To use the Integral Curves utility, choose the command "Integral Curves" from the Utilities Menu. The menu bar will disappear, and the Integral Curves utility will fill the screen.
This utility deals with differential equations of the form dx/dt = F(x,y), dy/dt = G(x,y). A solution of such an equation is a curve, (x(t),y(t)) in the x,y-plane. (These curves are known as "integral curves.") At each point through which the curve passes, its velocity is given by (F(x(t),y(t)), G(x(t),y(t)). In particular, the curve is tangent at the point (x,y) to the vector (F(x,y),G(x,y)). Once you specify any point on the curve, the curve is completely determined.
In the Integral curves utility, you specify the functions dx/dt and dy/dt. Then, you can start drawing a curve either by clicking at the initial point, or by typing in the coordinates of the initial point and clicking a button, "Draw Curve". You can repeat this even while a curve is being drawn. In fact, you can have up to 32 curves on the screen at the same time. The program will continue to draw each curve until F(x,y) or G(x,y) becomes undefined or until the curve gets very far off the screen. You can also stop all curve drawing by clicking on a button, "Stop Curves". (There is no way to start up existing curves again once you have stopped them in this way.)
The Integral Curves Utility can also draw a "field" of direction arrows. When you ask it to do this, by clicking on the button "Show Direction Field", it will draw arrows at each point in a grid. The arrow at the point (x,y) shows the direction of the vector (F(x,y),G(x,y)). Note that no indication is given of the length (or "magnitude") of this vector. (If (F(x,y),G(x,y)) is zero or undefined, no arrow is drawn.)
Ordinarily, integral curves are drawn forwards in time from their initial points. However, it is also possible to extend a curve backwards in time. There is a check box that you can check if you want to extend curves in both directions in time.
The computer uses numerical methods to solve the differential equations. Three methods are available. In order of increasing accuracy, they are Euler's Method, Runge-Kutta order 2, and Runge-Kutta order 4. (Euler's Method is actually quite inaccurate.) You select the method to be used by clicking on a radio button. The method used for a curve is the one in effect at the time you first start drawing it. All of the methods work by repeatedly trying to estimate where the curve will be after a small time interval, delta T. You can specify a value for delta T. The smaller it is, the more accurate the results will be but the longer it will take to draw.
When you save an example in the Integral Curves utility, all the initial points for any displayed curves are saved, along with the other data. When you load an example, the drawing of all the saved curves will be started at the same time. Also, a direction field will be drawn if a direction field was visible when the example was saved.
