I spent some time demonstrating, with a 3-D Maple plot, that the theta and phi which appear at the top left of the screen are the same theta and phi which are used in spherical coordinates. In Maple they represent the angular coordinates of the observer. For example, phi=0 means the observer is on the z-axis looking down towards the origin.
Unit vectors were defined in terms of gradients [Transparancy 2]. Coordinate lines of theta turn out to be lines of latitude, and coordinate lines of phi turn out to be lines of longitude.
Orthogonal Curvilinear Coordinates were introduced [Transparancy 3]. The intersection lines of isotimic surfaces of constant coordinate determine the coordinate lines for the nonconstant coordinate. Gradients to these surfaces determine the unit vectors of the coordinate system. If the unit vectors are orthogonal, then they are also parallel to the tangent vectors of the coordinate lines.
The textbook presents spherical coordinates before presenting curvilinear coordinates. In the next lecture, I will do things the other way around by first presenting various vector identities in curvilinear coordinates, and then showing them in spherical coordinates as a special case.
(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.)
