Observation 7 - Tangent Planes

They are covering:

• Lifting Matrix
• Difference between $\mathrm{\nabla }f$ and $\mathrm{\nabla }g$ for 2 and 3 variable functions respectively
• Finding tangent planes for a surface in both ways (using $f$ and $g$).

Did some examples of finding the gradients and normal vectors to find tangent planes and such.

• He actually did the "plane cutting a cylinder" problem.

A good idea for practice is to just have them list the steps they need to do, like:

1. Set $x^2+y^2=2$ and $y+2x=1$ and set one equal to the other
2. Find the velocity vector $\overrightarrow{v}$ at the given point $P$
3. Find the normal vector to the cylindar $\overrightarrow{n}$ at point $P$
4. ...

He's really just going through some of the harder textbook problems to get these ideas nailed down.