Application Center - Maplesoft
  Submit your work
Application Center Applications Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 3 - Vector Calculus Preview

App Preview:

Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 3 - Vector Calculus

You can switch back to the summary page by clicking here.

Learn about Maple
Download Application



 

 

Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 3 - Vector Calculus

 

Robert J. Lopez

Emeritus Professor of Mathematics and Maple Fellow

Maplesoft

Introduction
 

In the preceding two articles (Stepwise Solutions in Maple - Part 1, Stepwise Solutions in Maple - Part 2 - Linear Algebra), we detailed the Maple tools for obtaining stepwise solutions of problems in algebra, calculus of one and several variables, and linear algebra.  In this sequel, we detail the stepwise tools available in vector calculus.  

 

These tools include eleven task templates for computing the flux of a vector field through curves and surfaces, nine task templates for iterated integration, and eight task templates for computing surface integrals. In addition, there is a task template for integrating a planar vector field, and a task template that provides access to the Multiple Integration Maplet, yet another interface to the modified int command in the VectorCalculus packages. Finally, there are two tutors, one for visualizing the Frenet-Serret formalism for a space curve, and one for visualizing a vector field.

 

But again, we point out that Maple "does" symbolic math, and can therefore be directed to implement the steps of nearly any algorithmic computation.

 

Integrate a Planar Vector Field
 

Table 1 shows the default application of the task template that integrates a planar vector field.  Given the field, an integral curve (or flow line) is drawn through each point clicked on with the mouse.  For any coordinate system chosen, the Cartesian coordinates of the point selected are written in the relevant window.

Tools_Tasks_Browse: Vector Calculus_Vector Fields_Integrate Planar Vector Field

Integrate Planar Vector Field

Plot Window                      

 

,     

 

Vector Field

Component 1:

Component 2:

 

Coordinates

System:          Variables:

 

Path Parameter           

 

                  

                  

 

Table 1   Interactive integration a planar vector field - flow lines drawn through each point clicked on

 

Flux through a Plane Curve
 

The Flux command in the VectorCalculus package computes flux of a vector field through curves and surfaces. The syntax for this command varies for each of the four distinct curves it recognizes (circle, ellipse,  line segment, polygonal line), and for the general parametrically defined curve.  The following four task templates provide simplified interfaces to the Flux command in each of these cases.

Circle
 

Table 2 shows the calculation of the flux of the Cartesian vector field  through the circle with center  and radius 2.

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_2-D_Through a Circle

Flux through a Circle

 

 

 

 

 

            

           

 

Center :

 

 

Radius:

 

             

 

 

 

 

 

 

 

 

 

 

 

Table 2   Flux through a circle computed by task template

 

Ellipse
 

Table 3 shows the calculation of the flux of the Cartesian vector field  through the ellipse whose equation is .

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_2-D_Through an Ellipse

Flux through an Ellipse

 

 

 

 

 

            

           

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

Table 3   Flux through an ellipse computed by task template

 

Line Segment or Polygonal Line
 

Table 4 shows the calculation of the flux of the Cartesian vector field  through a polygonal line determined by three nodes. If only two nodes are given, the flux through the line segment these nodes determine will be computed. The integral that is displayed shows how the line segments are parametrized. If the Simplify button is pressed, a single integral results, which is simpler to evaluate but harder to interpret.

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_2-D_Through a Polygonal Line

Flux through a Polygonal Line

 

 

 

 

 

            

           

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4   Flux through a polygonal line computed by task template

 

General Plane Curve
 

Table 5 shows the calculation of the flux of the Cartesian vector field  through the curve .

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_2-D_Through a Plane Curve

Flux through a Plane Curve

 

 

 

 

 

            

           

 

  

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5   Flux through an arbitrary plane curve computed by task template

 

Flux through a Surface
 

The Flux command in the VectorCalculus package computes flux of a vector field through curves and surfaces. The syntax for this command varies for each of the two distinct closed surfaces it recognizes (box, sphere), and for the general parametrically defined surface. For such surfaces there are three specific domains the command recognizes (disk, ellipse, triangle), in addition to the general planar domain.

 

The following seven task templates provide simplified interfaces to the Flux command in each of these cases.

Box
 

Table 6 shows the calculation of the flux of the field  through the box defined by .

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_3-D_Through a Box

Flux through a Box

For the Vector Field:

 

 

 

            

            

            

 

     

         

     

           

     

            

   

 

 

 

 

 

 

 

 

 

 

Table 6   Flux through a box whose sides are parallel to the coordinate planes

 

Sphere
 

Table 7 shows the calculation of the flux of the field  through the sphere with center  and radius 3.

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_3-D_Through a Sphere

Flux through a Sphere

For the Vector Field:

 

 

 

            

            

            

 

                      

 

 

 

 

 

 

 

 

 

 

Table 7   Flux through a sphere computed by task template

 

Parametric Surface
 

Table 8 shows the calculation of the flux of the field  through the surface  whose domain is the planar region defined by . In this formulation, the surface parameters are  and .

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_3-D_Through a Parametric Surface

Flux through a Parametrically Defined Surface

For the Vector Field:

 

 

 

            

            

            

 

 

 

 

   

   

 

 

 

 

        

         

       

 

 

 

 

 

 

 

 

 

 

 

Table 8   Flux through a parametrically defined surface

 

Surface Defined over a Disk
 

Table 9 shows the calculation of the flux of the field  through the surface  whose domain is the disk with center  and radius 3. In this formulation, the surface parameters are  and , but in the flux integral, coordinates are automatically changed to polar to accommodate the domain of the surface.

Tools_Tasks_Browse: Vector Calculus_Integration_Flux_3-D_Through a Surface Defined over a Disk

Flux through a Surface Defined over a Disk

For the Vector Field:

 

 

 

            

            

            

 

    

 

   

   

   

  

   

    

           

 

 

 

 

 

 

 

 

 

 

 

Table 9   Flux through a surface defined over a disk

 

Surface Defined over an Ellipse
 

Table 10 shows the calculation of the flux of the field  through the surface  whose domain is the interior of the ellipse . In this formulation, the surface parameters are  and , but in the flux integral, coordinates are automatically changed to polar to accommodate the domain of the surface. Unfortunately, although the Flux integral is written in polar coordinates, Maple uses  for  and  for .

Tools_Tasks_Browse:

Vector Calculus_Integration_Flux_3-D_Through a Surface Defined over an Ellipse

Flux through a Surface Defined over Interior of an Ellipse

For the Vector Field:

 

 

 

            

            

            

 

    

 

   

   

   

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

Table 10   Flux through a surface defined over the interior of an ellipse

 

Surface Defined over a Triangle
 

Table 11 shows the calculation of the flux of the field  through the surface  whose domain is the interior of the triangle whose vertices are . In this formulation, the surface parameters are  and .

Tools_Tasks_Browse:
Vector Calculus_Integration_Flux_3-D_Through a Surface Defined over a Triangle

Flux through a Surface Defined over a Triangle

For the Vector Field:

 

 

 

            

            

            

 

    

 

   

   

   

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 11   Flux through a surface defined over the interior of a triangle

 

Surface Defined over a General Planar Region
 

Table 12 shows the calculation of the flux of the field  through the surface  whose domain is the planar region defined by . In this formulation, the surface parameters are  and .

Tools_Tasks_Browse:

Vector Calculus_Integration_Flux_3-D_Through a Surface Defined over a Planar Region

Flux through a Surface Defined over a General Planar Region

For the Vector Field:

 

 

 

            

            

            

 

 

 

    

   

   

   

 

 

 

             

 

 

 

 

 

 

 

 

 

 

 

Table 12   Flux through a surface defined over an arbitrary planar region

 

The astute reader will notice a similarity between the calculations in Tables 12 and 8. However, the task template in Table 8 is more general in that it provided for the definition of the surface in more than just the Cartesian coordinate system.

Surface Integration
 

Closely related to surface flux calculations implemented by the Flux command, Maple's VectorCalculus packages have a SurfaceInt command for computing surface integrals.  Surface flux is just the surface integral of the normal component of a vector field on a surface. The Flux command determines both that normal component and the element of surface area. The SurfaceInt command computes the surface integral of a scalar, and to do so it determines the appropriate surface-area element.

As for the Flux command, the SurfaceInt command recognizes the box and sphere as closed surfaces, and provides for integration over surfaces defined on a disk, ellipse, rectangle, triangle, or arbitrary planar region.

Box
 

Table 13 contains a task template used to compute the surface integral of the scalar function  over the surface of the box . There are six surfaces to consider, and the underlying SurfaceInt command pairs parallel faces of the box, so only three separate integrals appear.

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Box in  

  Integrand

   

 

 

           

           

           

 

 

 

                     

 

Table 13   Surface integral of a scalar field over the surface of a box

 

Sphere
 

Table 14 contains a task template used to compute the surface integral of the scalar function  over the surface of the sphere with radius 3 and center . The SurfaceInt command automatically writes the integral in spherical coordinates where  is the angle measured down from the positive -axis, and  is the angle measured around that axis. Note that , the surface-area element in spherical coordinates, is automatically inserted into the integrand.

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Sphere

  Integrand

   

 

Sphere Radius: 

 

Sphere Center:

 

 

 

                                    

 

Table 14   Surface integral of a scalar field over the surface of a sphere

 

Parametric Surface
 

Table 15 contains a task template used to compute the surface integral of the scalar function  over the surface  whose domain is the planar region bounded by the curves  and .

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Surface Defined Parametrically

  Integrand

   

 

   

   

 

   

 

   

   

 

 

 


 

 

 

 

 

                                

 

Table 15   Surface integral of a scalar field over a surface defined parametrically

 

Notice the surface-area element  in the integrand. Because of this, Maple can evaluate the inner integral analytically, but the resulting outer integral has to be evaluated numerically. Surface integrals are some of the most difficult integrals in calculus to evaluate in closed form.

Surface Defined over a Rectangle
 

Table 16 contains a task template used to compute the surface integral of the scalar function  over the surface  whose domain is the rectangle .

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Surface Defined over a Rectangle

  Integrand

   

 

 

Surface

Rectangle

 

 

 

 

                                       

 

Table 16   Surface integral of a scalar field over a surface defined on a rectangle

 

Notice the surface-area element  in the integrand. Because of this, Maple can evaluate the inner integral analytically, but the resulting outer integral has to be evaluated numerically. Surface integrals are some of the most difficult integrals in calculus to evaluate in closed form.

Surface Defined over a Disk
 

Table 17 contains a task template used to compute the surface integral of the scalar function  over the surface  whose domain is the disk with center  and radius 3.

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Surface Defined over a Disk

  Integrand

   

 

   Surface

   

 

 

Center :

Radius:

                    From   to                

 

 

 

                                  

 

Table 17   Surface integral of a scalar field over a surface defined on a disk

 

Maple spends at least five minutes trying to evaluate this integral in closed form, but never succeeds. The calculation was stopped by pressing the Interrupt icon (hexagonal stop-sign with hand superimposed) in the toolbar. The numeric approximation of the integral was obtained by pressing the Floating-Point Approximation button.  As has been noted earlier, surface integrals are some of the most difficult integrals in calculus to evaluate in closed form.

Surface Defined over an Ellipse
 

Table 18 contains a task template used to compute the surface integral of the scalar function  over the surface  whose domain is the interior of the ellipse . As with the computation of flux, the surface parameters are  and , but in the SurfaceInt integral, coordinates are automatically changed to polar to accommodate the domain of the surface.

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Surface Defined over an Ellipse

  Integrand

   

 

   Surface

   

 

 

Equation of Ellipse:   

                     From   to                

 

 

 

                                  

 

Table 18   Surface integral of a scalar field over a surface defined on the interior of an ellipse

 

The inner integral can be evaluated analytically, but the resulting outer integral can only be evaluated numerically.

Surface Defined over a Triangle
 

Table 19 contains a task template used to compute the surface integral of the scalar function  over the surface  whose domain is the interior of the triangle with vertices .

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Surface Defined over a Triangle

  Integrand

   

 

   Surface

   

 

 

Triangle

Vertices

 

 

 

                                      

 

Table 19   Surface integral of a scalar field over a surface defined over the interior of a triangle

 

The inner integrals can be evaluated exactly, the resulting outer integrals can only be evaluated numerically. The underlying SurfaceInt command writes the integral as a sum because the triangular domain cannot be swept with a single multiple integral.  From the limits of integration on the displayed integrals, it would be possible to infer the equations for the lines making up the sides of the triangle.

Surface Defined over a General Planar Region
 

Table 20 contains a task template used to compute the surface integral of the scalar function

 

  

 

over the surface  whose domain is the first-quadrant region bounded by the curves . The task template provides the option for integrating in either order, that is, , or .

Tools_Tasks_Browse: Vector Calculus_Integration_Surface Integration_Over a Box

Surface Integral on a Surface Defined over a General 2-D Region

  Integrand

   

 

   Surface

   

 

 

 

 

 

 

 

                                

 

Table 20   Surface integral of a scalar field over a surface defined on a general planar region

 

The inner integral can be evaluated exactly, but the resulting outer integral can only be evaluated numerically.

Double Integrals
 

In the VectorCalculus packages, the int command was modified with special syntax to make writing multiple integrals easier. Thus, the iterated integral

 

 

can be constructed at top-level with either of the commands shown in Table 36.

 

Table 21   Top-level ways for writing an iterated double integral

 

Integration carried out with the syntax on the right in Table 21 is more robust that that on the left. Corresponding to the older usage on the left of Table 21, we have the special VectorCalculus syntax

 

The inert integral is generated by adding the optional parameter "inert" to the modified int command. In addition, the special domains of rectangle, disk, ellipse, and triangle are recognized, along with provision for formulating a double integral over a general planar region.  These integrations are implemented in the following task templates.

 

Rectangle
 

Table 22 contains a task template used to compute the integral of the function  over the rectangle .

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_2-D_Over a Rectangle

Integrate  over a Rectangle

          

Inner Integral:                Outer Integral:   

                                  

 

Table 22   Iterated double integral over a rectangular domain

 

Disk
 

Table 23 contains a task template used to compute the integral of the function  over a disk with center  and radius 3.

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_2-D_Over a Circle

Integrate  over a Circle

          

Center :   ,                Radius =    

                    From   to                

                                  

 

Table 23   Iterated double integral over the disk with center  and radius 3

 

Ellipse
 

Table 24 contains a task template used to compute the integral of the function  over the interior of the ellipse .

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_2-D_Over an Ellipse

Integrate  over an Ellipse

          

Equation of ellipse:  

    From     to            

                                  

 

Table 24   Iterated double integral over an elliptic domain

 

Triangle
 

Table 25 contains a task template used to compute the integral of the function  over the interior of the triangle whose vertices are .

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_2-D_Over a Rectangle

Integrate  over a Triangle

          

Vertices

 

                                  

 

Table 25   Iterated double integral over the interior of a triangle

 

General Planar Region
 

Table 26 contains a task template used to compute the integral of the function  over that portion of the first-quadrant bounded by the curves  and .

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_2-D_Over a Rectangle

Integrate  over a General Region

          

 

 

 

                                  

 

Table 26   Iterated double integral over a planar region with curved boundaries

 

Triple Integrals
 

In the VectorCalculus packages, the modified int command supports integration over a cube, sphere, and tetrahedron, and over a general three-dimensional region. The following task templates implement these integrations.

Cube
 

Table 27 contains a task template used to compute the integral of the function  over the cube .

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_3-D_Over a Cube

Integrate  over a Cube

          

 

 

                                  

 

Table 27   Iterated triple integral over a cube in

 

Sphere
 

Table 28 contains a task template used to compute the integral of the function  over the sphere with center  and radius 4.

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_3-D_Over a Sphere

Integrate  over a Sphere

          

Center :   , ,                Radius =    

                                  

 

Table 28   Iterated triple integral over a sphere in

 

Tetrahedron
 

Table 29 contains a task template used to compute the integral of the function  over the interior of a tetrahedron with vertices .

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_3-D_Over a Cube

Integrate  over a Tetrahedron

          

Vertices                              

  

  

  

  

  

  

  

  

  

  

  

  

 

                                  

 

Table 29   Iterated triple integral over a tetrahedron in

 

General 3-D Region
 

Table 30 contains a task template used to compute the integral of the function  over the first-octant region bounded by the surfaces  and , lying inside the cylinder whose cross-section is bounded by the curves  and .

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration_3-D_Over a Cube

Integrate  over a General Region

                         

 

                                  

 

Table 30   Iterated triple integral over a general region in

 

Multiple Integration Maplet
 

The Multiple Integration Maplet provides alternative access to the int command as modified in the VectorCalculus packages. Table 46 contains the "minimal content" of the task template. The "default content" would include the Extended Help section, which we have placed here, outside of Table 46.

 

Extended Help
 

The Multiple Integration Maplet provides simplified access to the int command as modified in the VectorCalculus packages.  These modifications allow int to create multiple (iterated) integrals, both active and inert.  A number of pre-defined regions of integration are known to the command, which also admits user-defined domains.

The Maplet, accessed from the task template shown in Table 31, contains three panels.  The initial panel is shown in Figure 1.

 

Figure 1   Initial panel for the Multiple Integration Maplet

 

 

An integrand is entered in the field designated as "Function =".

 

By default, the Maplet expects a triple integral, but this can be changed by clicking on the 2D button to the right of "Dimensions."

 

The variables of integration are entered in the fields to the right of   (or in the 2D case).  Thus, if integration is over a disk, the underlying int command will switch to polar coordinates, with the first variable being the radial coordinate.  (For such cases, int will automatically include the appropriate Jacobian.)

 

Regions of integration are accessed by clicking on the "Edit 3D Region" or "Edit 2D Region" buttons.  The panels that open are shown in Figures 2 and 3, respectively.

Figure 2   Regions of integration for triple integrals

Figure 3   Regions of integration for double integrals

 

 

Three pre-defined 3D regions are recognized, namely, the sphere, parallelepiped, and the tetrahedron.  The general 3D region is entered by providing appropriate bounding functions in the section designated by "3D Region."

 

Four pre-defined 2D regions are recognized, namely, the triangle, rectangle, circle, and the ellipse.  Clearly, by "circle" is meant the closure, or more appropriately, the disk.  Both the circle and ellipse can be subdivided by choosing the Sector option.  The general 2D region is entered by providing appropriate bounding functions in the section designated by "2D Region."

 

Briefly, there are two panels that have to be filled out for a calculation by this maplet. However, the cases of two- and three-dimensional regions are separated, so there are three possible panels that can open when using this maplet.

 

Tools_Tasks_Browse: Vector Calculus_Integration_Multiple Integration Maplet

Multiple Integration Maplet

To launch the Multiple Integration Maplet, click the button to the right.

 

 

An abbreviated Help file is available from within the Maplet.

Extended help is available in the attached collapsed section.

Inserting Minimal Content excludes the Extended Help section.

Inserting Default Content includes the Extended Help section.

 

Table 31   The Multiple Integration Maplet accessed by task template

 

For example, to integrate  over the planar region bounded by the curves  and , fill in the secondary panel as shown in Table 32.

Table 32   Secondary panel for Multiple Integration Maplet

 

The initial panel for the maplet is show in Table 33. The function  is entered, the variables  and  are entered, the 2D radio button is selected, and the panel shown in Table 32 appears. The 2D Region radio button is selected in that panel, and the bounds describing the planar region are entered.

Table 33   Initial panel for the Multiple Integration Maplet

 

After the secondary panel is closed, the Display Integral and Symbolic Evaluation buttons are pressed. Note the explicit display of the relevant int command at the bottom of the primary panel.

There is one shortcoming of the Multiple Integration Maplet that the task templates in Tables 23, 24, and 28 overcome. Suppose, for example, the integral of  over the unit sphere centered at the origin is to be computed. The Multiple Integration Maplet would construct the integral

 

 

and display the command

 

int(z, [x, y, z] = Sphere(<0, 0, 0>, 1), inert)

 

The letters in the list  are interpreted by Maple as , respectively. The volume element  has correctly been included, and the variable  has correctly been converted to spherical coordinates: . Given that the integration variables are actually , the desired integral has been properly constructed.

 

To display the integral

 

 

requires the syntax

 

 

but unfortunately, the Multiple Integration Maplet does not have provision for the option of including a list of variable names within the definition of the region of integration.

 

Space Curves Tutor
 

The Space Curves tutor is an interface to the Frenet-Serret formalism for plane and space curves. Thus, given such a curve, the tutor draws the curve and plots the triad of tangent, principal normal, and binormal vectors. These frames can be animated to move along the curve. Any one of these three vectors can be isolated and graphed along the curve. For space curves, the osculating circle and the torsion are graphed, and for any curve, the curvature is graphed.

 

Table 34 shows the default space curve for this tutor.

 

Table 34   Space Curves tutor and its default space curve

 

We have modified the settings slightly. Instead of the default 5 frames, we have chosen just 1, and have used the Plot Options button to change the axis style to "frame" rather than the default "normal." The unit tangent vector is drawn in blue; the unit (principal) normal, in green; and the unit binormal, in magenta.

 

Of course, the Frenet-Serret formalism is also available in the Context Menu, which contains the options TNB Frame, Tangent Vector, Principal Normal, Binormal Vector, Curvature, Torsion, and Radius of Curvature.

It is also possible to implement these calculations stepwise, from first principles, as we summarize in Table 35.

 

Initialize

Loading VectorCalculus


Define curve

Unit tangent vector

Curvature

Principal Normal

Binormal

Torsion

Table 35   Frenet-Serret formalism stepwise, by first principles

 

Vector Fields Tutor
 

Table 36 contains an image of the Vector Fields tutor where we have changed the default three-dimensional vector field to a planar field.  The tutor draws the arrows of the field and one flow line, and integral curve for the field emanating from a given initial point.  The graph is generated by the FlowLine command of the Student VectorCalculus package.

Table 36   The Vector Fields tutor

 

To modify the graph produced by the tutor, it is necessary to invoke the FlowLine command and impose the options inherited from the fieldplot command of the plots package. Table 37 shows what can be done to the plot working outside the tutor.

 

 

Table 37   Illustration of the FlowLine command with options

 

 

Table 38 illustrates the integration of the same vector field via a task template.

 

Tools_Tasks_Browse: Vector Calculus_Vector Fields_Integrate Planar Vector Field

Integrate Planar Vector Field

Plot Window                      

 

,     

 

Vector Field

Component 1:

Component 2:

 

Coordinates

System:          Variables:

 

Path Parameter           

 

                  

                  

 

Table 38   Integration of a vector field via task template

 

 

After entering the data for the field and the window in which it would be displayed, a flow line is generated by clicking on the point through which it must pass.  Multiple flow lines can be drawn on the same direction field, and the coordinates of the point clicked on and through which the flow line passes are displayed.

 

It is possible to integrate the flow line in both directions, as is done in Table 38.

Legal Notice: © Maplesoft, a division of Waterloo Maple Inc. 2010. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.