Math 252 Lecture 14: Wednesday , February 3rd 1999.

What is the justification for defining the divergence of a vector field to be the dot-product of the del-operator with the vector field? This question was examined [Transparancy 1] by considering fluid flow through an infinitesimal surface in an infinitesimal time. A mass flow rate density vector field was defined, and an expression for its "Flux" through an infinitesimal area was derived. The divergence was then defined [Transparancy 2] to be the "net outflux per unit volume", in the limit of an infinitesimal volume.

By adding up the net outflux through the 6 faces of a rectangular box, and taking the infinitesimal limit, we obtain the familiar formula for the divergence of the mass flow rate density field, thereby justifying our definition of divergence [Transparancy 3]. (A similar argument can be made for a charge current density field.)

A few examples of vector fields and their divergences were given [Transparancy 4]. These examples were plotted in Maple using the "fieldplot" and the "fieldplot3d" commands. The first example appears to the right. It is a 2-dimensional vector field with a constant divergence of 2. Note that the divergence everywhere is 2, not just at the centre! (Just by looking at the picture, you can tell that th length of the vectors is increasing radially, and so the net outflux will always be positive.)

Here are the Maple commands which were used to produce the above plot:

  with(plots):
  fieldplot([x,y],x=-2..2,y=-2..2,arrows='THICK');
The plot to the right is a 3-dimensional vector vield with a constant divergence of 3. It was produced using the following Maple commands:

  with(plots):
  fieldplot3d([x,y,z],x=-2..2,y=-2..2,z=-2..2,
    arrows='THICK',axes='FRAMED');

To the right you'll see a vector field which resembles the field lines of a magnetic dipole. At the point (1,0), there is a positive divergence, from which the field flows. This is sometimes called a field source. At the point (-1,0), there is a negative divergence, which eats up the field lines. This is sometimes called a field sink.

This picture was produced using the following Maple commands:

  with(plots):
  xcomp:=(x-1)/((x-1)^2+y^2+0.1)-(x+1)/((x+1)^2+y^2+0.1);
  ycomp:=y/((x-1)^2+y^2+0.1)-y/((x+1)^2+y^2+0.1);
  fieldplot([xcomp,ycomp],x=-2..2,y=-2..2,
    arrows='SLIM',thickness=2);
Note that I threw in the "0.1" in the denominator of both terms in order to avoid infinities at (1,0) and (-1,0). If you are considering a point-like field source and sink, then the divergence becomes infinite at these places.

Next, we started talking about "Curl". I handed around a demo of fluid flowing in a vortex, which definitely has a "curl" to it! It was made from 2 pop bottles joined mouth-to-mouth.

The Curl of a vector field was defined [Transparancy 5] as the cross product of the del operator with the vector field. This definition was justified by considering an infinitesimal square path in the xy-plane and calculating the "swirl" of the vector field around this path. It was found [Transparancy 6], that the curl of a vector field in the direction normal to the square path is the "swirl per unit area" around the infinitesimal square.

Some examples were given [Transparancies 6, 7, and 8]. A couple of these were plotted in Maple. These plots are reproduced below, along with the Maple commands used to generate them.

This is example 3.18 of the textbook. I plotted the vector field using the following Maple commands: 
  with(plots):
  fieldplot3d([x*y*z,x^2*y^2*z^2,y^2*z^3],x=-1.5..1.5,
    y=-1.5..1.5,z=-1.5..1.5,arrows='THICK',axes='FRAMED',
    thickness=2,orientation=[20,30]);
It seems to have a curl to it, but it is kind of hard to tell from the plot.
I didn't show this in class, but plotted to the right is the curl of the above vector field. It was made using the following Maple commands: 
  with(plots):
  fieldplot3d([2*y*z^3-2*x^2*y^2*z,x*y,2*x*y^2*z^2-x*z],
    x=-1.5..1.5,y=-1.5..1.5,z=-1.5..1.5,arrows='THICK',
    axes='FRAMED',thickness=2,orientation=[20,30]);

The final example was that of a fluid with uniform mass density rotating with constant angular velocity about the z-axis. The plot shown to the right was made in Maple using the following commands: 
  with(plots):
  fieldplot([-y,x],x=-2..2,y=-2..2,arrows='THICK',thickness=2);
This vector field has a constant curl of "2" along the z-axis. Note that this curl is 2 everywhere in the field, not just at the origin. Think of the flow as though it were spinnning a little propellor in the fluid. The propellor will spin at the same angular speed everywhere in the fluid.


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.)

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Revised 04 February 1999 by John Hebron.