreme Value Determination Under Constraints -- 50.1 Extrema Under Constraints -- 50.2 The Substitution Method -- 50.3 The Method of Lagrange Multipliers. ; 50.4 Extrema Under Multiple Constraints.
• Titolo autorizzato: Calculus and Linear Algebra in Recipes
• ISBN: 9783662654583; 9783662654576
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910629289203321
• Lo trovi qui: Univ. Federico II