We started by asking why the orbits of the planets are planar. This was answered by using vector notation to find a constant vector which is always perpendicular to both the planet's radial position vector and its velocity vector. This constant vector is the normal vector to the plane of the planet's orbit.
Doing some vector algebra, and using vector idenities, we found an expression which could be integrated to give the equation of the planet's orbit. This is simplified without loss of generality by choosing an appropriate coordinate system. The equation turns out to be the polar equation of an ellipse, thereby proving Kepler's first law.
We finished with a short introduction to cylindrical and spherical coordinates. An example of plotting in spherical coordinates was given in Maple. More examples will be given on Monday, and a Maple worksheet will be posted with Monday's lecture notes.