I am currently teaching an introductory course on ordinary differential equations, and my students use GeoGebra to explore and visualize the solutions to ODEs. The SlopeField and SolveODE functions, together with an initial condition that can be moved around the graphics view, are great when there is only one dependent variable. We also spend a significant amount of time on systems (especially linear systems with 2 dependent variables). Although SolveODE still works, first-order systems should be thought of as vector fields (or sometimes called direction fields).
It would be great if there were a DirectionField or VectorField function that plots a 2-dimensional vector field, perhaps with parameters that controlled the density and scaling of the arrows.
I think this would help students to understand the solutions to systems better. For example, although two systems may give the same unparameterized curves as solutions, the arrows of vector field would make it apparent that the solutions are different parameterization, and hence have different speeds.
Thanks!
I like this idea 
Maybe these worksheets are useful for you:
Vector fields plotter:
https://www.geogebra.org/m/cXgNb58T
Velocity fields 2d:
https://www.geogebra.org/m/GXVqPCPV
https://www.geogebra.org/m/zVZm4KwY
https://www.geogebra.org/m/GxUwpUav
Velocity fields 3d:
https://www.geogebra.org/m/kYGsg3NS
More:
https://www.geogebra.org/m/wtdx457q
Regards.
Thank you for your suggestions, Juan. I have explored the possibility of creating a worksheet and distributing it electronically to my students. This would certainly serve the same purpose, , but there are other advantages to a simple DirectionField or VectorField function.
1. Time: I have limited contact time with the students and my goal at the start was to minimize the overhead needed to go from ODE to visualization of the solutions. This is why I chose to use GeoGebra in my class, and I hope that this will encourage students to use it outside of class. My students already learned how to generate slope fields and plot solutions on top of them. This would hopefully be a simple extension of what they already know.
2. I think there is pedagogical value in having the students enter the command to create a vector field, rather than have textboxes where the components of the vector field are entered individually. Understanding the input and output of function forces one to determine exactly what information is needed to generate the output. I.e. what parts of the system of ODEs determine how the direction field looks?
GeoGebra has served my purposes perfectly for this course, so far, and with this feature in place it would a complete tool. I will certainly make a worksheet for my class, if need be, but I am also open to any other suggestions (pedagogical or technical)! Thanks again.
Hi,
I understand your point. I think other people have requested that command in this forum before and I also think it will be useful. In the meantime, I suggest to make a custom tool. The two files I am sharing now have a custom Tool called Vector Field. First define a two-variable function, then choose a number for scale and another number for density.
Maybe these files are what you need.
Regards.
Thank you, Juan, for sharing your GeoGebra book!
Very interesting, and nice!
Sergio
Totally agree with you, that feature would be a big asset! If there'd be an additional tool for plotting scalar functions by equipotential curves, it would allow to examine and understand fields in a new visual & intuitive way and it will surely open new doors in teaching vector calculus.
In https://beta.geogebra.org/graphing (and other beta apps) you can now visualise functions R^2->R^2, for instance
f(u,v)=(u+v,u-v)
or C->C
f(x)=10/x+i
I know those are not vector fields, but maybe it helps.
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