Something that we didn't really go too deeply into, the Jacobian.
The Jacobian is where the r and the r^2sin(theta) com from in the transformation of coordinates from Cartesian to polar and spherical.
Here is the list of Millennium Problems posed by the Clay Mathematics Institute. The first (P vs. NP) is claimed to have been solved, but the jury is still out. The Poincaré Conjecture has been solved, rather famously, by Grigori Perelman a few years ago. The Riemann Hypothesis is probably the most elusive problem in math now that Fermat's Last Theorem and Poincaré have been knocked down.
Here is the background and list of Hilbert's 23 problems. Four have been called too vague to solve (bet you wish that would work on a test!) and three are still unresolved. One of them, the Riemann Hypothesis, is among the Milennium Problems.
This helped me when I had trouble visualizing what multi-variable function graphs would look like, without giving away too much like wolfram alpha sometimes does.
It's centered on Somerville, MA, but you can enter latitude and longitude or a ZIP code to focus on any point in the US.
The curved lines represent points with the same elevation. Lines close together mean a steep slope (either upward or downward depending on the actual elevations). Concentric curves mean a peak or valley. What indicates a saddle point?
Curvature and Torsion
Choose among several curves and see the rotation of the Frenet-Serret frame as you move the slider. From this you can perceive the curvature and torsion of the curve. Associated objects (such as the circle of curvature, evolute, and osculating sphere, as well as two views of the Frenet frames) may be displayed.
