2.2 The Geometry of Systems

September 25, 1998

The difference between vector fields and direction fields.

We get Maple to draw a vector field using fieldplot.

> with(plots):

> fieldplot([3*R-1.4*R*F,-F+.5*R*F],R=0..5,F=0..5);

[Plot]

The direction field is plotted using dfieldplot.

> with(DEtools):

> dfieldplot([D(R)(t)=3*R(t)-1.4*R(t)*F(t),D(F)(t)=-F(t)+.5*R(t)*F(t)],
[R(t),F(t)],t=0..1,R=0..5,F=0..5);

[Plot]

Some Important Examples

The Radial Vector Field (out)

> fieldplot([x,y],x=-1..1,y=-1..1);

[Plot]

> dfieldplot([D(x)(t)=x(t),D(y)(t)=y(t)],[x(t),y(t)],t=0..1,x=-1..1,y=-1..1);

[Plot]

The Radial Vector Field (in)

> fieldplot([-x,-y],x=-1..1,y=-1..1);

[Plot]

> dfieldplot([D(x)(t)=-x(t),D(y)(t)=-y(t)],[x(t),y(t)],t=0..1,x=-1..1,y=-1..1);

[Plot]

A Different Inward Vector Field

> fieldplot([-x,-2*y],x=-1..1,y=-1..1);

[Plot]

> dfieldplot([D(x)(t)=-x(t),D(y)(t)=-2*y(t)],[x(t),y(t)],t=0..1,x=-1..1,y=-1..1);

[Plot]

The Standard Circular Field

> fieldplot([-y,x],x=-1..1,y=-1..1);

[Plot]

> dfieldplot([D(x)(t)=-y(t),D(y)(t)=x(t)],[x(t),y(t)],t=0..1,x=-1..1,y=-1..1);

[Plot]

The Standard Hyperbolic Field

> fieldplot([-y,-x],x=-1..1,y=-1..1);

[Plot]

> dfieldplot([D(x)(t)=-y(t),D(y)(t)=-x(t)],[x(t),y(t)],t=0..1,x=-1..1,y=-1..1);

[Plot]

>