How are double integrals solved in Mathematica

Mathematica




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    Mathematica 3.0see table
  • Wolfram Research
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  • In contrast to Macsyma and Maple, Mathematica was not born in the university sector, but its roots are probably here. Prof. Dr. Stefan Wolfram founded his company Wolfram Research for Mathematica.

    Mathematica is an interactive, graphics-oriented mathematics system with an integrated high-level language that is programmed in a structured, procedural or rule-based manner. Over 850 integrated functions allow you to work quickly with mathematics. Solutions are given exactly numerically or symbolically. In this way, even complex differential equations can be solved step by step using numerical approximation. Create a calculation model with parameters and your PC will calculate all possible solutions for you. The hardware-optimized graphics routines enable Mathematica to generate contour plots as well as 2D and 3D representations in black and white as well as in color.

    The computer algebra system Mathematica combines symbolic and numerical arithmetic, graphics, animation, list processing and structured documentation with a powerful programming language. Mathematica enables the user to exhaust, visualize and solve the depths of mathematics without paper and pencil, calculator or complex software. With the optional accuracy and the possibility of matrix manipulation, Mathematica supports the user ideally as an interactive calculation tool and as a high-level language programming environment. It allows formulas to be entered and manipulated directly in algebraic form. In addition, Mathematica offers extended options for symbolic equation solution, integration, differentiation and resolution of higher-order equations. The hardware-optimized graphics routines enable Mathematica to generate contours and output 2- and 3-dimensional plots in black and white as well as in color.

    Numerical calculation

    Mathematica works with numbers of arbitrary (discretionary, arbitrary) size and precision. This gives the user complete control over numerical rounding errors. This allows a multitude of calculations with both theoretical and practical demands that would not be possible with normal scientific pocket calculators or standard programming languages.

    Mathematica's optional numerical precision enables a wide range of applications for mathematical functions: trigonometric and exponential functions, Bessel functions, hypergeometric functions, elliptic integrals, and much more. Of course, Mathematica also includes efficient numerical algorithms for matrix computation, numerical integration, and numerical equation solving capabilities.

    Symbolic calculation

    In addition to numerical computation, Mathematica can also work with algebraic formulas. All of the standard operational symbols used in algebra and computation, including integration, differentiation, and polynomial computation, have been implemented in Mathematica. The ability to manipulate and derive exact algebraic formulas enables the user to automate a wide range of calculations that previously could only be done by hand. Mathematica users can explicitly display formulas as equation solutions; this is sometimes easier than numerical results for understanding the complete structure of a solution.

    Graphical calculation

    The third element of Mathematica is graphics. Mathematica enables functions and data to be represented two-dimensionally and three-dimensionally, in color or black and white. It realizes application-oriented visualization of the results. Mathematica also generates three-dimensional color images of symbolic descriptions of arbitrary geometric objects. For example, Mathematica's symbolic and numerical computation capabilities can be used to generate a list of views of a complex polyhedron. From this a three-dimensional image can be created using Mathematica's graphic capabilities.

    Most versions of Mathematica support graphic animation. For the first time, the user sees a dynamic simulation or the changes in the graphic when one or more parameters are varied.

    Mathematica uses POSTSCRIPT for all graphical output. This size and resolution independent page description language produces reports in presentation quality, color negatives for books and magazines or even huge posters on media such as computer screens, laser printers, photo typesetting machines and other output devices. Mathematica also saves in the Encapsulated POSTSCRIPT format so that graphics can be easily inserted into documents from other programs, e.g. in desktop publishing programs.

    programming

    With over 860 built-in functions, Mathematica is a powerful programming language that can be expanded with your own functions. The Mathematica language supports various forms of programming, such as structured procedural programming in the style of C or PASCAL and functional programming in the style of APL. Since the vast majority of algorithms in mathematics were designed to transform instructions from one form of equation to another, Mathematica has allowed rule-based programming. This makes it possible to specify conversion rules for mathematical expressions. Each rule relates to a model and acts as an expression that represents the model. Most of the rules can be constructed by direct implementation of the formulas from books. Rules can be given a name and activated if necessary, or implemented in a global rule base in such a way that they are always available.

    The programming language in Mathematica is a high-level language that enables the user to create complex programs based on simple functions, which in turn directly access the primitive operations in Mathematica itself. Many such programs are available in the form of Mathematica packages. These packages have been created by Mathematica users and vendors over the years and are adapted to a wide variety of applications.

    System architecture, portability

    Mathematica consists of two parts: the kernel, which performs the mathematical calculations and executes the programs, and a "front end", which is the interface to the user.

    The kernel works the same on every computer platform. The result of this design concept is that all software packages developed with Mathematica are interchangeable on this computer platform. On the other hand, the "front end" is designed in such a way that it takes full advantage of the computer's capabilities.

    The "front end" and the kernel do not need to run on the same computer. All versions are compatible with each other. The "front end" can run on one computer and communicate via modem with another computer on which the kernel is installed. This enables users to access the speed of supercomputers with the usual presentation style of a PC.

    Mathematica's interface to other systems

    Mathematica offers three ways to make exchanging data with external programs simple and smooth.

    The easiest way to do this is to use the standard ASCII format. For example, Mathematica reads numerical data as produced by spreadsheet and database programs. Graphics can be exported in the file formats of desktop publishing systems such as Adobe Illustrator, Aldus Pagemaker or animation packages such as Video Works. Special "Print Forms" can be defined to export symbolic or numeric results that can be implemented in C or FORTRAN. Output in the typesetter language TeX is also possible.

    If you have a multitasking system that enables interdisciplinary communication, Mathematica can transfer data in real time (in UNIX this is done using pipes). Mathematica also offers the possibility of receiving data, e.g. from a laboratory measuring instrument or a programming language. Such programs can run from Mathematica or independently connect to the kernel.

    Ultimately, Wolfram Researchs offers registered Mathematica developers the specifications of the internal communication protocol MathTalk, via which the "front end" communicates with the kernel. This allows you to develop your own programs that replace the Mathematica "front end".

    Internal

    Wolfram Research developed the code of the Mathematica kernel as an object-oriented extension of C. The code comprises nearly 150,000 lines. The code is distributed almost evenly across the individual areas in Mathematica. The "front ends" for Mathematica on the Macintosh and the NeXT are each more than 50,000 lines long.

    The portability of Mathematica can be seen in over a dozen different versions for different hardware architectures. The main limitation is that you should have close to 6MB of addressable memory. This is not a restriction for UNIX workstations, but for the Macintosh it means that either sufficient physical memory must be provided or virtual memory management is supported. The Mathematica version for the 386 MS-DOS computer only requires approximately 4 MB of physical memory, as a virtual memory management system is integrated.

    Comparisons

    Like any intellectual product, Mathematica has a past life. As a programming environment, Mathematica is at a higher language level like C or FORTRAN, since it can proceed symbolically and interactively. In its interactive way of working, Mathematica is just as simple as BASIC, but by far more powerful. In the area of ​​numerical computation, Mathematica builds on systems such as APL and MatLab, but with greatly expanded, highly developed capabilities, such as freely selectable arithmetic precision. Equation solvers, like TK! Solver and Eureka, only include a handful of Mathematica's numerical functions. In the field of symbolic manipulation, algebraic computation systems such as Macsyma, Maple, Reduce, and SMP (a previous product by Stephan Wolfram) are closest to Mathematica. These systems share some of the capabilities of Mathematica, but have shortcomings in their general applicability or programmability. Mathematica appears to be unique with regard to the graphic possibilities, especially through the complete integration of three-dimensional surface graphics in resolution-independent POSTSCRIPT.

    Supported by the optional accuracy and the possibility of matrix manipulation Mathematica ideal for the user as an interactive calculation tool and as a high-level language programming environment. Enable the hardware-optimized graphics routines MathematicaTo generate contours and to output 2 and 3-dimensional plots in black and white as well as in color. The Mathematica User interface, the so-called FrontEnd, impresses with its hierarchical structure and versatile handling. It allows formulas to be entered and manipulated directly in algebraic form. It also offers you Mathematica extended possibilities for symbolic equation solution, integration, differentiation and resolution of equations of higher order. In the new version 3, the kernel and front end have been fundamentally improved. Mathematica represents expressions as mathematical formulas, including all mathematical symbols and the Greek letters; Entries can be generated with a click of the mouse using the symbol palettes. However, there is much more to these optical highlights than is apparent at first glance. Wolfram Research has succeeded in applying the complex structure of mathematical formulas to the Mathematica To map syntax.


  • Programming interfaces:


    TeX, C, FORTRAN, MathLink

    Data import / export:

    ASCII, binary (others by integrating FLT files)

    Graphics import / export:

    BitMap (BMP), Device Independent Bitmap (DIB), Macpaint (MAC), Postscript (PS, EPS), Windows Metafile (WMF), Tagged Image File Format (TIFF), Adobe Illustrator File (AL), Wave (WAV)

    • Dynamic formula editor
    • over 300 math symbols and operators
    • Any arrangement of symbols / operators in symbol palettes
    • Complete character sets: Greek, Latin, Gothic, script, double bar
    • Numerical calculations
    • Any accuracy
    • Symbolic bills
    • Graphics and sound
    • animation
    • programming language
    • External programming interface
    • Available on over 20 platforms
    • Cheap school licenses
    • Standard system
    • Flexible quantity licenses and campus licenses
    • Formula editor
      • over 300 math symbols
      • over 100 operators
      • Freely definable symbols and operators
      • Powers, indices, binomial coefficients, vectors, matrices
      • Any arrangement of symbols / operators in symbol palettes
      • Individual symbol palettes with just three clicks
      • Complete character sets: Greek, Latin, Gothic, script, double bar
      • According to the manufacturer, all currently existing mathematical symbols, letters and operators
      • The shape of the symbols automatically adapts to the mathematical structure (integral sign becomes larger or longer, ...)
      • Platform-independent presentation
      • Export to PostScript and TeX
    • Any exact numbers
    • Complex numbers
    • Special math functions (elliptical, hypergeometric, etc.)
    • Matrix operations (inverses, eigenvalues, etc.)
    • Fourier transforms
    • Linear and non-linear function adaptation (Fit)
    • Integration, differentiation
    • Determination of zeros
    • Differential equations
    • Optimization & linear optimization
    • Number theoretic functions
    • Algebraic simplification
    • Polynomial decomposition
    • Symbolic integration
    • Definite and improper integrals, multiple integrals
    • Sums, products and series, infinite series
    • Section integrals
    • Solving algebraic equations and systems of equations
    • Root objects offer arbitrary precision
    • Symbolic matrix operations
    • List processing
    • Function and data charts
    • 2D, 3D, contour and density diagrams
    • 3-dimensional object representation
    • Lighting models
    • Advanced, descriptive graphics language
    • PostScript output
    • Moving graphics
    • Digitized sound output of functions and data
    • Import of sound files
    • Interactive, symbolic language
    • Uniform presentation of lists, formulas, graphics, programs, etc.
    • Procedural and functional programming
    • Transformation rules
    • Object-oriented approaches
    • Pattern recognition for general expressions
    • Symbolic tracking of the program sequence
    • Symbolic formula representation in traditional mathematical form
    • Interactive documents with text, graphics, sound and math
    • Selection of mathematical sub-expressions with the mouse and recalculation (New)
    • Symbol palettes contain math symbols and Greek letters
    • Hierarchical structure
    • Any custom function palettes
    • Hypertext links
    • Help browser provides comprehensive explanations and examples
    • Word processing with self-defined character sets
    • Conversions in standard graphic and text formats
    • Possible connection to cores on other computers
    • Reading files and programs
    • Reading in any ASCII and binary formats
    • Reading in any graphic format
    • C, Fortran and TeX editions
    • Calling external programs
    • Speedup through integration of numerical libraries (NAG, ISML, etc.)
    • MathLink is a high-level communication interface to external programs
    • Linking Mathematica and Databases
    • File and text manipulation language
    • International character sets

    For an up-to-date overview of the platforms on which Mathematica versions are running, please select the Mathematica link on the manufacturer's homepage!

    platform
    Firmly-
    disk storage
    Working
    Storage
    (R.A.M)
    Motorola
    Intel
    HP PA-RISC
    SPARC
    IBM OS / 2 ** 2.0 or higher
    DEC OpenVMS ** 7.0