Math 252 Lecture 11: Wednesday , January 27th 1999.

We continued with our discussion of polar coordinates in the xy-plane [Transparancy 1]. The acceleration vector was obtained by differentiating the velocity vector, and being careful about differentiating the radial and angular unit vectors. Four different acceleration terms were obtained [Transparancy 2], corresponding to "pure radial acceleration", "centripetal acceleration", "pure angular acceleration", and the "Coriolis acceleration". By considering Newton's Laws, we obtained expressions for the radial and angular components of Force. In particular, the torque was shown to be the time-rate-of-change of angular momentum, and conservation of angular momemtum was demonstrated.

An example problem was worked out [Transparancy 3], which investigated the velocity and acceleration of a fly walking on a rotating disk.

At this point we finished with Chapter 2 of the textbook and started in on Chapter 3. Scalar Fields were introduced [Transparancy 4] and various examples were given. The concept of an "isotimic surface" was also introduced.

Taking a Maple break, I showed the isotimic surfaces for various scalar fields plotted in Maple. The graphs are reproduced below, along with the Maple code used to generate them.

Isotimic Planes:

At the right, you will see three planes, which are isotimic surfaces to the scalar field x + 2*y - 3*z , with values 1, 2, and 3. They were produced with the following Maple commands: 
  with(plots):
  plot1:=implicitplot3d(x+2*y-3*z=1,x=-2..2,y=-2..2,z=-2..2):
  plot2:=implicitplot3d(x+2*y-3*z=2,x=-2..2,y=-2..2,z=-2..2):
  plot3:=implicitplot3d(x+2*y-3*z=3,x=-2..2,y=-2..2,z=-2..2):
  display3d({plot1,plot2,plot3});

Isotimic Spheres:

At the right, you will see three concentric spheres, which are isotimic surfaces to the scalar field x^2 + y^2 + z^2 , with values 1, 4, and 9. They were produced with the following Maple commands: 
  with(plots):
  plot1:=implicitplot3d(x^2+y^2+z^2=1,x=-3..3,y=0..3,z=-3..3):
  plot2:=implicitplot3d(x^2+y^2+z^2=4,x=-3..3,y=0..3,z=-3..3):
  plot3:=implicitplot3d(x^2+y^2+z^2=9,x=-3..3,y=0..3,z=-3..3):
  display3d({plot1,plot2,plot3});

Isotimic Cylinders:

Here you will see three concentric cylinders, which are isotimic surfaces to the scalar field x^2 + y^2 , with values 1, 4, and 9. They were produced with the following Maple commands: 
  with(plots):
  plot1:=implicitplot3d(x^2+y^2=1,x=-3..3,y=0..3,z=-3..3,
    scaling=CONSTRAINED):
  plot2:=implicitplot3d(x^2+y^2=4,x=-3..3,y=0..3,z=-3..3,
    scaling=CONSTRAINED):
  plot3:=implicitplot3d(x^2+y^2=9,x=-3..3,y=0..3,z=-3..3,
    scaling=CONSTRAINED):
  display3d({plot1,plot2,plot3});

Isotimic Ellipsoids:

These are three concentric ellipsoids, which are isotimic surfaces to the scalar field (x^2)/4 + (y^2)/9 + z^2 , with values 1, 3, and 9. They were produced with the following Maple code: 
  with(plots):
  plot1:=implicitplot3d(x^2/4+y^2/9+z^2=1,x=-2..2,y=0..3,z=-1..1,
    scaling=CONSTRAINED):
  plot2:=implicitplot3d(x^2/4+y^2/9+z^2=4,x=-3..3,y=0..6,z=-2..2,
    scaling=CONSTRAINED):
  plot3:=implicitplot3d(x^2/4+y^2/9+z^2=9,x=-6..6,y=0..9,z=-3..3,
    scaling=CONSTRAINED):
  display3d({plot1,plot2,plot3});

Isotimic Cones:

These are three cones displaced along the z-axis, which are isotimic surfaces to the scalar field sqrt( x^2 + y^2 ) - z , with values 0, 1, and 2. They were produced using the following Maple code: 
  with(plots):
  plot1:=implicitplot3d(sqrt(x^2+y^2)-z=0,x=-2..2,y=-2..2,z=-2..4,
    scaling=CONSTRAINED,grid=[20,20,20]):
  plot2:=implicitplot3d(sqrt(x^2+y^2)-z=1,x=-2..2,y=-2..2,z=-2..4,
    scaling=CONSTRAINED,grid=[20,20,20]):
  plot3:=implicitplot3d(sqrt(x^2+y^2)-z=2,x=-2..2,y=-2..2,z=-2..4,
    scaling=CONSTRAINED,grid=[20,20,20]):
  display3d({plot1,plot2,plot3});
(Note that I increased the grid which Maple uses for plotting surfaces to 20x20x20 points, up from its default of 10x10x10, in order to make the apex of each cone less rounded. This still isn't a fine enough grid to take away the round appearance of the cone apex, however using too many points in the grid will cause memory problems in Maple.)
If you would like to play with any of these Maple structures, you can download the Maple Worksheet. Note, however, that the file size is 640 K and hence the file may take a while to download.

Having finished the Maple break, we launched into a discussion of the directional derivative of a scalar field [Transparancy 5]. I presented a long-winded derivation of the directional derivative [Transparancy 6], which unfortunately put half the class to sleep... (Sorry about that!)

The end result led to a definition of the gradient of a scalar field [Transparancy 7]. The directional derivative is then simply written as the dot product of the gradient with the unit vector in the direction of the derivative.

We also defined the DEL Operator, which provides a convenient shorthand for writing the gradient. (It will also later prove useful for writing the divergence and curl of a vector field.)

Various properties of the gradient were shown [Transparancy 8]. In particular, it was shown that the gradient points in the direction of maximum increase of the scalar field and that it is perpendicular to the isotimic surfaces. Furthermore, its magnitude is the maximum rate of increase of the scalar field per unit distance.

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 / lec11/index.html
Revised January 1999 by John Hebron.