Description of the Application Panel Components - Purpose and Usage.

The description of the buttons, text boxes, check boxes and drop down boxes present in the applet panel are described below.

Text boxes

. In the two upper boxes (named Input Vx and Input Vy), the user can type the equations for the x and y Cartesian components of the vector field. Below the two upper boxes is a set of four boxes where the user can input the range of the x and y variables for the vector field to be displayed. The box named Input x0 (top left) allows the user to input the minimum value for the x variable, the box named Input xN (top right) allows the user to input the maximum value for the x variable, the box named Input y0 (bottom left) allows the user to input the minimum value for the y variable and the box named Input yN (bottom left) allows the user to input the maximum value for the y variable. Below the set of four boxes mentioned above, we have a box (named Input t) to input the value of time at the particular instant of interest, and at the right of this box, we have a box (named Input number of points) where the user can type the mesh density value. The value the user types in this box is the number of lines of the grid in the x and y directions.

Check box

. The check box at the left side, below the text boxes, allows the user to chose the option to normalize the vector field (multiply all the vectors by the same constant). The purpose of this option is to obtain a better visualization of the vector field. If the user checks the box, this option is turned on.

Drop down box

. This feature allow the user to chose what he/she wants to do with the vector field. The options in this drop down box are streamline,path line, streak line and draw contour. The purpose of each of these options is described below.

1.

Streamlines. If this option is chosen, the user needs to click the Draw button at the far right of the application panel for this option to became operational. After the Draw button is clicked, the user can mouse click inside the display area, and at the mouse clicked point, the application will draw the streamline of the vector field which pass through this point. The user can draw as many streamlines he/she desires. The streamlines are solutions of the equations:

$$\frac{dx}{ds} = Vx(x(s),y(s),t)$$

and

$$\frac{dy}{ds} = Vy(x(s),y(s),t),$$

with initial conditions x(0) and y(0) (the point given by the user by mouse clicking in the display area), and s is the parameterization of the vector field. We integrate the equations above for s > 0 and s < 0 to obtain the streamline that pass through the point (x₀,y₀). Notice that the streamlines are given at a given value of time (the value of time inputed in the box Input t). We take a picture of the vector field at the particular instant we want, and we trace the lines tangent to the vectors. That is exactly what we do in the application with this option. For time dependent vector fields, different values of t give different vector fields, and therefore, different streamlines.

2.

Path lines. If this option is chosen, the user needs to click the Draw button at the far right of the application panel for this option to became operational. After the Draw button is clicked, the user can mouse click inside the display area, and at the mouse clicked point, the application will draw the path line of the vector field which starts at this point. The user can draw as many path lines he/she desires. The path lines are solutions of the equations:

$$\frac{dx}{dt} = Vx(x(t),y(t),t)$$

and

$$\frac{dy}{dt} = Vy(x(t),y(t),t),$$

with initial conditions x(0) and y(0) (the point given by the user by mouse clicking in the display area). To obtain the path line that started at the point (x(0),y(0)), we integrate the equations above forward in time (integrate for t > 0). Notice the difference between the equations defining streamlines and path lines.

3.

Streak lines. If this option is chosen, the user needs to click the Draw button at the far right of the application panel for this option to became operational. After the Draw button is clicked, the user can mouse click inside the display area, and at the mouse clicked point, the application will draw the streak line of the vector field which starts at this point. The user can draw as many streak lines he/she desires.

4.

Draw Contour. If this option is chosen, the user needs to click the Draw button at the far right of the application panel for this option to became operational. As a result additional buttons and a text box appear in the applet panel, just above the display area. The user can know draw a contour with the shape of a polygon in the display area by mouse clicking. The points inputed by the user by mouse clicking are the edges of the polygon. To close the polygon, just click the button close. If he wants another contour, he can click the button clear. After the user closed the contour by clicking the button close, another two buttons appear. The use interpolated button is used if the user wants a contour that is smooth and pass over the inputed points. This contour is smooth except at the first inputed point, as can be easily checked by the user. The circulation button when clicked gives the circulation on the inputed contour. The contour inputed by the user is always treated as a material contour by the application. If the user changes the value of time in the text box Input t, and then click the Draw button at the far right of the application panel, the inputed contour is convected by the vector field, and its position at the new time is calculated and its new shape and position displayed. The user can chose the number of points used to evaluate the new position of the contour, so the user can obtain the new position of the contour to the desired accuracy. In the text box named # of points added per side to advance in time, the user can input the number of points per side of the polygon (edges inputed by the user by mouse clicking) used to advance the contour in time.

Button

. The main button of the application is the Draw button at the far right of the application panel. After the user chose the desired option in the drop down box, he/she needs to click the Draw button to turn on the chosen option. When the option Draw Contour is chosen in the drop down box, another set of buttons appear. They are the Close and Clear buttons. After the user inputs the contour and clicks the Close button, other two buttons appear. They are the circulation button and the use interpolated button. If the user clicks the circulation button, the application evaluates the circulation on the inputed contour, and if the user clicks the circulation button, the application interpolate the user inputed points by a spline, and this becomes the contour considered by the application.