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mmafoundations - Mathematica Foundations Steven Tschantz A...

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Mathematica Foundations Steven Tschantz 1/18/11 A theory for Mathematica To understand a complex system like Mathematica , it helps to break it down into its basic components, formulate its basic underlying principles and rules, abstract and simplify its behavior. It helps to have more than a collection of examples and catalog of instances. In short, it helps to have a theory of that system, a model predicting how that system behaves. Operating with Mathematica Elements of the Mathematica system Mathematica is a computer algebra system, a program that lets you do symbolic and numerical computations, but it is also a program to facilitate recording and presenting such computations. The Mathematica front-end is a specialized editor that manages Mathematica files, called notebooks, containing text, input expressions, calculation results, and interactive graphics (and other dynamic elements). Actual calculations are carried out by a second program, the Mathematica kernel, controlled from the Mathematica front-end. Mathematica uses its own formal language to express calculations, similar to, but more precise and restricted than ordinary mathematical notation, eliminating ambiguity and enabling algorithmic manipulation. Mathematica will format output much as ordinary mathematical notation, and can interpret certain notations in input, but input is usually linear, using the Mathematica language, as in programming. Using notebooks The first idea for using Mathematica notebooks is simply as a recording of inputs to the Mathematica kernel and outputs returned, documentation of experiments like a scientist's lab notebook. If you select File/New/Notebook from the menu, you get a new blank untitled notebook. Type and your typing goes in an input cell. Press shift-enter and the input is evaluated, output placed under the input, and the insertion point moved below ready for typing another input. Inputs are labeled as evaluated, and outputs are given corresponding labels, so the whole history of a conversation with the Mathematica kernel is made apparent. This is useful since inputs to the Mathematica kernel often have side-effects. Assignments are remembered by the kernel, going back and editing the input does not change what the kernel is assuming, and reevaluating may not always change the state of the kernel in the way you expect. Once you have a satisfactory computation, you can go back and edit your notebok, deleting unnecessary inputs and outputs, and adding text comments. The second idea for using notebooks is to document your calculations. Your goal should be to give all of the inputs in the order needed to calculate your results, so that you could start over, reopen the notebook and reevauate all of the inputs to get the same outputs, a kind of proof of calculation. Inputs and outputs usually don't explain the meaning of what's been calculated, so you should annotate your calculations with text so that someone reading your notebook can understand your thinking and intent.
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