An Engineer's Guide to Mathematica

7 Interactive Graphics

7.1 Interactive Graphics: Manipulate[]

The Manipulate command is a straightforward way create an interactive environment for the manipulation of the parameters u₁, u₂, … of an expression f(x, y, u₁, u₂, …). The output of Manipulate can be numbers, symbolic expressions, and graphics. The Manipulate command has a very wide range of interactive capabilities. We shall discuss several of them in detail and follow their introduction with examples.

To illustrate the capabilities of Manipulate, the following general form is assumed

  Manipulate[expr, 
    text1, 
    Item[object1,pos1] 
    {spec1}, 
    Delimiter, 
    text2, 
    Item[object2,pos2] 
    {spec2}, 
    Delimiter, 
    …, 
    ControlPlacement->{pl1,pl2,…}, 
    Initialization:>(proc), 
    SaveDefinitions->True 
    TrackedSymbols:>{u1,u2,…}] 

Each of these terms will be discussed in what follows.

The Manipulate command creates two types of output. In our case, the first type will be primarily a graphical display of the results, which can be composed of one or more fully annotated figures with each figure containing one or more curves. The procedure to create the graphic is denoted expr, which represents f(x, y, u₁, u₂, …). If more than one expression is used to create expr, then each expression is terminated with a semicolon, except for the last expression, which is terminated with a comma. The second type of output is a set of control devices that provide the interactivity; that is, the ability to change the parameters u₁, u₂, …, (denoted uN) that appear ...