Full Description
This book provides a clear and easy-to-understand introduction to higher mathematics with numerous examples. The author shows how to solve typical problems in a recipe-like manner and divides the material into short, easily digestible learning units.
Have you ever cooked a 3-course meal based on a recipe? That generally works quite well, even if you are not a great cook. What does this have to do with mathematics? Well, you can solve a lot of math problems recipe-wise: Need to solve a Riccati's differential equation or the singular value decomposition of a matrix? Look it up in this book, you'll find a recipe for it here. Recipes are available for problems from the
· Calculus in one and more variables,
· linear algebra,
· Vector Analysis,
· Theory on differential equations, ordinary and partial,
· Theory of integral transformations,
· Function theory.
Other features of this book include:
· The division of Higher Mathematics intoapproximately 100 chapters of roughly equal length. Each chapter covers approximately the material of a 90-minute lecture.
· Many tasks, the solutions to which can be found in the accompanying workbook.
· Many problems in higher mathematics can be solved with computers. We always indicate how it works with MATLAB®.
For the present 3rd edition, the book has been completely revised and supplemented by a section on the solution of boundary value problems for ordinary differential equations, by the topic of residue estimates for Taylor expansions and by the characteristic method for partial differential equations of the 1st order, as well as by several additional problems.
Contents
Preface.- 1 Ways of speaking, symbols and quantities.- 2 The natural, whole and rational numbers.- 3 The real numbers.- 4 Machine numbers.- 5 Polynomials.- 6 Trigonometric functions.- 7 Complex numbers - Cartesian coordinates.- 8 Complex numbers - Polar coordinates.- 9 Systems of linear equations.- 10 Calculating with matrices.- 11 LR-decomposition of a matrix.- 12 The determinant.- 13 Vector spaces.- 14 Generating systems and linear (in)dependence.- 15 Bases of vector spaces.- 16 Orthogonality I.- 17 Orthogonality II.- 18 The linear balancing problem.- 14 The linear balancing problem. 14 Generating systems and linear (in)dependence.- 15 Bases of vector spaces.- 16 Orthogonality I.- 17 Orthogonality II.- 18 The linear compensation problem.- 19 The QR-decomposition of a matrix.- 20 Sequences.- 21 Computation of limit values of sequences.- 22 Series.- 23 Illustrations.- 24 Power series.- 25 Limit values and continuity.- 26 Differentiation.- 27 Applications of differential calculus I.-28 Applications of differential calculus I.- 28 Applications of differential calculus II.- 28 Applications of differential calculus I.- 28 Applications of differential calculus II. 28 Applications of differential calculus II.- 29 Polynomial and spline interpolation.- 30 Integration I.- 31 Integration II.- 32 Improper integrals.- 33 Separable and linear differential equations of the 1st order.- 34 Linear differential equations with constant coefficients.- 35 Some special types of differential equations.- 36 Numerics of ordinary differential equations I.- 37 Linear mappings and representation matrices.- 38 Basic transformation.- 39 Diagonalization - Eigenvalues and eigenvectors.- 40 Numerical computation of eigenvalues and eigenvectors.- 41 Quadrics.- 42 Schurz decomposition and singular value decomposition.- 43 Jordan normal form I.- 44 Jordan normal form II.- 45 Definiteness and matrix norms.- 46 Functions of several variables.- 47 Partial differentiation - gradient, Hessian matrix, Jacobian matrix.- 48 Applications of partial derivatives.- 49 Determination of extreme values.- 50 Determination of extreme values under constraints.- 51 Total differentiation, differential operators.- 52 Implicit functions.- 53 Coordinate transformations.- 54 Curves I.- 55 Curves II.- 56 Curve integrals.- 57 Gradient fields.- 58 Domain integrals.- 59 The transformation formula.- 60 Areas and area integrals.- 61 Integral theorems I.- 62 Integral theorems II.- 63 General about differential equations.- 64 The exact differential equation.- 65 Systems of linear differential equations I.- 66 Systems of linear differential equations II.- 67 Systems of linear differential equations II.- 68 Boundary value problems.- 69 Basic concepts of numerics.- 70 Fixed point iteration.- 71 Iterative methods for systems of linear equations.- 72 Optimization.- 73 Numerics of ordinary differential equations II.- 74 Fourier series - Calculation of Fourier coefficients.- 75 Fourier series - Background, theoremsand application.- 76 Fourier transform I.- 77 Fourier transform II.- 78 Discrete Fourier transform.- 79 The Laplacian transform.- 80 Holomorphic functions.- 81 Complex integration.- 82 Laurent series.- 83 The residue calculus.- 84 Conformal mappings.- 85 Harmonic functions and Dirichlet's boundary value problem.- 86 Partial differential equations 1st order.- 87 Partial differential equations 2nd order - General.- 88 The Laplace or Poisson equation.- 89 The heat conduction equation.- 90 The wave equation.- 91 Solving pDGLs with Fourier and Laplace transforms.- Index.
