Calculus I: Cartography lecture
After doing implicit differentiation and the condition for two curves to
intersect perpendicularly (slopes are negative reciprocals) it's nice to
do something with maps. Here is a brief outline of my lecture on this
subject.
- Show that the curves xy=2 and x2 - y2 = 3
intersect perpendicularly at (2,1). Show that this computation is
independent of the constants 2 and 3. Show transparencies of these
curves intersecting orthogonally. Give other examples, eg cartesian or
polar coordinate grids.
- Bring in a globe (there's one in Clap 327) and discuss the problem
of mapping a round surface to a flat one. (Theorem Egregium!)
- Discuss the classical map projections (cylindrical, azimuthal,
conic, Mercator, etc). Point out advantages and disadvantages of these
projections. Point out that sometimes the latitudes and longitudes on a
map intersect orthogonally, and sometimes not. (See any book on
cartography; I have transparencies.)
- Give some homework.