Math 252 Lecture 13: Monday, February 1st, 1999.

We started with a discussion of the flow lines of a vector field [Transparancy 1]. Reference was made to the paper airplanes thrown in Friday's class, and how the paths of the airplanes represented the flow lines of a velocity field. An equation for the flow lines of a vector field was derived. As an example, we considered the flow lines of the gradient vector field of a scalar field with ellipsoidal isotimic surfaces [Transparancy 2].

The parametric equations of the flow line going through the point (1,1,1) were derived [Transparancy 3], and this flow line was plotted as a spacecurve in Maple. To illustrate the isotimic surface and the flow line on the same graph, the two plots were put together. The combined plot appears at the right, where the red line is the flow line going through (1,1,1). This picture was made with the following Maple commands:


 
   with(plots):
   plot1:=implicitplot3d(x^2/4+y^2/9+z^2=1,x=-2..2,y=0..3,z=-1..1):
   plot2:=implicitplot3d(x^2/4+y^2/9+z^2=4,x=-3..3,y=0..6,z=-2..2):
   plot3:=implicitplot3d(x^2/4+y^2/9+z^2=9,x=-6..6,y=0..9,z=-3..3):
   plot4:=spacecurve([t,t^4/9,t^4],t=0..1.5,color='RED'):
   display3d({plot1,plot2,plot3,plot4},axes='FRAMED',style='WIREFRAME',scaling='CONSTRAINED',orientation=[-35,75]);
I didn't show this in class, but here is how to put three different flow lines on the graph:

  with(plots):
  plot1:=implicitplot3d(x^2/4+y^2/9+z^2=1,x=-2..2,y=0..3,z=-1..1):
  plot2:=implicitplot3d(x^2/4+y^2/9+z^2=4,x=-3..3,y=0..6,z=-2..2):
  plot3:=implicitplot3d(x^2/4+y^2/9+z^2=9,x=-6..6,y=0..9,z=-3..3):
  plot4:=spacecurve([t,t^(4/9),t^4],t=0..1.5,color='RED'):
  plot5:=spacecurve([t,2*t^(4/9),t^4],t=0..1.5,color='GREEN'):
  plot6:=spacecurve([t,(t/2)^(4/9),(t/2)^4],t=0..3,color='BLUE'):
  display3d({plot1,plot2,plot3,plot4,plot5,plot6},axes='FRAMED',
    scaling='CONSTRAINED',style='WIREFRAME',orientation=[-55,70]);
The red flow line goes through (1,1,1), the green flow line goes through (1,2,1), and the blue flow line goes through (2,1,1). The graph produced by Maple appears to the right.

We then moved on [Transparancy 4], to the defintion of the Divergence of a Vector Field. Roughly speaking, the divergence is the "amount by which flow lines out of a unit volume exceed the flow lines into a unit volume." (The definition will be made more precise in Wednesday's lecture.) A few example were given, including [Transparancy 5] a vector field with parallel flow lines which nonetheless has a divergence.

Here are the scanned-in transparancies, in full-colour JPEG format:

(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 60 and 75 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 225 K, which takes a bit longer to download.)

SFU / Math & Stats / ~hebron / math252 / lec_notes / lec13 / index.html
Revised 11 February 1999 by John Hebron.