Math 252 Lecture 12: Wednesday , January 29th 1999.

Today, we looked at vector fields. Examples of vector fields were given. It also turns out [Transparancy 1] that the gradient of a scalar field is a vector field.

Various examples were given [Transparancy 2] of the gradient vector fields corresponding to scalar fields. The scalar fields used were the same ones considered last lecture, and for which we plotted isotimic surfaces in Maple.

Maple's "plots" package also has a nice command for plotting gradient vector fields, called "gradplot3d". It has various options which you can read about in Maple's on-line help, including different styles of arrowheads for the vectors. I used this command to plot the gradient vector fields for all the scalar fields we considered last lecture. These plots are reproduced below, along with the Maple commands used to generate them. For comparison, I also include the isotimic surfaces we plotted last lecture.

Gradient Vector Field for Isotimic Planes:

Below and to the left, you will see the plot of isotimic planes which we made last lecture. To its right, you will see the corresponding gradient vector field plotted. The gradient vector field was produced using the following Maple commands:

  with(plots):
  gradplot3d(x+2*y-3*z,x=-2..2,y=-2..2,z=-2..2,arrows=SLIM,grid=[5,5,5],axes=FRAME);

Gradient Vector Field for Isotimic Spheres:

Below and to the left, you will see the plot of isotimic spheres which we made last lecture. To its right, you will see the corresponding gradient vector field plotted. The gradient vector field was produced using the following Maple commands:

  with(plots):
  gradplot3d(x^2+y^2+z^2,x=-3..3,y=-3..3,z=-3..3,arrows=SLIM,grid=[5,5,5],axes=FRAME);

Gradient Vector Field for Isotimic Cylinders:

Below and to the left, you will see the plot of isotimic cylinders which we made last lecture. To its right, you will see the corresponding gradient vector field plotted. The gradient vector field was produced using the following Maple commands:

  with(plots):
  gradplot3d(x^2+y^2,x=-3..3,y=-3..3,z=-3..3,scaling=CONSTRAINED,arrows=SLIM,axes=FRAME);

Gradient Vector Field for Isotimic Ellipsoids:

Below and to the left, you will see the plot of isotimic ellipsoids which we made last lecture. To its right, you will see the corresponding gradient vector field plotted. The gradient vector field was produced using the following Maple commands:

  with(plots):
  gradplot3d(x^2/4+y^2/9+z^2,x=-6..6,y=-9..9,z=-3..3,scaling=CONSTRAINED,arrows=SLIM,axes=FRAME);

Gradient Vector Field for Isotimic Cones:

Below and to the left, you will see the plot of isotimic cones which we made last lecture. To its right, you will see the corresponding gradient vector field plotted. The gradient vector field was produced using the following Maple commands:

  with(plots):
  gradplot3d(sqrt(x^2+y^2)-z,x=-2..2,y=-2..2,z=-2..4,scaling=CONSTRAINED,arrows=SLIM,axes=FRAME);

If you would like to play with any of these Maple structures, you can download the Maple Worksheet. The file size is 125 K.

At the end of the lecture, the whole class made paper airplanes and threw them in unison, in order to demonstrate the flow lines of a velocity field. I took a photograph of the spectacle, and when it is developed I will scan the image into the class web site. Stay tuned!

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 / lec12 / index.html
Revised January 1999 by John Hebron.