Math 252 Lecture 28: Wednesday, March 10th 1999.

At the start of the lecture, one of the students asked a question about how to plot parametric surfaces in Maple. Here is the example which was given:

  x:=u^2;
  y:=sqrt(2)*u*v;
  z:=v^2;
  plot3d([x,y,z],u=-2..2,v=-2..2,axes='FRAMED',
   orientation=[170,135],style='PATCH');
The resulting plot appears to the right.

(Tip: If you use style='PATCH' and print on a b/w printer, the whole picture will come out black. For printing, it is best to use the default style, which is 'HIDDEN', meaning the plot is a grid in which the surface is assumed to be opaque and therefore parts of the grid will be hidden.)

We continued with surface elements from last lecture. The area cosine principle was introduced [Transparancies 1 and 2], which applies to cases in which the surface is given by z = f(x,y). This principle enables us to avoid calculating the cross product of the parameter tangent vectors on the surface.

An example of the area cosine principle was given [Transparancy 2], in which the area of an ellipse can be obtained by assuming it is a circle viewed at an angle.

The area cosine angle can also be calcuated by taking the gradient of z - f(x,y). It was shown [Transparancy 3] that this method yields the same expression that was obtained before.

We then moved on to define surface integrals, in terms of an infinite sum over an infinitely fine grid on a surface [Transparancy 4]. Using this definition, one can integrate a scalar field over a surface. One can also integrate a vector field over a surface by taking the dot product of the vector field with the unit normal to the surface. This can be interpreted as the flux of the vector field through the surface.

An example of such a flux calculation [Transparancy 5] was given for a specific vector field through a specific surface.

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 / lec28 / index.html
Revised 11 March 1999 by John Hebron.