ماتماتیکا به‌وسیلهٔ مثال‌ها

__Mathematica by Example, Sixth Edition__ is an essential resource for the Mathematica user, providing step-by-step instructions on achieving results from this powerful software tool. The book fully accounts for the changes to functionality and visualization capabilities and accomodates the full array of new extensions in the types of data and problems that Mathematica can immediately handle, including cloud services and systems, geographic and geometric computation, dynamic visualization, interactive applications and other improvements. It is an ideal text for scientific students, researchers, and aspiring programmers seeking further understanding of Mathematica. Written by seasoned practitioners with a view to practical implementation and problem-solving, the book's pedagogy is delivered clearly and without jargon using representative biological, physical and engineering problems. Code is provided on an ancillary website to support the use of Mathematica across diverse applications and subject areas. Contents Preface 1 Getting started 1.1 Introduction to Mathematica 1.1.1 Getting started with Mathematica Preview Basic rules of Mathematica syntax 1.2 Getting help from Mathematica Mathematica help 2 Basic operations on numbers, expressions, and functions 2.1 Numerical calculations and built-in functions 2.1.1 Numerical calculations 2.1.2 Built-in constants 2.1.3 Built-in functions A word of caution 2.2 Expressions and functions: elementary algebra 2.2.1 Basic algebraic operations on expressions 2.2.2 Naming and evaluating expressions 2.2.3 Defining and evaluating functions 2.3 Graphing functions, expressions, and equations 2.3.1 Functions of a single variable 2.3.2 Parametric and polar plots in two dimensions 2.3.3 Three-dimensional and contour plots; graphing equations 2.3.4 Parametric curves and surfaces in space 2.3.5 Comments 2.4 Solving equations 2.4.1 Exact solutions of equations 2.4.2 Approximate solutions of equations 3 Calculus 3.1 Limits and continuity 3.1.1 Using graphs and tables to predict limits 3.1.2 Computing limits 3.1.3 One-sided limits 3.1.4 Continuity 3.2 Differential calculus 3.2.1 Definition of the derivative 3.2.2 Calculating derivatives 3.2.3 Implicit differentiation 3.2.4 Tangent lines Tangent lines of implicit functions Parametric equations and polar coordinates 3.2.5 The first derivative test and second derivative test 3.2.6 Applied max/min problems 3.2.7 Antidifferentiation 3.2.7.1 Antiderivatives u-Substitutions 3.3 Integral calculus 3.3.1 Area 3.3.2 The definite integral 3.3.3 Approximating definite integrals 3.3.4 Area Parametric equations Polar coordinates 3.3.5 Arc length Parametric equations Polar coordinates 3.3.6 Solids of revolution Volume Surface area 3.4 Infinite sequences and series 3.4.1 Introduction to sequences 3.4.2 Introduction to infinite series 3.4.3 Convergence tests 3.4.4 Alternating series 3.4.5 Power series 3.4.6 Taylor and Maclaurin series 3.4.7 Taylor's theorem 3.4.8 Other series 3.5 Multivariable calculus 3.5.1 Limits of functions of two variables 3.5.2 Partial and directional derivatives Classifying critical points Tangent planes Lagrange multipliers 3.5.3 Iterated integrals Area, volume, and surface area Triple-iterated integrals 4 Introduction to lists and tables 4.1 Lists and list operations 4.1.1 Defining lists 4.1.2 Plotting lists of points 4.2 Manipulating lists: more on Part and Map 4.2.1 More on graphing lists; graphing lists of points using graphics primitives 4.2.2 Miscellaneous list operations 4.2.2.1 Other list operations 4.2.2.2 Alternative way to evaluate lists by functions 4.3 Other applications 4.3.1 Approximating lists with functions 4.3.2 Introduction to Fourier series Application: the one-dimensional heat equation Application: the wave equation on a circular plate 5 Matrices and vectors: topics from linear algebra and vector calculus 5.1 Nested lists: introduction to matrices, vectors, and matrix operations 5.1.1 Defining nested lists, matrices, and vectors 5.1.2 Extracting elements of matrices 5.1.3 Basic computations with matrices 5.1.4 Basic computations with vectors 5.1.4.1 Basic operations on vectors 5.1.4.2 Basic operations on vectors in 3-space 5.2 Linear systems of equations 5.2.1 Calculating solutions of linear systems of equations 5.2.2 Gauss–Jordan elimination 5.3 Selected topics from linear algebra 5.3.1 Fundamental subspaces associated with matrices 5.3.2 The Gram–Schmidt process 5.3.3 Linear transformations Application: rotations 5.3.4 Eigenvalues and eigenvectors 5.3.5 Jordan canonical form 5.3.6 The QR Method 5.4 Maxima and minima using linear programming 5.4.1 The standard form of a linear programming problem 5.4.2 The dual problem Application: a transportation problem 5.5 Selected topics from vector calculus 5.5.1 Vector-valued functions 5.5.2 Line integrals 5.5.3 Surface integrals 5.5.4 A note on nonorientability 5.5.5 More on tangents, normals, and curvature in R3 6 Applications related to ordinary and partial differential equations 6.1 First-order differential equations 6.1.1 Separable equations 6.1.2 Linear equations 6.1.2.1 Application: free-falling bodies 6.1.3 Nonlinear equations 6.1.4 Numerical methods 6.2 Second-order linear equations 6.2.1 Basic theory 6.2.2 Constant coefficients Application: harmonic motion 6.2.3 Undetermined coefficients 6.2.4 Variation of parameters 6.3 Higher-order linear equations 6.3.1 Basic theory 6.3.2 Constant coefficients 6.3.3 Undetermined coefficients 6.3.4 Variation of parameters 6.3.5 Laplace transform methods Application: the Convolution Theorem Application: the Dirac delta function 6.3.6 Nonlinear higher-order equations 6.4 Systems of equations 6.4.1 Linear systems 6.4.1.1 Homogeneous linear systems 6.4.1.2 A(t) constant Application: the double pendulum 6.4.2 Nonhomogeneous linear systems 6.4.3 Nonlinear systems 6.4.4 Linearization 6.5 Some partial differential equations 6.5.1 The one-dimensional wave equation 6.5.2 The two-dimensional wave equation Bibliography Index